An eigenvalue algorithm based on norm-reducingtransformationsCitation for published version (APA):Paardekooper, M. H. C. (1969). An eigenvalue algorithm based on norm-reducing transformations. Eindhoven:Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR41102

DOI:10.6100/IR41102

Document status and date:Published: 01/01/1969

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AN EIGENV ALUE ALGORITHM BASED ON NORM-REDUCING

TRANSFORMATIONS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EINDHOVEN OPGEZAG VAN DE RECTOR MAGNIFICUS DR.IR. A.A.TH.M. VAN TRIER,HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIEUITDESENAATTEVERDEDIGENOPDINSDAG

2 DECEMBER 1969 TE 16.00 UUR.

DOOR

MICHAEL HUBERTUS CORNELIUS PAARDEKOOPER

GEBOREN TE ZOETERWOUDE

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR

DE PROMOTOR PROF. DR. G.W. VELTKAMP

Aan Wil Aan Evert, Wouter en Liesbeth

0. Introduction

0.0. Introductory remarks

0.1. Notations, definitions and elementary theorems

0.2. A survey of Jacobi-like

0. 3. Summary

1. Real Norm-Reducing Shears

1.0. Introduction

1.1. Row congruency and Euclidean parameters of a

shear

1. 2. transformations by real unimodular

shears

for the real unimodular norm-

1 .4. The particular case D = F 0,

1.5. The commutator in relation to shear transfor­

matioas

2. Complex Norm-Reducing Shears

2.0. Introduction

2. 1 • Row congruency and Euclidean oa.ra,llle c

shear

of a

2.2. The unimodular norm-reducing shear

transformation

for the complex unimodular norm-

reducing shears

2.4. The case D = F == 0

2.5. The commutator in relation to shear transfor­

mations

1

1

9 18

24

29

29

46

49

53 53

53

55

61

64

5

3. Convergence to Normality

3.0. Introduction

3.1. A lower bound for the optimal norm-reduction

b,r shear transformations

3.2, The convergence theorem

4• Jacobi-like Methods for almost Diagonalization of

almost Normal Matrices

4.0. Introduction

4.1. Almost diagonalization of a complex almost

normal matrix

4.2. Almost block diagonalization of a real almost

normal matrix

4·3· The real diagonali representative of

Jn,tm(A) 4.4. The complex diagonalizing representative of

5. Numerical stability and the norm-reducing process

5.0. Introduction

5.1. Input and output perturbations related to

rounding errors

5.2. Error analysis of similarity transformations

5.3. The general error analysis applied to shear

transformation

5.4. A numerically stable transformation by the dia­

gonalizing representative of >Sntm(A)

5.5. Diagonal dominance and shear transformations

References

Sa.menvatting

Curriculum Vitae.

6

69 69

73

76

76

77

90

94

97

1.01

101

104

112

118

125

128

139

145

CHAPTER 0

INTRODUCTION

0. 0. Introductory remarks

Since the rise of the program-stored digital computer it has been

possible to master effectively the bulk of work necessary to solve

numerically the algebraic eigenvalue problem, i.e. the approximate

calculation of the eigenvalues and eigenvectors of a linear trans­

formation represented by a given matrix. The advent of this appa­

ratus has stimulated the construction new algorithms for this

problem. As concerns the Hermitean eigenvalue problem we mention

the numerically stable methods of Givens and Householder.

But also for the non-Hermi tean problem several new methods are

proposed •. Since this problem can be very ill-conditioned, the

construction of the latter algorithms presents serious difficul­

ties. Inexact arithmetic, the reverse of the computer's speed,

makes therefore the problem mathematically interesting. Research

on the numerical solution of the non-Hermitean eigenvalue problem

is very active at present. At the moment it is not yet clear which

of the algorithms proposed by several authors is preferable. The

QR-algorithm, developed by Francis in 1961-1962, attracts much at­

tention and inspires confidence as to speed of convergence and ac­

curacy of the results.

The method for the non-Hermitean eigenvalue problem which we pre­

sent in this book, is of what is known as the Jacobi-like type,

i.e. an extension of the classical Jacobi-method to non-normal

matrices.

The Jacobi-algortihm is based on the use of rotations, the

original matrix A A being recursively transformed into matrices 0

7

A , A , •••• , which tend to a diagonal form. In each step of the

p~ode~s the plane rotation is chosen to minimize the sum of the

squares of the moduli of the non-diagonal elements. In principle,

each normal matrix A can be transformed into a diagonal form by

these unitary Jacobi-transformations; the Euclidean norm of the

matrix A is invariant under these transformations and for normal .!.

matrices this norm equals ( .E !A.I 2) 2 , where A, A~, .••• , A

J=1 J 1 "" n

are

For

the eigenvalues of A. 2 n 2

non-normal matrices 1fAIIE > .. E I A. I (!IAIIE ~J=1 J i,

called the Euclidean norm of A); hence it is not possible to

transform these matrices unitarily into diagonal form, and so the

Jacobi-method is fruitless to achieve this end.

In 1962, Eberlein [4] suggested the use of non-unitary plane

transformations in order to diminish the Euclidean norms of the

matrices in the sequence thus obtained. It is not impossible that

this may lead to diagonalization of non-normal matrices since [22]

inf f!T-1 A Tl!E 2

T regular

n E

j=1

In the first part of this thesis (chapter 1, 2, 3 and 5) we con­

struct and investigate an algorithm to normalize non-normal ma­

trices. In chapters one and two an algorithm is described to re-.. duce (in some sense optimally) the EUclidean norm of a real,res-

pectively complex, non-normal matrix qy a plane non-unitar,y trans­

formation. In chapter three we prove that the sequence-{~},

generated by the successive application of this algorithm, con­

verges to the class of normal matrices with the same eigenvalues

as A0

, and,finally, in chapter five we show that computation of

these plane non-unitary similarity transformations can be per­

formed in a stable way.

In the second part (chapter four) an algorthm is described b.y which,

using unitary plane transformations, an almost normal matrix- let

8

us say the result of our norm-reducing process - can be transformed

into almost diagonal form. F·rom the diagonal elements of this form

we may read approximations of the eigenvalues we have aimed at.

0.1. Notations, definitions and elementary theorems

Q. 1 •. 1 • We

and we

our preliminaries with the definition of a normal

a list of well-known theorems concerning these

matrices.

Let A be a linear transformation on R to R , where R n n n is a

unitary or a Euclidean space of dimension nand let~* be the ad-

of A .

Definit:ion 0.1. A (=>./hA ""AA*. Let A be the matrix representation of ..4 on some orthonormal basis

of Rn; the conjugate transpose A* of A is the matrix representation

of fi * on the same basis.

In the we deal with square matrices of order n over the field

of complex numbers, unless mentioned otherwise. The eigenvalues of

the matrix A will be denoted by A.. (j ""1,2, ••• , n) where A.=fJ..+iV.o J J J J

Defini ti.on 0. 2. A normal ~ A*A AA*.

Theorem 0.1. The matrix representation A of /cJ on an orthonorrnal

basis is normal if and only if JJ is normal ([21], p.56).

A matrix A is normal if and

similar to a diagonal matrix ([21], p.165).

if A is unitarily

A real matrix A is normal if and only if A is orthog-

similar to a matrix that is the direct sum of matrices of the

9

form (A.), where

( Re(.\)

- Im( A.)

A. is a real eigenvalue of A and of

Im(A.))

Re( A.)

2 x 2 matrices

where A is a complex eigenvalue of A. This direct sum is called

Murnaghan 1 s canonical form of A.

Theorem 0.4. Let A be a normal matrix. The real parts of the eigen­

values of A are eigenvalues of' the Hermitean part i(A + A *)of A,and

the imaginar,y parts of the eigenvalues of A are eigenvalues of the . * skew-Hermitean part i(A- A ) of A.

Proof.

U*AU = Since A is normal,

diag ( ll· +i v.) , where J J

there exists a unitar,y matrix U so that

IJ.. + iv. (j = 1, ••• , n) are the eigen-J J

values of A. We see that

U* A ; A* U = diag ( ll j) , j 1, ... , n

and

U* A- A* U d' (' ) . 2 ~ag ~ vj ' J 1 , 2, ••• , n. 0

0.1.2.

Theorem 0.5. (Schur's lemma). For any matrix A there exists a

unitar,y matrix U for which holds U*AU = T, where T is of upper

triangular form; T is diagonal if and on~ if A is normal ([21],

p. 158).

Theorem 0.6. For any real matrix A of order n with k pairs of

complex conjugate eigenvalues

1J. £ ± i V£ (V£ >0, 1 ..;;; £ ..;;; k, 0 ..;;; k ..;;; [~ ])

and n-2 k real eigenvalues llj(j = 2k+1, ••• ,n) there exists an

10

orthogonal matrix Q so that

triangular form

has the following block upper

2

( \)

1 u u1n f.l1 -a- 13 1

a IJ.1 3 "- 2n

1

'

-

IJ.2k+4

0 0

0 0

and the elements a£ of QTAQ are positive,

In the proof of Schur's lemma we have to add the part of

the inductive proof corresponding to a pair of complex

Let X ;!: i y1 be eigenvectors corresponding to 1

!l1 + i \) . Since \) -fo X and y are linearly and we 1 1 1 1 have

A [x : y1 J [x1 ~) ( ~1 -\!1) 1 ' ' 1 ll1

Then sin

:) (cos q> -A [x y1] . 1 cos Sln q>

Let

(cos {jl - sin • xos• =Lx,:y] 1 1 . Sln {jl COS cp Sln cp

(cos• - sin~) ( "' =lx:y] 1 1 1 sincp COS (jl V

. 1

u 1

x cos cp + y sin cp 1 1

u x sin cp + y cos cp • 2 1 1

Then A [ u u ] 1 2

and T

u u 1 2

sin •) ("' cos cp v

1

- v1 )~~s • -sin •)

j.J.1

ln <p COS<p

-v,) IJ.1

T If X y

1 1 0 then we take cp O, otherwise we determine cp by

T Taking either value of <p, u1

u2

0. Finally to obtain orthonor-

mal vectors, u and u have to be standardized to length one. 1 2

-1 -1 Let v

1 ·- l!u 11 u V := llu2

1!2

u2

• 1 2 1 2

Then !lv 1[ = llv 11 1 ' T 0 and V V

1 I 2 2 2 1 2

A [v1 V ] = [v V ] ( IJ.1 -v/cx) (0.1.1)

2 1 2 cxv IJ.1 ' 1

where a = l!u ll /llu IT • 2 2 1 2

Let v1 and v2 be the first two columns of an orthogonal matrix ~; ~,

let B .- p- AP 1 1 1 •

Then from (0.1,1) we derive

B P TAP = 1 1 1

with a1 = ·ex v > 0. 1 1

fl1

a 1

0

0

- v 2 /a 1 1 ~13 ~,n

111 ~23 ~2n 0 ~33 ~3n

0 ~n3 ~nn

For the rest, l.he formal inductive i:'roof of the theorem is anal-

ogous l.u l.hat of Schur's lemma. 0

Definition 0.3.

I!AIIE is called the E"L.tcJ.idean norm of A.

Theorem o. 7. The Euclidean norm IIAI!E of A is invariant relative

to unitary similarity transformations, i.e.

U*U = I =) I!U*AU11 = IIAII • E E

Theorem c.s~_ Let A be a complex matrix with eigenvalues

\ , !..2

, ••• , t..n. Then

n ~ L:

i=1 I t...l 2

~

Eq'J.ality holds in (0.1.2) if and only if A is normal.

(0.1.2)

Proof. This iE a direct consequence of the theorems 0.5 and

n . 1

Definition~L',(A) := (IIA!I: - L: jt...j 2)2 • i=1 ~.

6(A) i£ calJ.ed the departure-of normality of A [14]

Next we give some theorems about the departure of normality which,

1 3

as the Euclidean norm, is invariant relative to unitar,r trans~or­

mation of A.

Theorem 0.9. For any matrix A

inf o. T regular

The infimum is assumed if and only if A is diagonalizable [22].

Theorem 0.10. For each matrix A there exists a normal matrix N

which has the same eigenvalues as A and is. such that I!A-NffE .,;;L~(A).

Proof. According to Schur 1 s lemma there exists a unitary matrix U

for which holds

U*AU = diag ( 11.:.) + T, J

where T is strictly upper triangular.

Then A = N + UTU*, where N = U diag(X.)U*. J

This matrix N is normal and

0

Corblla;y. Let A= diag( X.), A. being the eigenvalues of A. Then J J

there exists a unitary matrix U for which holds

!IU*AU - All ..; /:::,(A).

Theorem 0.11. For any real matrix A of order n there exists a

real normal matrix N of order n for which holds that A a:nd N have

the same eigenvalues and llA-NilE .,;;; 6(A).

Proof. Let QTAQ be the block upper triangular matrix indicated

in theorem 0.6. Then QT AQ = M + T, where

14

2

0 V

1.!1 -v 0 V _ _!_

1 1 a 1

V 1.!1 a -v 0

1 1 1

Ilk -\: !

M= ' T= vk Ilk

0 ll2k+1 ,j 0

with U an upper triangular matrix of which

u2

. 1 2

. = 0 , i = 1 , 2, ••• , k. ~- ' ~

2

0 \l.k

vk --ak

~-vk 0

0

0

Thus QT AQ is the sum of a Murnagban canonical form M and a pertur-

bation matrix T. Then

T T A = QJVIQ + QTQ = N + P. (0.1.3)

The matrices A and N have the same eigenvalues and N is a normal

matrix. Since the Euclidean norm and the departure of normality

of a matrix are invariant relative to unitar.y similarity transfor­

mations, we have

(0.1.4)

where (M,T)E is the inner product of the matrices M and T (consid­

ered as elements of the n2 -space) corresponding to the Euclidean

norm. As we see from the matrices M and T, written in full above,

15

k l!A-r-rn~ = 1'12 (A)- 2 z:: (a[v,.e) 2 v/a,.e..:; t'1

2 (A). o .e=t

Corollary. Let M be Murnaghan 1s canonical form corresponding to

the eigenvalues A, A, ••• , A of the real matrix A. Then there 1 2 n

exists an orthogonal matrix Q for wh:i.ch holds

IIQT AQ - Mlf ..,; 1'1(A).

Definition 0.5. Let~ (k = 0,1, ••• ) be similar to A= A0

and

n(A) the class of normal matrices with the same eigenvalues as A. The sequence~ is said to converge to normality (or to converge

to ?1.(A)) if there exists a sequence {\;} where \: E '!/.(A) (k = 0,1, ••• ), so that

Theorem 0.12. Let {~} be a sequence of similar matrices. It con­

verges to normality if and only if lim ~(~) = 0. k- 00

Proof. The sufficieney follows immediately from theorem 0.10

(for real matrices from theorem 0.11).

The convergence to normality of {~} implies that there exists a

sequence {Nk}, NkE:il.(A) so that lf~-\;IIE -o. Then there exists a

sequence of unitary matrices {vk} for which holds that

Then n 2 _ * 2 _ 1 2 _ I (k)l2 I (k)la lll\:lfE - I!Vk ~UJIE - 11 A+ Ekl E - .~ A.+ e.. + .L: e~ 0

J=1 J J J Jl.t .,yv

16

Hence n

0.1. 3.

In contrast to 6(A) the measure of non-normality of the matrix A

which we define in this subsection, is effectively computable.

Definition 0.6.

commutator of A.

* * C(A) := A A -AA • The matrix C(A) is called the

Theorem 0.13. Let 6(A) be the departure of normality of the ma­

trix A and C(A) the commutator of this matrix.

Then [14] 1

62 (A),.; [(n3 -n)/12]2 [IC(A)IIE (0.1.5)

and if A I 0 then [5]

(0.1.6)

Corollar,y. Let {~} be a sequence of similar matrices. It con­

verges to normality if and only if C(~) - 0 •

Finally, we define here some notions which are used in the descrip­

tion of Jacobi-like algorithms.

Definition 0.7. A shear matrix Tn is a non-singular matrix which --- .-vm

differs from the unit matrix I on~ in one of its two-dimensional

submatrices. In that one submatrix the elements are t,e,e' t,em, tm.e

and t • The indices ,e and m, 1 ~ ,e <m ~ n are called the pivot-mm pair of T,em and the elements t,e,e' t.em' tm£ and tmm are called the

Jacobi.;..parameters of T.em• The class of shear matrices with pivot­

pair (,e,m) will be denoted by if .£m· [.em , ~m and U .£m are

the classes of shear matrices with pivots .£and m which are uni­

modular (i.e. jdet(T£m)1 = 1), orthogonal and unitary respectively.

Definition 0.8. The matrix

will be called the (..e,m)-restriction of A.

Definition 0.9. 1

S(A) := ( 2! la. ·12Y2. ifj J.J

S(A) will be called the departure of dia,gonal form of A

0. 2. A survey of Jacobi-Hke algorithms

In a Jacobi-like procedure for the computation of the eigenvalues

A1

, A2

, ••• , An of a matrix A of order n a sequence

A=A,A,A, 0 1 2

is constructed in which the matrices ~

18

(k) (aij ) are recursively

defined by the relation

-1 Ak+1 := Tk ~ Tk (k 0,1,2, ••• ).

The matrix Tk is a shear matrix with pivot-pair(..ek, ~) and Jacobi­

parameters

(k) (k) t = p

..ek,..ek k t ..ek,~ qk

(k) (k) t ~,..ek rk t

~·~ = sk.

The indices ..ek and ~, 1 ~ ,ek < ~ ~ n constitute the pivot-pair

of the k-th iteration of the Jacobi-like process. The choice of

the successive pivot-pairs (..ek' ~) is called the pivot-strategy

of the process. In several Jacobi-like processes the pivot-pairs

are selected in some cyclic order. We mention especially the se~

pivot-strategy indicated by the scheme:

(£ ' m ) 0 0 ( 1 '2)

[<'k·V1l (..ek+1'~+1) ~ (..ek+1 ,£k+2)

( 1, 2)

The method of Jacobi ([16], 1846) is

' if

,if

if

one

..ek < n-1, ~ <n

,ek < n-1, ~ n (0.2.1)

,ek n-1, ~ n.

of the few efficient

methods of solving the Hermitean problem which existed before 1950.

After its rediscovery in the late forties several modifications

and generalizations of this method have been proposed,

1. Hermitean matrices

In the Jacobi-procedure for the computation of the eigenvalues of

a Hermitean matrix the shear matrices Tk are unitary and the

Jacobi-parameters pk' qk' rk and sk are chosen to minimize the

* departure of diagonal form of ~+1 = Tk ~Tk. By minimizing this

19

departure the element a~k+1 ) of A 1

is annihilated. Therefore ""k'~ le+

the decrease of the departure of diagonal form equals 2ja(k) f2 •

~·~ a) In the classical Jacobi-process ([9], [12]) the pivot-pair

(tk, ~) is chosen so that

I a ( k) I = max ( I a~~) I ) . £k,~ i < j ~J

b) In the serial Jacobi-process the pivots are chosen in confor­

mity with rule ( 0. 2. 1 ) [ 13] •

c.) In the serial Jacobi-method with threshold t the pivots run

serially through all superdiagonal positions of the matrix,

except those for which /a~~)j < t ([24], [29]). l.J

If all off-diagonal elements of A(k) are smaller in modulus

than t, then the threshold t is lowered.

Each of these pivot-strategies gives rise to a convergent process:

if the values of the Jacobi-parameters for which a}k+1) = 0 are

chosen in a reasonable way, then lim \: = diag(A.)k'~ [s]. k-oo J

Moreover, the asymptotic convergence is quadratic ([17], [18],

[ 3 3]).

2. Normal matrices

In the extension of the Jacobi-procedure to the case of normal

matrices, as proposed by Goldstine and Horwitz [10], the shear

matrices Tk are unitary and for each k the Jacobi-parameters of

of Tk are chosen so as to minimize ~(\:+1 ) = ~ ja~~+1 )j 2 • if j l.J

Although at each step the decrease of the departure of diagonal

form is optimal,Voyevodin has exhibited a class of matrices, for

which, independently of the pivot-strategy, this process is sta­

tionary before the diagonal form is reached [32]. To prevent sta­

tionarity, Goldstine and Horwitz have modified their algorithm,

and they have shown that with this modified algorithm

20

lim k -+C'O

(Aj). Ruhe [25] has proved that if such modifica-

tions are superfluous, the convergence is quadratic.

3. Triangularization by unitary shears

Greenstadt [11](1955) and Lotkin [19] (1956) have generalized the

Jacobi-procedure to arbitrary matrices. Their algorithms,according

to a made by J. von Neumann, are based on Schur's lem-

ma (theorem 0.5): for each matrix A there exists a unitary ma­

trix U for which holds that U*AU is triangular. In both general­

izations the shear matrices Tk are unitary. Greenstadt determines

the Jacobi-parameters at the k-th stage of the process in such a

way that a(k+i) = 0, Lotkin, on the contrary, determines them so ,ek'~

that . Z. la~~+1 )1 2 is minimal. For some matrices, however, the ~ <J ~J

the sequences {~} generated by these methods are not convergent

whatever pivot-strategy is used([ 1 ], [2]).

4. Norm-reducing by non-unitary shears

In 1962 in her paper "A Jacobi-like Method for the Automatic Com­

putation of Eigenvalues and Eigenvectors of an Arbitrary Matrix''

[4] P.J. Eberlein introduces a norm-reducing process by transfor­

mations with non-unitary shears Tk, The underlying idea is that

(in conformity with theorem 0.9)

inf T regular

n z (0.2. 2)

Moreover, if and only if A is diagonalizable, there exists a non­

singular matrix T such that T-1AT is normal; in that case the in­

fimum in (0.2.2) is assumed. So the aim is to construct recurs­

ively a sequence A0

=A, A1

, A2

, ••• , where ~+i := T~1 ~ Tk, so

that um n~n~ k .... eo

n =r.IA·I~

j=1 J

In terms of definition 0.5 and according to theorem 0.12 this se­

quence {Ak} converges to normality.

21

a) In Eberlein 1s method ~+1 is produced from~ in two steps:

"' -1 -1 -1 ~ ~ := ~ ~~ , ~+1 := ~ sk ~ sk ~ (0.2.3)

with

~and ~unitary shears and Sk a norm-reducing non-unitary

shear, all having the same pivot-pair (tk'~). ~is chosen

such that C0 n c where ,..k,A;k mk,~

c

This pre-treatment facilitates the construction of a suit-·

able norm-reducing Sk. The Jacobi-parameters of Sk are chosen

in the following way:

(

Pk qk) (cos""' i e

1

"'sinh~ ' rk sk -ie -itj,ksinhcpk cosh<pk }

~ and <)lk real.

In order to minimize ~~~+1 11E , considered as functions of

and ~k' two simultaneous quartic equations have to be solved.

Since this is not easy, Eberlein gives an approximation

(~k' ~k) of the solution (~k' $k). The norm-reducing shears

Sk corresponding to (~k' $k), in combination with the plane

rotations ~ suffice - independently of the choice of the

unitary shear~- to obtain a sequence {Ak} which converges

to normality, provided the pivot-pairs (,ek'~) are chosen

appropriately.

b) In Rutishauser's norm-reducing algorithm [28]the transforma­

tion with Tk is also performed in two steps as in (0.2.3).

The unitary shear 1\: annihilates the element c,e of f"'V f"V *,......, ,.._.J ""'* k' ~ C = ~ ~ - ~ 1\· The non-unitary shear Sk is a diagonal

matrix, which scales-~ in the following manner.

If r~t t I ~ I~ I. ' then the lengths of the t-th column k' k ~'~

and the t-th row are made equal, else this operation is ap-

plied to the m-th column and the m-th row. Rutishauser states

22

that the sequence {~} obtained in this way converges to nor­

mality.

c) Voyevodin [ 31] proposes to use the following Jacobi-parameters.

-1 The parameter \:is chosen to minimize i[Tk A TklrE

The convergence to normality has been proved.

d) Osborne 1s equilibration [23], too, is based on the principle

of norm-reduction. The aim of Osborne's algorithm is to im­

prove the condition of the eigenvalue problem. In each step

of the process the Euclidean lengths of a certain row and its

corresponding column of the transformed matrix are made equal,

Osborne has proved that if A is irreducible, then there exists

a non-singular diagonal matrix ~ such that the diagonal ele­

ments of C(D-1AD) are zero. With sequential pivoting the pro-

cess mentioned above constructs iteratively such a

matrix D. The matrix D-1AD is called equilibrated,

diagonal

In the

class of matrices similar to A by diagonal transformation, -~ the eq,Jilibrated matrix D 'AD has a minimum Euclidean norm.

5. Diagonalization and combination of no:t'lll-reduction and

diagonalization

If !:J.(A) is small in relation to ILil,.'l~ then the matrix A is called

almost normal. According to the corollary of theorem 0.10 such an

almost normal matrix can be unitarily transformed into an almost

diagonal matrix. The "almost-diagonalization" of an almost normal

matrix is the second stage of a Ja9obi-like process for

arbitrary matrices and follows the process of norm-reduction. In

Eberlein 1s process the diag\)nalization is already promoted during

the norm-reduction stage. For that purpose the unitary shear~ in

(0,2.3)is chosen such that the departure of diagonal form of

~1 (s~1 ~ Sk) ~ is minimal. However, Voyevodin 's counterexample

23

[32] shows that the global convergence to diagonal form of the se­

q·.:ence f-\:}, n1:"':dned with these {~}, cannot be proved sin.:;e the

Eberletn algorithm is a generalization of the Goldstine-Horwitz

proced:;re. Ruhe [26] has shown that if the sequencE {.I\}, gener­

ated ty Eberlein 1 s norm-reducing diagonalizing algorithm, con­

verges to diagonal form, then the convergence is quadratic.

0. 3. Summary

In chapter we investigate the norm-reducing shear transformations

on the pivot pair (-e ,m), applied to a real matrix A. It is shown

that, in consequence of the invariance of the Euclidean norm of a

matrix under orthogonal transformations, each shear in a class of

w1~at will be called row congruent shears brings about the same norm

reduction. This class is determined by what will be called its Eu­

clidean parameters (x,y,z), x> 0, y > 0 (definition1.2). With the

Euclidean parameters of a shear Tim in such a class we find a simple

expression describing the Euclidean norm of the transformed matrix

(theorem 1.2). If the shears are restricted to be unimodu­

lar then :1 T;~AT );mll~ is a quadratic function of x, y and z defined

on the hyperooloid xy- "" 1. In theorem 1.4 it is shown that, ex­

cept for a number of particular cases, this quadratic attains its

infim~m on the hyperboloid for finite values of x, y and z.

:r:r: section 1. 3 we describe the algorithm to compute the Euclidean

(x,y,z) of the class'»1m(A) of row congruent unimodular

optimal norm-reduc shears (theorem 1.5). For the computation of

tLese parameters we have to determine a real root of a quartic

equation, which is uniquely localized by an inclusion theorem (lem­

ma 1.5).

TLe particular case that the infimum of the quadratic on the hyper­

b:;loid is not assumed for finite values of the Euclidean parameters

is fully described in section 1.4.

24

In section 1. 5 it is shmm that after optimal norm reduction by a

unimodular shear on tile pivot-pair (t,m) the commutator C 1 of the

transformed matrix has the properties c Jm = O, c tt = cr:rrn.

In chapter 2 we the complex norm-reducing shear trans-

formations on the pivot-pair (t,m) applied to a complex matrix A.

As in the real case, each shear of a class of what will be called

row congruent shears brings about the same norm reduction. This

class is again determined by its Euclidean parameters (x,y, z), where

now x and y are real and positive, but z is complex (definition

2. 2). the Euclideaz: parameters of a shear T ,em in such a class

we find an expression, less simple as in the real case, describing

lj _, 1[ ( ) 1 f h , T .£mAT tm: E theorem 2. 2 • In order to s imp i y this express ion, t e

matrix A is pre-treated by a unitary shear U.£m' so that

(U~~AUEm)m.£ = 0. In theorem 2,) it is shown that if A is a pre­

treated matrix (i.e. am.£ 0) and T t is a unimodular shear on the

pivot-pair (t,m), then IIT;~AT,emlli isma quadratic function of x, y,

z and;;, defined on xy-jzj 2 1, x> 0, y> 0, z complex,

As in the real case, this quadratic attains its infimum on

xy- I zl 2 = 1 for finite values of x, y and z unless the matrix A

satisfies particular conditions.

In section 2,) we ciescribe the algorithm to compute the Euclidean

parameters of the class irltm(A) of complex unimodular optimal norm-

reduc shears on the pivot-pair (t,m) (theorem 2.5).

TI1anks to the pre-treatment of the original matrix, it is possible

to transfer the algorithm for real matrices to the complex case.

In section 2.5 it is proved that for a real matrix A the Euclidean

parameters of the class of complex unimodular optimal norm-reducing

shears are real.

0.3. 3.

In chapter 3 the effect of consecutively applied unimodular optimal

25

norm-reducing shear transformations is investigated.

Let the pivot-pair (£,m) be chosen so that

C equal to A*A-AA*. Let T.R.m E~m(A) and A' = T~~ATtm' Then

theorem 3.2 gives a lower bound for the decrease of the Euclidean

norm effectuated by Ttm:

Our proof of this result is essentially the same as that of Eber­

lein [4]. Since WB make use of the Euclidean parameters (x,y,z) of

the shears involved, our calculation of Eberlein 's estimate for the

optimal decrease of the Euclidean norm is considerably more simple

than her ovm.

We use this estimate in the proof of the convergence theorem. Let

(,e ,m ),(.e ,m), ••• be a sequence of- pivot-pairs. Let A :=A and 1 1 2 2 0

~ := T:1

\: T 0 (k"' 0,1, •.. ), where T .e is a unimodular ""k'~ _, ""k'~ k'~

optimal norm-reducing shear on the pivot-pair (,ek'~). In theorem

0.3 we prove that if the pivot-strategy is so that for each k:

then the sequence {~} converges to normality in the sense of def­

inition 0.5.

0. 3 ·4.

As a consequence of the convergence theorem of chapter 3 we find

that for each e > 0 and each matrix A there exists an integer k and

a normal matrix N with the same eigenvalues as A, so that for the

matrix \: obtained after k norm-reducing similarity transformations

26

where ~(~) is the of normality of Ak.

Therefore, in 4 we consider a Jacobi-process which almost

diagonalizes an almost normal matrix A.

In the first part of this process t!:e Hermi tean part of A is almost

diagonalized. The resulting matrix, A1

(say), is shown to have ar:

almost block stru.cture, the Hermitean part of a

block being almost a multiple of the unit matrix (lemma 4.1). As a

consequence, it is shovm that the skew- Hermitean parts of these

diagonal blocks can be diagonalized by a second sequence of Jacobi

rotations without disturbing the "almost diagonal" character of the

Hermitean parts of the diagonal blocks (lemma 4. 2). Let be the

resulting matrix after this second half of our process. 'Ihe depar­

ture of diagonal form S (A2

) of this ultimate matrix proves to be

bounded by a function of

(i) the

(ii) the

of normality of the original matrix A;

of diagonal form of the Hermitean of the

matrix A, obtained after almost diagonalizing the Hermitean 1

part of A;

(iii) The departures of diagonal form of the skew-Hermi tean parts

of the diagonal blocks of

This function tends to zero if each of these quanti ties tends to

zero (theorem 4.2). If for real matrices we want to use only real transformations, a

somewhat more complicated result is obtainable. of

the symmetric part of a real almost normal matrix A results in an

almost block diagonal matrix which is an almost canonical

form, unless, if ~ ~ iv is a

of A, there exist yet other

orem 4.6).

of complex eigenvalues

(s) with real part(s) !J. (the-

Already the norm-reducing stage of the eigenvalue procedure,

diagonalization can be promoted by executing the norm-reducing shear

27

transformation with that element T.£m E m.£m(A) that, moreover, min­

imizes the departure of diagonal form of the transformed matrix.

This element will be called the diagonalizing representative of

~m (A).

TLe problem of the numerical stability of a single optimal norm­

reducing shear transformation, executed in floating point arithme­

tic, is considered in chapter 5. It proves to be possible to per­

form the transformation with the diagonalizing representative

TimE rtl:em(A)

mation e

in such a way that the actual result of the transfor­

T~~(A +F)Tim + G, where IIFIIE is small relatively to

is small relatively to llTi~ATimiiE' This result may contribute to explain the accuracy of the solution

of the eigenvalue problem which was observed during our numerical

experiments with procedures based on the algorithms described in

this thesis.

28

CHAPTER 1

REAL NORM-REDUCING SHEARS

I. 0. Introduction

In this way

one step of the norm-red:.wing Jacobi-like process, and we shall

determine tf,e norm-reducing unimodular shear similarity

transformation for the real case. Since for one transformation

the pivots £ and m are fixed, we shall omit the subscripts when

no ambiquity arises.

L L Row congruency and Euclidean parameters of a shear Let To be a shear rr,atrix with pivot-pair (,e,m) and Jacobi-

)'"m parameters p, q, r and s. So the (,e,m)-restriction of T hn is

(: :) . Let d := det(T£ ),

m_1 non-singular T£m

pivot-pair (£,m).

(1.1.1)

thus d ps-~r. Since T£m is supposed to be

exists. Tim- 1 is also a shear matrix with

The (i,m)-restriction of Tfm-1 is

(1.1.2)

In this chapter we asswne that the matrix A and the matrix T.£m'

by which A is transformed, are real matrices. In the description -1 of the Euclidean norm of A T im' we shall try to take into

aoco~~t the invariance of this norm under orthogonal transforma­

tion. In particular, if Q.£m is an orthogonal shear then we have T -1 -1

for each shear T n : IIQ, 1' n A •J' a QD liE liT 0 AT 0 11. Hence -vm .zm ,vm "'m ,m ,vm ,vm the optimal norm-redc1cing shear is determined except for an ortho-

gcnal factor Q n • xm

29

Definition 1.1. The matrices S and T will be called row congruent

if S = TQ for some orthogonal matrix Q.

Theorem 1 .1. S and T are row congruent if and only if SST TTT.

Proof, If S = TQ, with QQT

For the proof of the sufficiency of the condition, we make use of

the polar factorization ([20], page 74) of the matrices S and T.

Let S = PU, T =RV, where P and Rare positive semi-definite ma­

trices, U and V orthogonal m:ttrices .Since P and R are complet

determined by SST and TTT respectively and the latter are equal,

P R. So S = PU = RU = TV-1u. Hence S and•T are row congruent. D

The theorem shows that the class of matrices row congruent to T

is uniquely determined b,y the elements of ; they p~rametrize

the equivalence classes into which the full linear group of non­

singular matrices is d·ecomposed by row congruency.

Now we consider row congruency for shear matrices with pivot-pair

(t,m), If p, q, rand s are the Jacobi-parameters of Ttm then the

(t,m)-restriction of Ttm T~ is

pr + qs)

r2+ (1.1.3)

Pefinition 1.2. The quantities

x := x(Ttm) := p2+ q2

y := y(Ttm) := r2+ s2

z : z(Tfm) pr + qs

will be called the Euclidean parameters of Ttm'

According to theorem 1.1. the Euclidean parameters (x,y,z) of T1

m

determine the class of shears row congruent to T£m' 'rhis class

will be denoted asl't_gm(x,y,z).

Lemma 1 .1, The Euclidean parameters (x,y,z) of a shear Tfm

30

the inequalities

X > 0, y > o, (1.1.5)

Conversely, if x, y and z ( 1 .1 • 5) , then they determine the

class ~" (x,y, z) of shears on the pivot-pair Ce ,m). This class ,vill

has an upper and a lower representative, B£m and

, with (t,m) restrictions

yi y-iz) A ()

le_ and L£m = (1.1.6) 0 y2 X

a) From (1.1.5) and definition 1.2·we find for the (£,m)­

restriction of T£mT£;

(

+ q2

pr + qs

Since is non-singular T, T"T is ,c;ill hill

x > 0, y > 0, xy - z2 > 0.

b) If (x,y,z) satisfy (1.1.5) then the shear

(x+>~

(x+y+2'/ xy-z2 ) --i z

restriction

H£m

z

Cx,y,z). This shear has Euclidean parameters

representative of~" (x,y ,z). "''m

definite, and hence

with (£,m)-

) is the s;;rmmetric

c) From (1.1.6) we see immediately trJBt Bn E 1(," (x,y,z) and ,m .«ill

Corollary 1. For each shear Tlm E ~tm(x,y,z) there exist ortho­

gonal shears Q£m and Rim such that

T£m = B£m Q.£m = 1tm R£m'

where B£m and L£m are triangular shears with (£,m)-restrictions as

31

mentioned in (1.1,6).

Corollary 2. If T 0 E~" (x,y,~), then det2 (T") = xy­~ . ~

In the sequel we mostly use unimodular shears. Then the (t,m)­

restrictions of the triangular represr.::ntatives are

Bern (y-> y~z) and L;;m = ( )'

0 l 0 y2 X ""2z X

L 2. Similarity transformations by real unimodular shears In this section we shall consider the similarity transformation

by a real shear T£m with pivot-pair (£,m) and Jacobi-parameters

p, q, r and s.

Let

and

d := det(T;;m) = ps- qr

A '·. T-1 AT .em ,em'

(1.2.1)

(1.2.2)

The elements of Ar will be denoted by a!., i = 1,2, •• ,n,j=1, ••• ,n. ~J

Only the elements of A in the ;;-th and m-throws and columns are·

affected by the similarity transformation with T1m.

For the elements of A1 we find with (1.1.1) and (1.1.2)

and

a! J.ffi

p + r a aJ~= im' "'""

aJ,e (ps au"" qr amm + rs a;;m- pq am;;)/d

a' = fia - q2 a_ 0 + qs(a. 0 - amm)}/d :.em ' J!m uw "'"'

a'm:t {p2

ame- :t'a£m- pr(a;;;;- amm)}/d

a~ (ps amm- qr au- rs atm+ pq am;;)/d

a!. = a ... lJ lJ

otherwise.

(1.2.3)

(1.2.4)

2 I 2 order to simplify the formulae for IIAI!E and. IIA liE we introduce

the followir..g notation.

n c z: a. aik

i=1 ~j

if £,m

n ·- z: a ..

i=1 Jl

if'x,m

n cr : z: a~

i, j=1 l.j

ilt,m jf,e,m

I The same functions of the transformed matrix A

I i be denoted by Cjk' Rjk' e' and cr 1 respectively.

(1.2.5)

-1 T .£m AT .£m will

For convenience and for simplicity of the formulae we will not men­

tion the dependence of these parameters on A (resp.T,e;1 AT,em)'

,e and m.

We now find

(1.2.6)

Obviously, a is an invariant of A under similarity transformations

b;y shears with pivot-pair (,e,m). Since e = (71.~ 1 )) 2 + (A.~2 ))2 (where (11 (2\ hill hill

A., ' and A, 1 are the eigenvalues of the (,e,m)-restriction of A), ,,;m Lill

e is also such an invariant : e e 1 • In order to determine !lA 1 !1 2

E we (a 1 - a 1 f and C t + C I + R t + R 1 • tm mt U mm U mm

TheorEilm 1.2. I!T.£;1 A'r,emll~ is in terms of the functions

of A defined in (1.2.5) and the Euclidean parameters of T.£m'viz.

Proof. From (1.1.4), (1.2.3) and (1.2.4) we see

C1 + C1 = Le mm

and

n 2: { i=1

if'..e,m

n

;(.

)a. + 2(pr+qs)a. 0a. } 1m l-v 1m

(1.2. 7)

R 1 + R1 = ,e,e mm z i"'1

if'..e,m

- 2(pr+qs)a 0 .a . }/d2 "'1 m1

(Rmmx + R,e,eY - 2R,emz)/(xy-z2).

Since L ~

IIA 1 ll CJe + cr:nn + Rj,e + Rr:nn + (aJm- a~) + u + e,

formula (1.2.7) is obtained. 0

In order to determine inf

T..eJ ~m (liT~: AT ,em liE), the rational function

in the right-hand part of (1.2.7) has to be minimized in the r.alf-

cone

x > 0, y > 0, xy - z2 > o.

Since the determination of the values of x, y and z minimizing

this rational function, is rather complicated, we shall henceforth

restrict ourselves to unimodular shear matrices T;,m• With this

restriction on T 0 we may restate theorem 1.2 as : hm

Theorem 1.3. If T1m is a-unimodular shear with Euclidean para­

meters (x,y,z), then

(1.2.8)

34

where

· f(x,y,z) := ax + ~y + 2yz +(-XX+ ~y

with n z (a~i+a2 .), ~:=

i=1 ~ ffil

n 2: (a~.+a~ ),

i=1 hl lm

n y:= Z (a, 0a. -a

0.a .)

i=1 lh lm hl ml

,m if£,m if.£, n; (1.2.10) and

A := a£m' (1.2.11)

In order to determine inf (liT 1;1 AT );milE), we have to mi.nimize

T ..emE ~£m the function f(x,y,z) with side conditions

X> 0, y > o, xy- = 1. (1.2.12)

Definition 1.3. The subset ~:={(x,y,z);x>O,y>O,xy-z2 = 1} of

R will be called the positive sheet of the hyperboloid xy-z2 = 1. 3

Previous to presenting in section 1.3, an algorithm to compute

the values of x, y and z minimizing f on '1{ , we shall demonstrate

some properties of the coefficients of f and then we :?.hall estab­

lish a sufficient condition for which the infimum of f, on i.he

surface'J/ , is assumed for finite x, y and z.

In the we make use of the notations introduced in (1.2.10)

and (1.2.11) and, moreover, we define

D ·- a~ - - yv,

E 2 V + 4il.fh (1.2.13)

F 2 ·- - y

Lemma 1.2. The quantities a, ~, D, E and F defined in (1.2.10)

and (1.2.13) have the following properties

(i) o::;;. O, ~ :;;. O, F :;;. 0;

(ii) D2 + E F :;;. 0;

(iii) if E.< 0 then D 0 implies F = 0, o: = 0, ~ = 0;

(iv) iff E > 0 (E = O, E < 0), then the (,e,m)-restriction

of A has two different real (two equal real, two com­

plex conjugate) eigenvalues A~~) and A~!) respectively.

35

Proof, (i) a and pare non-negative since they are sums of

squares, In order to show F ~ 0, the elements in the ~ -th column

and the m-th row of A, not belonging to the (t,m)-restriction of

A, are considered as components of a vector in R2n_4

• In the

same way we consider the elements in the m-th column and the ~-th

row, not belonging to A£m' as components of a vector in R2n_4

•

From the inequality of Cauchy-Schwarz follows for these vectors

2 F = ap-y = n n n

[ ~ (a~ o+a2.) ][ ~ (a: +a~.)]-[ ~ (a. a. -a a; a .)J~o.

i=1 lh m1 i=1 1m ..vl i=1 u 1m hi. m 1

if't,m il,e,m i;h,m

(ii) If ap f 0 then

D2 + EF = (afl-pA-yv) 2 + ( }+4At.t) (o:~/) 2

o:p{v- ~ (o:~J-pA.) f+ a~? (o:IJ+0A.)2 ~ 0.

If a0 0, then F = o, thus D2+ EF = ~ 0.

(iii) If E < 0 and D = o, (i) and (ii) imply F = 0, hence a0=y2•

In order to prove o: = p 0 we make use of the fact that a!J-phyv.

This implies that (o:ll +0A.) 2 =(o:IJ-pfl.)2 +4a!34 "'¥ 2 ( l +4A.JJ.)

Hence E < 0 implies that y ~ 0 and O:!J.+pA.= O.Since ~-pA.=yv= O, we

also have: A.=O.Since E <0 implies :\j.!<o;t follows that o:=p= 0.

(iv) E = (a£.£-amm)2

+ 4.-emam.-e= (a££+amm)2

- 4(a£.£amm~ a£mam£)

( A(1) + i\(2))2- 4 i\(1) A.(2)= (;\.(1) _ i\(2))2 • • em £m £m .£m ' £m .£m

A

Hence iff E > 0 (E O, E < 0), A£m has two different real (two

equal rPal, two complex conjugate) eigenvalues A.~!) and 11.}!) re­

spectively. D

We shall now demonstrate that if D and F are not both equal to

zero, there exists a compact set Qc~such that the minimum of

f(x,y,z) on;( is assumed in the interior of Q. To prove this

theorem we need

Lemrrra 1.3. If D and F are not both equal to zero, then for

(x,y' x+y- implies f(x,y, z)- eo.

Proof. We start by remarking that

D and F wonld be equal to zero.

>0, for otherwise both

Let 'f2 be the subset of R defined by 3

xy- o, X ? 0 1 y ? 0.

Then~ c 1l. In 1:<. we have

f(x,y,z) = ~(x,y,z) + w2 (x,y,z),

where cp : we + i3Y + 2yz, w := -.\x + !J.Y + v z.

In fl. we find for the linear part cp of f:

2cp=( ex+~) (x+y)+(cx-~) (x-y)+4yz?( ex+~) (x+y)-1 ex-~ llx-y l-4lrzi

2 2 l 2 o:+~)(x+y )- {( o:-13) +4y }2 {(x-y) +4

2 2 2 l =(cx+~)(x+y)-{(cx+p) -4F {(x+y) -4(xy-z) }2

?{x+y) {cx+f3-V(cx+~) 2-4F2 }?0.

IfF> then in~ and a fortiori on:;(, cp ?a(x+y), where 1l >0.

Thus if F > 0, then on{('

2 f = cp + w - co for x+y - =.

Now we conside:J? the particular case of F 0, D I 0. Then in;fl.:

2cp :;, ( {x+y-V(x+y) 2- 4(xy-z2

) }.

Hence ondf : cp >0 and in 14: cp?O.

(x,y, z)E 1< and cp 0 implies cxx+py = -2yz, hence

(cxx+py)2

= 4/ = 4cxpz2

..; 4cx(3xy.

Hence in~ cp 0 if and only if (x,y,z) = t((3,cx,-y), t ~ o. Along this line,~, the plane cxx+py +2yz = 0 is tangent to the

boundary a k of 11,. On t we find w = tD. Hence., since D I o, in tz - {(o,o,o)} : cp + I wl>o.

Let E be the intersection of the plane x+y = 1 and 12 • Since C is

a compact set and cp + lwi>O in t, continuous function cp+lwl

attains on't_ a positive minimum, say o(o>O). From the linear

homogeneity of cp and w it now follows that in 1<, : cp + lw r:;;;. o(x+y).

37

A fortiori cp+lw I~ o(x+y) on:(. Consequently

cp+w2~ cp+ lw I - i ~ o(x+y)- Hence also in the case F O, D f O,

on ~ q~+w2- oo for x+y .... oo • D

Theorem 1.4. If D and F are not both equal to zero, then the

infimum of f(x,y, on{ is assumed for finite x,y and z.

Proof. Lemma 1.3 asserts the existence of a. YJ.umber M > 0 such

that on~f(x,y,z) >f(1,1,0) if x+y >M. The theorem now follows

from the continuity of f and the fact that the subset of 1{ for

which x+y .;;; M is a ccmpact set. o

In the next section we make also use of 1

Lemma 1.4. (i) inf (ax+f3y+2yz) (x,y, z)E'/(

2F2 • The infimum is assumed

for finite x,y ,and z if and only if F f 0 or a + f3 = 0.

) inf ,,) -A.x-+tJ.y+vz)2

(x,y,z)Eq max(O,-E). The infimum is

assumed for finite x,y and z if and only if E I 0 o~ A. = ~·

Proof. (i)If F > 0 then, as we have seen in the proof of lemma 1.3,

on?{ a:x-r(3y+2y~~ .... oo for x1~ - 00• Hence, in this case the infimum

on :t{ of the non-negative function ax+f3y+2y z is assumed for finite

x, y and z. Using a Lagrange multiplier we easily find that 1

~n(ax+f3y+2yz) = 2F2r The coordinates of this unique stationary

1

point on:{are F'"'2 (f3,a,-y). If a+f3=0, then a=f3=y=0; then

ax+f3y+2yz = 0 for each (x,y,z)EJf.

If F = o, cx+f3fO,then on-:( ax+f3y+2yz ~ (a:+f3){x+y-V(x+y) 2 -4}~ 0.

We now consider the curve r c:(, d.efined in the following way :

:X:= -(a-f3)t+{1+(a+f3) 2 t 2 }~, y=(a-p)t+{1 +(a:+f3)2 t 2 }~, , t > o. On this curve r , using the fact that F = 0, we find

a:x+f3y+2yz=(a:-rp) {V 1+(a+f3) 2 t 2 -(a+f3 )t} .... 0 for t - 00•

d. Hence ~ (ax+f3y+2yz) = 2F2

•

(ii) If E > 0, then the plane -A.x~+vz=O intersects~. The point

with coordinates

:x:= . E+21l(J.L-A.) _ E-2A(u-A.) {E2+E(A.-!.!.)2}f' y- {E2+E(A.-!.!.)2}f

is an element of this intersection.

If E ,.. 0 then Afl ,..o and so we can apply, with appropriate modifi­

cations, the reasoning of (i)

~ (-A:x:+!.!.Y+vz) 2 = -E. D

Definition 1 .4. The class of unimodular row-congru.ent shears

for which the Euclidean norm of T 0-1A T

0 is minimal, will be

...vm ..vm

m

call~d the class of minimizing shears corresponding to A, £ and m,

and will be denoted by ~,em (A),

The Euclidean parameters (:x:,y,z) of the shears in ~,em (A) minimize

f on"!(. If D and F are not both to zero, then theorem 1.4. shows that mt .tm(A) is non-void, The particular case D = F = 0 will

be discussed in section 1.4.

1. 3. An algorithm for the real unimodular norm-reducing shears ars

In this section we suppose that D and F are not both equal to zero.

According to theorem 1.4.this is a sufficient condition for

f(:x:,y,z) to attain a minimum on~. We apply Lagrange 1s method of

multipliers to determine the point (x,y, z) on<?t where f is statio­

nery. We consider

g(x,y,z;p):=ax+py+2yz+(-A.x+uy+vz) 2+p(xy-z2 -1) (1.3.1)

In the stationary of f on'?( the partial derivatives ·Of

g(:x:,y,z;p) with respect to :x:,y,z and p are zero. Hence x,y,z and

satisfy

a - 2A.w + PY 0 (1.3.2)

€Sy p + 2/.l.w + pX 0 (1.3.3)

1 2gz = y + \'W - pz 0 (1.3.4)

and

=xy- - 1 0 (1.3.5)

39

where

w :~ -Ax + ~ + vz. (1.3.6)

We now eliminate x, y and z from the equations (1.3.2) to (1.3.6)

incl. Since px -0-211w , pY = -a+2J.. w and pz = y+vw, (1.3.5)g:Lves

On the other hand, we multiply (1.3.2), {1.3.3) and {1.3.4) by jl,

-A. and v respectively and add results. Then we find

ajJ - rn. - Y v + P w - (l +4 All) w ~ o.

With the notation (1.2,13), the equations

become 2 ?

p + E w- - 2Dw- F 0

( 0-E)w+D 0.

(1.3.8)

(1.3.7) and (1.3.8)

Elimination of w from (1.3.9) and (1.3.10) gives

(1.3.11)

The value of the Lagrangean multiplier p corresponding to the

minimum of f on~ will be called the feasible multiplier. This

multiplier satisfies (1.3.11). The next lemmas make it possible to

locate the feasible multiplier among the zeros of the quartic

equation,

Lemma 1.5. The Lagrangean multiplier p, corresponding to a

stationar,y point of f on~, satisfies the inequalities

p ~ - + min(O,E),

p < min (o,E)

(1.3.12)

(1.3.13)

Proof. We multiply (1.3.2), (1 .3.3) and (1.3.4) by x, y and z

respectively and add results. Then we find

o:x+t3y+2y z+2 ( -A.x-11J.y+v z )2 + 2 p 0.

According to lemma 1.4

~ (ax+t3y+2y z)

i~ ( -AX+iJY+ vz)2

40

max(O,-E).

Therefore

p 1 2 ],_

""2(a:x-t(3y+2y z)-( -l..x-ljly+vz) ~ -F2 +min(O,E). (1. 3.12)

If F I O, then (1.3.13) follows immediately from (1.3.12), which

for the present case proves the theorem.For the case F = O, D I 0

we have to show that E < 0 implies p < E and that E ;:;, 0 implies

p <0. Now let F 0, If E > O,then (1.3.12) shows p ~O.Since p 0

would imply w = D O, as is seen from (1.3.9) and (1.3.10), we

have p <0. If E ~ 0, then (1.3.12) implies p~E. Now p = E again

implies D = 0, as is seen from (1.3.10), hence p <E. This proves

(1.3.13). D

Ler:nma 1.6. +E 2 -2D F=O The equations { w w - have one and (p-E)w + D = 0

and only one solution (p,w) for which holds

p < min(O ,E).

For this root holds 1

-(F+rf /EY2 ~p~- if E > o,

if E = O,

(1.3.14)

(1.3.15)

(1.3.16)

E- 1 nl I E I P if E< o. c 1. 3 .n) Proof. We investigate the intersection of the quadratics (1.3.9)

and (1.3.10). To that end we distinguish three cases.

I E > 0. In this case the graph of (1.3.9) in the ~,w)-plane is

an ellipse with centre p 0, w = D/E. The half-axis parallel 1

to axis w = 0 is of length (F+ n2/E)2 • This ellipse intersects 1

the a:xis w 0 in the points having as coordinates p =£F2 , w=O.

The graph of (1.3.10) in the (p,w)-plane is an hyperbola with

asymptotes p = E and w = o. The hyperbola passes through the

centre cf the ellipse. In figure 1 the quadratics are sketched

for the case D > o. If D < 0 then the appropriate sketch is

obtained from figure 1 by reflection with respect to w = 0.

If D = 0, then the hyperbola degenerates into the lines p =E

41

42

and w = o. If Il I 0 then the quadratics have a unique point

of intersection S (see figure 1) in the half-plane p < o. The

-coordinate of S satisfies the inequalities l. l.

-(F+D 2/E)2 < p<- F2 ..;; O.

p

E

D,E> 0.

fig. 1.

If Il = 0 and F I 0, the degenerate hyperbola and the ellipse

have a unique point of intersection in the half-plane p < 0. . l.

The coordinates of this point are p = - F2 , w = o.

II -E o. In this particular case the graph of (1.3.9) in the

(p,w)-plane is a parabola which intersect the axis w = 0 in le

the po].nt having as coordinates p = :±:. p2, w 0. If D I 0,

then the parabola and the hyperbola (graph of (1.3.10)) r~ve a unique point of intersection S in the half-plane p < 0. The

p -coordinate of this point S satisfies the inequality .1.

p < F-2 .;;;; 0.

p

E=O I D>O .

fig. 2.

If D = 0 and F I o, then the parabola and the degenerate

hyperbola have a unique point of intersection in the half -r;lane 1

p < 0. The coordinates of this point are p = - F2 , w = o.

43

III E < 0. In this case the graph of (1.3.9) in the (p,w)-plane

is a r.yperbola with centre p "' 0, w D/E. This hyperboia

intersects the axis w 0 in the points having as coordinates ~ ~

44

p .:!:. F2, w "' o, and its asymptotes are p .:!:. I El 2 (w-D/E).

These asymptotes and the graph of (1.3.10) have a unique

of intersection in the half-plane p < E. The coordinates 1 1

of this point are p = D! !E!-2 , w "' (D) !E!2 •

We conclude from lemma 1.2 (iii) that D r O, forD= 0 would

F = o. Hence the

is not degenerated.

of (1.3.10), being an hyperbola,

As we see from figure 3 the byperbolae have a unique point of

intersection S in the half-plane p < E. The p -coordinate of S

satisfies the inequality

< p <E.

p

0>0 . E <0 .

fig. 3.

If F I 0 1 we can sharpen the upper bound of the p -coordinate

of s. For that purpose we :1

of p "" E - F2 and the :1

consider T,the point of intersection

of (1.3.10). The coordinates ofT

are p = E - F2 , w D Simple calculation show that the :1

left-hand part of ( 1 • 3. 9) in T equals (E-2F2) (D2 +EF) /F < 0, .l.

whereas in the point with the coordinates p = E-F2, w 0 the

.l. same function has the value -2EF2 > o. Hence the

of S is smaller than that of T if F >0. That means

:1 :1

E-IJI t.Eil-2 E;p ,;; E-F2 • D

From the lemmas 1.5 and 1.6 we that there exists one and

one Lagrangean multiplier that corresponds to a stationar,y

point of f on t( . With theorem 1 we find that in this unique

stationar,y point the minimum of f on ?I( is reached. Using the fea­

sible multiplier p, i.e. the root of (1.).11) which satisfies

(1.3.13), we find with (1.).2), (1.3.3) and (1.3.4) that f(x,y,z)

is minimal on ~ in the point

x= 2f.!D - @~p-E) p(p-E ' (1.3.18)

y -2XD - f 12:E) p(p-E (1.3.19)

z -vD + 1 12:E) p(p-E

. (1.3.20)

We summarize the results of the lemmas in

Theorem 1.5. If the quantities D and corresponding to the ma-

trix A and the pivot-pair (£,m) are not both equal to zero, then

the Euclidean parameters x, y and z of ~m(A) (being the values

of x, y and z which minimize f on :t ) may be computed from the

formulae (1.3.18), (1.3.19) and (1.3.20), where p is the unique

root of the quartic equation (1.3.11) for which root holds

p < min(O

45

1. 4: The particular case D = F , 0

In this section we investigate the properties of the

shears in the case that the functions D and F of the

matrix A and the pivot-pair (2,m) (see (1.2.13)) are to zero.

Theorem 1.6. Let D = F = 0. Then

}t'{o:x-tpy+2yz+(-A.x-tp.y+vz?} = max (0,-E).

This infimum is assumed for finite (x,y, z):: d( if and

a = 0 A [3 = 0 A (E .j 0 V A.= 1J.).

Proof. We two cases.

I a = [3 = 0 . Then y 0.

if

For this case the theorem has already been proved in lemma

1.4 (ii).

The infimum is assumed for the following values of x, y and z:

x= E+2u(u-A.) 1

, y= E-2A.(u-A.)1

, z• - (u-A.)v 1

,if E>O

{:Ff' +E(A.-tJ.f r2 {Ef +E(A.-tJ.l ra {:Ff'+E(A.-tJ.)2 }2

X= 1 y = 1 ' z = 0 if 0 1 1

X = 2j Ej-211-l.r , z =lEI-2 vsign(A.)if E < 0.

II a I 0 V [3 I o. According to lemma 1.2 (iii) this situation does not occur if

E < O.

Since D = F = 0, the line which the plane ax+fly+2y z 0

is tangent to the cone xy-~ O,ooinoides with the intersection

of the planes a x-tf3y+2y z 0 and - A.X-f].J.y+v z = 0. This line, whi oh

we shall denote by l , is in parameter form by

(x,y,z) = 2t([3,a:,-y ), t ~ o. Now we describe a curve r on:( of

which t:l is the asymptote. 1 1

r : x=- ( a-[3 )+{ 1 +(a+fl )2 t 2 }~' y=(a-[3 )t+{ 1 +(o:+fl i t 2 rr' z=-2yt' ~0. On this cure we find, using the fact that D = F = 0,

o:x+[3y+2yz = (o:+P)N1+(o:+[3)2 t 2-(o:+[3)t}- 0 fort- eo

and

Since on <1( ax+py+2y z > 0,

\Pf{ax+py+~z+(-Ax~y+vz) 2 } = o,

but this infimum is not assumed for finite x, y and z. ::l

Remark 1 • In the case D = F = 0 the

(p-E)2 (p2 (2p-E) = 0 has solutions p O,O,E,E. Thus the for­

mulae (1.3.18), (1.3.19) and (1.3.20) for the Euclidean parameters

x, y and z of the optimal norm-reducing shears are not usable.

Remark 2. If a = 0 A ~ = o, then each affected element of A not

belonging to A£m equals zero. Hence the investigation may be con­

fined to the (l,m)-restriction of A. If a 0 A p = 0 A E > 0,

f on//( determines a

class of shear similarity transformations. Each transformation of

this class symmetrizes the (l,m)-restriction of A, i.e. a£m a~.

If a 0 A p 0 A E = 0 t\ A. t ll, then the infimum of f on !( is

not assumed. This situation is connected with the defeotness of

Alm: this matrix of order two has two equal real eigenvalues and

is not symmetric.

If a 0 A p = 0 A E < 0, then the Euclidean parameters ~

(x,y,z) = rEI 42(2riJ.r, 2IA.I·, A.)) correspond to shears T£m'

such that the (l,m)-restriction of Tt~1A Ttm has Thfuxnaghan 1 s

canonical form:

where A.(1 ) is a complex tm

Remark 3. The particular case a + p f o, D = F = 0 will be il­

lustrated by an example ([3], pag. 272).

47

A

31 6

2

5

3C

7 2

5

-3 3

13

16

If (x,m) = (1,2), then a= p y 54, A= 6, ~ = 30, v = 24. Thus

D F 0, E = 1 296. Since a + p I 0, D = F = 0,

~ (flT1 ,;1

A T12fl~ = o + e = 1868.

This irSimum is not assumed as is seen from the solution

(x,y,z) = t(1,1,-1) of the equations a:x+py+2yz = o, -Ax+p.Y+vz = 0,

thus of the equations x + y + 2z o, -x+5y+4z = o. This line ,(3 :

(x,y,z) = t(1,1,-1) is generator of the as,ymptotic cone ~-z2 0

which for large (x,y,z) approximates the hyperboloid ~-z2 1 •

.l is an asymptote of the curve r c :(, defined by 2 _l. I

x y = {1+(108t) }2 , z -108t , (t ~ o).

Now we consider the uppertriangular shear T1 2 with Euclidean

' parameters (x,y,z) E r • Thus

A " (y~ ey4) T1,2

Then -1 6(y+z) (5y-z):Y1 1 1.

31-6zy -3(y+z)? -4(y+zr?

6y-1 +6 -1 _l.

7 zy 3y-2 -1 -=A( t). T1,2AT1,2 1 1

2y""2 2 (y+z )y ""2 13 -8

l 1

5Y""2 5 (y+z )y ""2 16 -3

For (x(t), y(t), z(t) r we find

1 ( -1) ~ ( -2) ;r(t}+ z(t2 0 (t-3/2) YTtT = 0 t ' y t = -1 + 0 t ~y(t)

and [y(t)+ z(t);~~)(t)- z(t)]

Hence

37 0 0 0

c 1 0 0 1 A(t)

0 0 13 -8 +O(t'"'2) , for t-oo

0 0 16 -3

;;;_;;;;;.::::;;,:;.::._.;I.:. From the theorems 1 .4 and 1.6 it follows tbat on 1r(' the

infimlUll of f(x,y,z) o:x-!f3y+2yz+(-AX"'J.!Y+vz? is assumed iff

D ~ 0 V F ~ 0 V ( a = 0 fl p = 0 f\ (E ,) 0 V A. ll ) ) ; then the minimum

is also the unique stationary point unless ex = 0 fl p = 0 f\ E > 0

(then the minimum value is assumed on the intersection of~ and

the plane -A.x-fjly+vz 0) or a = 0 fl p == 0 fl E = 0 fl A = ll (then f

is zero everywhere on:{ ) • The infimum of f(x,y,z) is not assumed onf'iff

D == 0 fl F = 0 fl (a I 0 V f3 I 0 V (E = 0 f\ A I ll ) ) • In this case f

bas not even a stationary point on 'f' .

1. 5. The commutator in relation to shear transformations

In this section we demonstrate that a stationary value of

1!T "-1

A T £ If~ 2 , considered as function of the fuclidean pararne1>e1:s ,~:;m m ~~- _

1 of T 0 , corresponds to annihilating a part of C(T 0 A T0 ) • .vm .vm .vm

Theorem 1.7. Let f(x,y,z) be the function in the right-hand part

of ( 1. 2. 7), thus f(x,y, z) = liT ,e~1A T .£milE 2 ~ (x,y, z) being the

Euclidean of T 0 • Let C(T-1A T0

)= (c!.). For the 1 .vm ,tm .vm lJ

elements of C (T ~' A T ..em)holds c };(' 1 =c lm 0 if and if the

Euclidean parameters x, y and z ofT~ are such that f is station­

ary in (x,y, z).

Proof. We assume that p, q, rand s are the Jacobi-parameters of

the shear T~. ~calculating the elements of c(T~1 AT~) we

find

49

c' mm

2c 1 £m

Since p2

2pq

= f X

2rs

we obtain c t-t

2pq

pr

qs

ps+qr

+ f y + f z

pr

qs

ps+qr

c' mm c1m 0 if and only if fx f y

f = o. 0 z

We would remark that the elements of the (.t,m)-restriction of C(A)

are obtained from (1.5.1) by substituting p = s = 1, q = r o. If C(A) = (cij), then we find c U fx(1,1,0), cmm fy(1,1,0)

and c-tm ifz(1,1,0). Thus the element of the (t,m)-restriction of

C(A) are the components of grad fin (x,y,z) = (1,1,0).

Now we restrict ourselves again to unimodular shears Tim" We in­

vestigate the properties of C(T;1 AT ) for the case ToE: ?'Jl;n (A), Nm m Nm Nm i.e. the Euclidean parameters (x,y,z) of T..tm are such that

f(x,y, z) o:x+~y+2yz +( -Ax+f.ty+vz) 2 is minimal on If.

Lemma 1 • 7. Let T £m be a unimodular shear w:i th Euclidean para-

meters (x,y,z). I,et C(T~1 A T 0 ) (c! .). For the elements of

-1 kill lJ C(T£m A Ttm) holds c£..e- c~ ctm = 0 if and if f is station-ary on/{ in (x,y,z).

Proof. We introduce new variables: t and w

X = t + w, y = t - w.

Since on~ xy - z2 = 1, x > o, y > 0, t!( is now given by

2 ~ . t = (1+w )2

• With the variables wand z we find a new expression

( ) -1 2 kw,z foriiT..em AT-tmHE:

50

Partial differentiation of k(w,z) gives

kw(w,z)= (a~) ~ +a-0+2{(~-A)t-(~+A)w+vz} {U~A w-(~+A) },

Simple calculations produce the relations between these

partial derivatives of k(w,z) and the elements of the (t,m)­

restriotion of C(T£~1 A T£m).

(

1z-(p2 +s2 -q2 -:I- )

pq-rs (

c' - c1

) ££ mm

2c 1 £m

pr-qs) (kw)

ps+qr k z

where p,q,r and s are the Jacobi-parameters of Ttm•

Since 1z-(p2 +s2 -q2 ) (ps+qr) -(pq-rs) (pr-qs) = t(ps-qr )(p2 +q2 +:I- +s2 )=t,

we conclude that cl£- c~ = elm = 0 iff kw = kz = o, i.e. iff f is stationa:cy on#( • D

Remark. If C(A) = (c .. ), then, by substituting ~J

( 1 • 5 • 3) , we find c" n.. c = k ( 0, 0) , c n = ik ( 0 , 0) •

Theorem 1 .8.

inf I! T..emE ~,em

NN mm W ,vill Z

Let C(A) = (c .. ). Then lJ

-1 A T_emiiE = I!A[E

if and only if c - c = c = o. ££ mm £m

, q=r=O in

Proof. If et£- cmm = c£m= o, then, as is seen from lemma 1.7,

f(x,y,z) is stationa:cy oncf'in (x,y,z) = (1,1,0).

There, as observed after theorem 1.6, f is a minirrum; hence

min (l!T£~1 A Ttmf!E) = lfAI[E •

T£mE t tm

Conversely, if inf (ffT£;1

AT£m!IE) [AI!E' thenf(x,y,z) is TtmE ~,em.

51

stationary on -:( in ( x,y, z) =( 1 , 1 , 0). From ~emma 1. 7 it now follows

that C 00 - c = c = o. 0 "'"' mm tm

If inf. nT -1

A T "m \lE = ffAffE' then it follows from Remark 4 T re ,em N

,em'- ,em

Remark.

after theorem 1.4 that Ttm Otm unless

a = 0 A~ 0 A(E > 0 V(E 0 A X=~)).

Finally we shall prove that if a matrix is non-normal, then for

some pivot-pair (t,m) et£- cmm and c£m are not both equal to zero.

Theorem 1.9. Let C(A) =(c .. ) be the commutator of A. A matrix A lJ

is normal if and only if for each pivot-pair (t,m) c,e,e- cmm= ctm = 0.

Proof. The necessity of the condition follows immediately from

the definition of a normal matrix. If for each (t,m) ct,e= cmm and

c£m = 0, then obviously C(A) = ai. However, tr(C(A)) = 0, hence

C(A) = 0 and then A is normal. 0

CHAPTER 2

COMPLEX NORM-REDUCING SHEARS

2. 0. Introduction

In this chapter we shall consider the norm-reducing process

for complex matrices. An algorithm analogous to that for the real

case as described in the preceding chapter, may be used to obtain

the optimal unimodular norm-reducing shear transformations. We make

use of the formulae (1.2.3) and (1.2.4) for the elements of the

transformed matrix

t -1 A Ttm A Ttm

but now Ttm and A are complex matrices.

2. 1. Row congruency and Euclidean parameters of a shear

Let Ttm be a complex shear with pivot-pair (t,m); p, q, rand s

are the Jacobi-parameters ofT. , thus . ..,m

(: :) (2.1.1)

and the (.e,m)-restriction of T£~1 is

( ~ y) .=.!:. :2. d d

(2.1.2)

where d:= det(T.em) = ps - qr.

As in the preceding chapter, we shall investigate the consequences

of the invariance of the Euclidean norm under unitary similarity

transformation.

53

Definition 2.1. The matrices Sand Twill be called row congruent

if S TQ for some unita~ matrix Q.

Analogously to the preceding chapter we can prove

Theorem 2.1. The matrices S and T are row congruent if and only

if SS* = TT*.

The row congruency is now considered for shear matrices with pivot­

pair (t,m). If p, q, rand s are the Jacobi-parameters of T$m,

then the (t,m)-restriction of T$mT£; is

( 1:1' _· lql'

pr+qs

Definition 2.2. The quantities

x := x(T£m) : JpJ2 + JqJ2

Y := y(T£m) := JrJ2 + JsJ2

z := z(T ) := pr+qs tm

will be called the Euclidean parameters of T$m'

(2.1.3)

(2.1.4)

LeiiliilB. 2.1. The Euclidean parameters (x,y,z) of a shear T$m satis-

fy the inequalities

X > 0, y > 0, (2.1.5)

Conversely, if x, y and z satisfy (2.1.5) then they determine the

class ae$m(x,y,z) of shears on the pivot-pair (t,m).

This class has an upper and a lower triangular representative

B$m and L$m respectively, with (t,m)-xestrictions

and Lim= (2,1.6)

54

Corolla:cy 1. For each shear T tmE t£ ,em (:x:,y, z) there exist unitary

shears Qtm and Rtm such that

where Btm and Ltm are triangular shears with (.£, -restrictions as

mentioned in (2.1.6).

Corollary 2. If TtmE ~ tm(x,y,z), then !det(T,em)l 2 == :x;v-lz! 2

•

In the .sequel we mostly use unimodular shears. Then :x;v-!zl 2 = 1

:: ~(ey~~,m)re;tric)ti: :~t:e(~~lar :)presentatives are

0 .) :x: 2 z x 2 •

2. 2. The complex unimodular norm-reducing shear transformation

In this section we shall consider the similarity transformation by

a complex shear Ttm with pivot-pair (t,m) and Jacobi parameters p,

q, r and s.

Let and

d := det(T,em) = ps-qr

I -1 A': T,em AT.£m.

(2.2.1)

(2.2.2)

The elements of A' are denoted b,y a!., i 1 ,2, ••• ,n, j 1,2, ••• ,n. ~J

We make use of the formulae (1.2.3) and (1.2.4) for the elements

of the transformed matrix.

In order to simplify the formulae for IIAII~ and IIA' 11~ we introduce

the following notations:

n

cjk := E a .. aik i=1 ~J

i,lt,m

n Rjk := E a.; ajk

i=1 Jl. (2.2.4)

i,lt,m

55

e ·- laul2 + 1amm12

n (2.2.5) a ·- 2:! la .. j2

. . 1 ~J l.,J= if.£,m jf..e,m

The corresponding functions of T..e:1 A T..em will be denoted by

C 'jk, R' jk, e 1 and o 1 • For convenience, we do not mention the

dependence of these parameters on A, Since .£ and m are fixed

during one step of the process, these indices are omitted where

no ambiguity arises. We now find

(2.2.6)

Theorem 2.2. I!T ..e:1 A T ..em!!~ is expressible in tenus of the

functions of A defined in (2.2.3), (2.2.4) and (2.2.5) and the

Euclidean parameters x, y and z of Ttm' namely

-1 R x + R ,eY-2 Re(R z) I!T.£mAT,emiiE=C,e_ex+CJ+2Re(C.£mz)+ mm :.e .£m +

:x;v-lz12 + la,emy+(a££-amm)zj jam.£x-(a..e..e-amm)zl

2- !z(a..e..e-amm)j

2 +

:x;v-1 z 12

With (1.2.3) we find

n 2:!

i=1 if£,m

+ (pr + qs)ai,taim+ (pr + qs) ai.£ aim }

56

(2.2.7)

+ qs)ao.a .+(pr + qs)an. a . "'~ nu "'~ m~ }

= {R X + mm

After some

( 1.2.4)

IPs - qrl2

- 2Re(R1mz) }/(xy- lzl 2).

~~·u~~.c but time-consuming calculations we find from

I a~,.el 2 +1 alm12

+1 a~1 2 +1 a~l 2 + la,.emY + (a££- amm)zl2 +

xy- lzl2

larntx-(a££- amm)il2-lz(a11- amm)1

2- 2Re(z2a1mam£)

+ • xy- lzl2

Since

I!T -1 AT 112 = C' +C' +R' +R' +la' r2+1a' 12+1a' 12+1a' 12+ a £m £m E U mm t . .e mm ££ ..em m£ I · mm '

we obtain (2.2.7). 0

We now have an expression of x,y and z describing I[T - 1 A T n 00 • tm ,.,mE This function has to be minimized in the domain x > 0, y > 0,

xy- lzl o. I~ order to simplify the variational problem we

the matrix A a pre-treatment. We transform A with a shear

Utm so that (U1m A U..em)m£ = o. Since the shears with pivot-pair

(..e,m) constitute a group, this pre-treatment causes no loss of

in the norm-reducing process. Let A be the ~~a-,c~c,~

matrix, then A = U1m A U£m" Let T ,em = U£m T £m" Then -1 rv "" -1 ""' "' rv

Ttm A T..em = T..em A Ttm and Ttm T1m U..em T£m T1m· U1m·

Consequently, if (x,y,z) and (x,y,z) are the Euclidean parameters

of T£m and T,em respectively, then

57

From the non-singularit,y of U~m it follows that the pre-treatment

corresponds to a non-singular transformation of the Euclidean ,..., -1 ""

parameters. Hence, liT~ A T JJm!lE' consi~ered a;:_ a function of

(x,y, i), is stationar.v if and only if liT ,e;1 A T ,em liE' considered

as a function of (:x:,y,z), is stationar,y in the corresponding point,

Since~- lzl 2 = 1 is equivalent to i y- fZI 2 1 (as is seen by . -~ -taking determinants in (2.2.7a)), !IT.em A T.em!IE' considered""'as a

function of the Euclidean parameters of a unimodular shear T,e ,

is stationar.v if and only if 1fT .e;1

AT .emi!E' considered as a f~c­tion of the Euclidean parameters of the corresponding unimodular

shear Ttm' is stationar,y.

The Jacobi-parameters (p,-r,r,p), where !PI unitar.y shear u)Jm have to be such that am.e = o, or,

(1.2.4), am£ p2

-(a.e.e- amm)pr- a.em r2 = o. This

1

1, of the

to

p : r = a ,e[ a mm .:t {a ,e[ a mm+ 4a £m am£ }13 2allli (2.2.7b)

The sign in pis chosen in such a way that p r is maximum.in

modulus. The relation (2.2.7b) which we have derived for p and r

is used to c(ompute a unitar.yi:h~ar)U£m with (JJ,m)-restriction

,. cos cp -e SJ.nq> •

u.em = -ie e sin cp cos q>

In the sequel we suppose that A is the result of such n~o-,c~~'~

with UJJm' Thus am£ = 0, Then theorem 2.2. can be re-stated as

If am£ = 0 then

!IT ~;1 AT mD~ = CUx + CJ + 2 Re(C.emz) +

+ Rp,j! + R~- 2 Re(R,emz)+ ia,emY + (ap,[ amm)zl2

~ - jzl2 + a + e.

In order to determine inf (!IT - 1 A T "m1I!E), the rational

T E Cf £m "' . Jlm £m

function in the right-hand side of (2.2.7b) has to be minimized

in the half-cone

X > 0, y > 0,

As in chapter one, we shall restrict ourselves to unimodular

shears T$m' With this restriction on Ttm we may re-state theorem

2. 2. a. as

Theorem 2.3. If amt= 0 and T$m is a unimodular shear with

Euclidean parameters (x,y,z), then

(2.2.8)

where

(2.2.9)

with

n n (X ·- l:: Clau12+ jami12) ~:= .E (I a im12 + I at i 12 ) ·-

i=1 i=1 if-t ,m irft,m

n (aiiim- a,e' a . )

(2.2.10) y := .E

i=1 ~ m1

i~.£,m and

ll == atm ' \1 := at.£ -a mm (2.2.11)

In order to determine int (ffT_e;1A T mffE), we have to minimize

T£mEv..em

the function f with side conditions

X> 0, y > 0, (2.2.12)

Definition 2.3. The subset#(:= {(x,y,z); x>O,y>O,:x;y-j z! 2 1}

of (x,y,z)-space (x and y real, z complex) will be called the

positive sheet of the surface xy- I z! 2 = 1.

In the sequal we make use of the notations introduced in (2.2.10)

and (2.2.11) and, moreover, we define

59

D ·- aj..l. .. yv

E ·- I vl2 (2.2.13)

F :=a~ - lrl2

Analogous~ to chapter one we can prove

Lemma 2.2. The quantities a, ~ , E and F defined in (2.2.13)

have the following properties

(i) a ~ 0, ~ ~ o, F ~ 0,

(ii) ~ = lx~~)- x~!)l 2 ,where x~:) and 1~!) are the eigenvalues of

A_em·

We shall now staroe a sufficient condition for the requirement that

the infimum of f on the surfacer is assumed for finite x, y and z.

Lemma 2.3. If D and Fare not both equal to zero, then for

(x,y, z)E: df, x + y- oo implies f(x,y, z) - oo •

The proof of this lemma is analogous to that of lemma 1.3.

Consequently we are able to state

Theorem 2.4. If D and F are not both equal to zero, then the

infimum of f(x,y,z) on~ is assumed for finite' x, y and z.

Analogous~ to.lemma 1.4 we can prove _;!,_

Lemma 2 • 4. ( i) 2F2• The infimum is inf (ax + ~y + 2Re(yz))

f' assumed for finite x, y and z if and only if F I 0 or a = ~ = 0.

(ii) ~ (lilY + vzl) = o. The infimum is assumed for

finite x, y and z if and only if E f 0 or ll - o.

Definition 2.4. The class of complex unimodular row congruent -1 shears T

1m for which-the Euclidean norm of T

1m A T1m is minimum

will be called the class of minimizing shears, corresponding to A,

£ and m and will be_ denoted by ~£m(A).

The EUclidean parameters (x,y,z) of the shears in ?n (A) minimize £m f on :t(. If D and F are not both equal to zero then theorem 2.4

60

shows that 17Z £m(A) is non-void. The particular case D = F = 0 will

be discussed in section 2.4.

2. 3. An algorithm for the complex unimodular norm-reducing shears

In this section we suppose that the functions D and F of the pre­

treated matrix A are not both .equal to zero. According to theorem

2.4 this is a sufficient condition for f(x,y,z) to attain a mini­

.mum ono/'. We apply Lagrange 's method of multipliers to determine

the point(s) (x,y, z) on r Where f is stationary.

We consider

g(x,y,z;p):= ax + py+ 2Re(yz)+ I~Y + vzl 2 + p(xy-lzl 2 -1). (2.3.1)

Let u := Re(z), v := Im(z).

In the stationary points off on~the partial derivatives of g with

respect to x, y, u, v and p are zero. Hence x, y, u + iv) aad p

satisfy

and

gx a + pY = o,

gy ~ + px + 2j~j 2y + 2 Re(~vz) = O,

i~) = y + ~~y + (jvj 2- p)z = 0

gp = xy - I zl2- 1 = 0.

Let w be defined by

w := ~y + vz. (2.3.6)

We now eliminate x, y and z from the equations (2.3.2) through

(2.3.6). Since px = - ·p - 2j ~~ 2Y- 2Re(~vz), py=-rx and pz=Y+IJ.vY+! vj2 z,

(2.3.5)

On the other hand we multiply (2.3.2) and (2.3.4) by~ and -v re­

spectively and add the results. Then we find

61

(2.3.8)

With the notation of (2.2.13) the equations(2.3.7) and (2.3.8) be-

come

p2 + Eiwl 2 - 2 Re(Dw)- F = O,

(p - E)w + D = O,

p real, w complex, (2.3.10) complex, E :;:;. 0.

Elimination of w from (2.3.9) and (2.3.10) gives

(p - E) 2(p2 -F) + IDI 2 (2p- E) = o. (2.3.11)

The value of the Lagrangean multiplier p corresponding to the mini­

mum of f on f will be called the feasible multiplier. This multi­

plier satisfies (2.3.11). The following lemmas make it possible to

locate the feasible multiplier among the zeros of the quartic equa­

tion.

Lemma 2.5. The Lagrangean multiplier p corresponding to a statio­

nary point of f ono/ is negative and satisfies the inequality

p <E; -

Proof. We multiply (2.3.2) and (2.3.3) by x and y respectively

and we take the real part of (2.3.4) which is previously multiplied

by 2z. Adding these results, we obtain, using (2.3.5),

a:x: + f3Y + 2 Re(yz) + 2 lilY+ vzl 2 + 2p = 0,

According to lemma 2.4, .J,_

~ (ax + f3Y + 2 Re(yz)) = 2 F2,

Therefore 1

p = -t( a:x: + !3Y + 2 (Re ( yz) ) - lilY + vz !2 < - i 2• (2.3.13)

If F > 0 then indeed p < 0. In order to prove p < 0 also in the

case F O, D r O, we assume p = O, F = O. Then, according to

(2.3.13) also 1JY + vz = w = 0 and we conclude from (2.3,10) D = 0,

Then F = O, D r 0 implies p < O. 0

62

Analogously to lemma 1.6 we state

Lemma 2.6. The equations (2.3.9) and (2.3.10) have one and only

one solution (p,w) for which p is negative. For this holds 1 1

-(F + ID I2/E)2 ,.;;;; pE;; - F2 ' if E > 0

1

p .;;;; - F2 , if E = 0.

_;!,.

(2.3.15)

(2.3.16)

Proof. If D = 0 then (p ,w) = (-F2 ,0) is a solution of the equa-

tions (2.3.9) and (2.3.10). This is the unique solution with neg­

ative p. If D I 0 we substitute in (2.3.9) and (2.3.10) 1

i arg D w = w e • Then we obtain the following equations for p and

w':

p 2

+ E I w 1 [

2 - 21 D [Re ( w' ) - F 0

(p-E) w1 + [D[ = 0. (2.3.17a)

(2.3.17b)

Since E ~ 0, we derive from (2.3.17b) that in the restriction

p < o,w' is real and positive. Thus for the case p < o,the qua­dratic equations become

p2 + E(w 1

)2

- 2[Diw'- F 0

(p-E) w' +ID[ = 0.

Consequently, the reasoning applied in the proof of lemma 1.6 can

be applied to prove the present lemma. D

From the lemmas 2.5 and 2.6 we conclude that there exists one and

only one Lagrangean multiplier that corresponds to a stationary

point of f on :If. With theorem 2.4 we find that in this unique

stationary point the minimum of f on~ is reached. Using the fea­

sible multiplier, i.e. the negative root of (2.3.11) which satis­

fies (2.3.12), we find with (2.3.2), (2.3.3) and (2.3.4) that

f(x,y, z) is .minimum on :fin the point

X=

y =

2 Re(~D)- §(p-E) p p-E)

ex --p

(2.3.18)

z z - vn + x~p-E) . p(p-E

We summarize the results of the preceding lemmas in

Theorem 2.5. If the quantities D and F, corresponding to a pre­

treated matrix A (arnt 0) and pivot-pair (,e,m), are not both

equal to ~ero, then the Euclidean parameters x, y and z of ~,em(A)

may be computed from the formulae (2.3.18), (2.3.19) and (2.3.20),

where p is the unique negative root of the quartic equation

(2.3.11).

It is now clear that on account of the pre-treatment performed on

the original matrix it is possible to carry over the algorithm

for real matrices to the complex case. Without this pre-treatment

the analogy with the real algorithm is lost and then it is much

more difficult to obtain the Euclidean parameters of the unimodu­

lar shears Ttm for which IIT,e~1 A T.eJI E is minimum.

2. 4. The particular case D F = 0

Analogously to the real case we investigate the properties of the

optimal norm-reducing shear similarity transformations in the

particular case that D = F = o.

Theorem 2.6. Let D = F = 0. Then

}1ft {a:x + !3Y + 2 Re( yz) + !IJY + vzl 2} = 0.

This infimum is assumed for finite (x,y, z) E cf' if and only if

ex = 0 11 ~ = 0 II(E > 0 V 11 = 0).

Proof. We distinguish two cases.

I. ex ~ = o. Then y = 0. For this case the theorem has already

been proved in lemma 2.4 (ii). The infimum is assumed for the

following values of x, y and z :

E + 21~1 2 E

x= (Ff+ El~f2)f' y= (E2+ Ej~l2)2 t z= 1!. E - V (E2+ El~12>* t

if E > 0.

X= 1 ' y = 1 ' z = o, if E ~ = O.

On r we find,

ax+~y+2Re(yz)

and

the assumptions about D and F:

(a+~){V1+(a+~) 2 t 2 -(a+~)t}-o fort- eo

~ + vz ~JV1+(a+~) 2 t 2 - (a+~)t}- 0 fort ..... oo.

Since on.( ax+~y+2Re(yz)+ ~~ + vzl 2 > 0 (the intersection of

the planes ~y + vz = 0 and ax + ~y + 2 Re( yz) = 0 is generator

of the cone xy - r z r 2 = 0)'

~ {ax + ~y + 2 Re(yz)+ ~~ +vz[2

} = 0

and this infimum is not assumed for finite x, y and z. D

Remark. From the theorems 2.4 and 2.6 ii; follows that onlf the

infimum of f(x,y,z) = ax + ~y + 2 Re(yz)+ j~y + vz[ 2 is assumed

iff D I 0 V F I 0 \I{ a = 0 A ~ = 0 A (E I 0 V l.t = 0)); then the min­

imum is also the unique stationary point unless a 0 A p= 0 A E >0

(then the minimum value is assumed on the intersection of~ and

the plane ~ + vz = o) or a = 0 A p = 0 A E = 0 A l.t 0 (then f

is zero everywhere onJ( ) • The infimum of f(x,y,z) is not assumed on~ iff

D = 0 A F 0 A (a I 0 V p I 0 V (E = 0 A ~ :j 0)). In this case f

even has no stationary points on f'.

2. 5. The commutator in relation to shear transformations

As in the real case, the reaching of a stationary point of

'I!T ;::;1A T J::mliE' considered as a function of the Euclidean para-

65

meters of T0

, corresponds to the annihilating of a part of ' ,vm

C(T,e~1 A T£m). Without proof (which is entirely similar to the

proof of theorem 1.7),we state

Theorem 2. 7. Let T ,em be a shear wi t:i Euclidean parameters x, y

and z = u + iv. Let f(x,y,z) be the function in the right-hand

( ) ,, ) I -1 2 side of 2.2.7, i.e. flx,y,z IT ,em A T,emiiE• 1 Let C(T 0 T 0 ) = (c! .). For the elements of C(T,- A T,.J holds

Nm ,vm lJ Nm hlil

ct,e = c~ = c~m = o, if and only if for the Euclidean parameters

(x,y,z) of T,em holds

af af = af ax (x,y,z) ay (x,y,z) au (x,y,z)

= o. av (x,y, z)

Now we restrict ourselves again to unimodular shears T,em•

We shall investigate the properties of C(T,e~1 A Ttm) when

T £m E: 11b,em (A), i.e. for the Euclidean parameters (x,y, z) of T £m

holds f(x,y,z) is minimum on:t( in (x,y,z).

Lemma 2.7. Let T0 be a unimodular shear with Euclidean para-"'m 1 ,

meters (x,y,z). Let C(T0

- A T0

) = (c! .). For the elements of _

1 ,.,m "'m lJ

C(T,em A T,em) holds c£;;- cr:un = c£m 0 if and only if f is sta-

tionary on:!( in (x,y,z).

We first consider a pre-treated matrixA (for which

a me =' 0) and we introduce the new variables t and w:

X = t + W, y t - W •

Since on xy-1 zl 2 = 1 , x > 0, y > 0, :r' is now by ..!.

t = (1+vf+l z[' 2 )2

• With the new variables wand z which para-

metrize :f', we find from (2.2.9) a new expression k(w,z) for

1fT -1fT 1! 2 : ,em ,em E

f(x,y, k(w,z):= (a-ttJ)t+(a-~)w+2 Re(yz)+!!l(t-w)+vz[ 2•

From simple calculations, we find the relations between

the elements of C(T,e;1A T,em) and the partial derivatives of

k(w,z):

66

o' -c' IPI 2+1 si

2-[ql

2-frl

2 - - k .e.e mm pr-qs pr-qs w

-0 lm =i - (2.5.3) pq-rs ps qr k +ik u V

c' - - -pq-rs qr ps k -ik me u V

where u = Re(z), v = Im(z); p,q,r and s are the Jacobi-parameters

of Tx,m• Since the determinant of the coefficient matrix in (2.5.3) equals i-1 ps-qrf 2 (I pl' 2 +f qf +! rj 2 +I sf2

) = (x+y) /2 =J 0, we find

o 1 - c 1 = c 1 = 0 if and if k = k = k = 0, thus iff .. et mm ..em w u v

f(x,y,z) is stationar,r on~. It is easy to see that (ou- omm) 2 + 4lo,.em[ 2 is an invariant of

C(A) under unitar,r shear similarity transformation of A. Hence, "" if A is a general matrix and A the result of the pre-treating of

A, the quantities o£..e - c~ and ctm corresponding to A are zero

iff the same quantities corresponding to A are zero. . ~ -1 "'

On the other hand we have seen that 11T ,em A T ,em liE is stationar,r

I ("' ) I . -1rv under the restriction det T .em = 1 ~ff liT ..em A T .emlfE is

stationar,r under the restriction jdet(T..em)l 1 (see section 2.2). Hence we find also for a general matrix A: c!n - c 1 = c 1 = 0 iff

""""' mm £m f(x,y,z) is stationar,r ondf'. 0

Remark. Let C(A) = (ci)' T tm E t)l,m and k(w,z) ftTx,:1A Ttm!IE,

where w = x-y. Then it is a simple matter to prove that

c - c = akj ; c = t (ak + i ak) J

)1,)1, mm aw (w,z)=(O,O) )l,m au av (w,z)=(D,O).

As in chapter one, a direct consequence of lemma 2.7 is

Theorem 2.8. Let C(A) = (c .. ). Then inf (ftT "-1

A T ,_ftE)=IIAIIE lJ T E t ...,m """'

J,m J,m

if and only if c££ - cmm = c,em = o.

Finally we. prove that for a real matrix A the Euclidean parameters

of the complex optimal norm-reducing unimodular shears are real.

67

Theorem 2.9. Let ~tm(A) be the class of complex unimodular op-

timal norm-reducing shears. If A is a real matrix, then the

Euclidean parameters of the row-congruent shears in 1n ,em (A) are

real.

Proof. As we have seen in lemma 1.7 a real optimal

unimodular shear T£m (determined except for row congruency) trans­

forms the matrix A in such a way that if C(T~1 A T£m) (oij), then c£~- c~ = c~ = 0. But then, as we have seen in theorem 2.8,

it is not possible to decrease the Euclidean norm of T~1A T,em by

a complex unimodular shear transformation on the pivot-pair (~,m).

Thus the real unimodular optimal norm-reducing shears are also op­

timal ones of the complex algorithm. D

68

If A is a real matrix, then

( I!T ~1 A T £m HE) = min

T£mEl£m T ,em complex

(liT ~1 A T ,emiiE).

CHAPTER 3

CONVERGENCE TO NORMALITY

3. 0. Introduction

In this chapter we shall prove that the unimodular

optimal norm-reducing shear transformations described in the pre­

ceding chapter, and using a well-chosen pivot-strategy, a sequence

{Ak} of similar- matrices is obtained which converges to normality

in the sense of definition 0.5. Our proof of the convergence theo­

rem is essentially the same as that of Eberlein [4]. Since we make

use of the Euclidean parameters of the norm-reducing shears, our

calculation of Eberlein 1s estimate for the decrease of the Euclid-

ean norm(see theorem 3.2)is considerably more than her own.

3.1. A lower bound for the optimal norm-reduction by shear transformations

In order to prepare the proof of the convergence theorem we would

call to mind some properties of unimodular optimal norm-reducing

shears as indicated in sections 1.5 and 2.5. Ex:cept for row con­

gruency, a shear T n is uniquelY determined. by its Euclidean para-• A;ll

meters x, y and z of which x and y are real and z is complex. In-

stead of (x,y,z) we introduce the real variables t, u, v and w :

X = t + w, y t - w, z u + iv. (3.1.1)

The conditions x > 0, y > 0, XY -[zj 2 >0 which are necessa:cy and

sufficient to characterize a shear (see lemma 2.1) are equivalent 1

to the condition t >(u2 + v2 + w2fl,

The conditions x > 0, y > 0, :xy -l'z[ 2 = 1, which characterize uni­

modular shears are equivalent to

(3.1.2)

We shall consider u, v and w as independent variables, determining,

·except for row congruency, a unimodular shear T,em•

69

Let 1

k(u,v,w' A) :=HT~ A Thnll~.

The determination of the unimodular shears Thn such that

is equivalent to that of (u , v , w ) for which holds 0 0 0

k(u ,v ,w 'A)= inf k(u,v,w;A). 0 0 0 ( ) u,v,w E R

3 Let P be the point in the (u,v,w)-space with coordinates

u = v·= w = 0. This point P corresponds to the Euclidean para1ne1ceJ:s

x = y = 1, z = 0 of the class Uhn of unitar.y shears on the pivot­

pair (t,m). Let (c .. ) = C(A). we have seen in the remark after :I.J

lemma 2.7

akl aw p

akt + i akl c ,e[ cmm .au P av P

In the proof of the convergence theorem we make use of a gradient

method to approximate the minimum of k(u,v,w;A). In order to sim­

plify the application of this method it is advantageous to have

c hn = o. For unitar.y shears U Jkn holds uzmc(A)U hn = C(UlnA U tm).

Hence the (t,m)-element of C(UlmA Ulrn) can be annihilated by

transformation of A with the shear which diagonalizes the

(,e,m)-restriction of the Hermi tean matrix C(A). Let A 1= UtmA U,em•

Then the u and v components of grad k( u, v, w;A ') in u = v = w = 0

method that is used for the

approximation of the minimum of k(u,v,w;A 1 ) it suffices to vary

along the w-axis. This corresponds to considering only those shears

that are row congruent to a diagonal matrix DJkn = diag(dj), where

d. = 1 for j I£ A j I m and dn d = 1. J N m

Since ItA nE = l!A I nE we easily find

70

We .now introduce a new variable -r : 1,_

1 +'!; 2

(1--r) ' -1 < ... < 1.

Further we define

Then /"L I EO K, /NI' EO M, L + 2N == -t( c }.e - c~-r).

With this new variable " and these notations we find

__ 2 _ (L+2N) 1i+(K+4M) , 2-(L-2N) , 3 +

1-1:'2 1--t2 (3.1.5)

If K =M= o, each affected non-diagonal element of A1 is equal to

zero, hence o = 0 and mJJn (At) = 1{.£ m"

If L + 2N o, then et.£- c~= 0. Since elm also equals zero, we

conclude from theorem 2.8 ?n.£m(A 1 ) 1t.£m' In both cases norm­

reduction is impossible on the pivot-pair (,e,m).

We now consider the case L + 2N f 0. Then K + 4 M > 0.

It is easy to see that

~ 0 ~ = -2 (1 + 2N) , "-r==O

A gradient approximation of the minimum of b('>) is obtained by

taking as a trial value of 'r:

'T 0

_ d 0 1 1 1

d 2

0 r = -i L + 2N d-. _0 d 2 0 K + 4M

... - 't' '>=

(3.1.6)

(It follows from (3.1.4) that I 't" rEO ) Using this value T we 0 0

obtain frcim (3.1.5)

71

-21: t(L + 2N)-(L - 2N)-r 2 2K" 4 o ________ ..;..o + _..;;o_

1-'r2 -'];2 (1-1:2)2 0 0 0

2K 4

=i(L+2N)2

[ 1 +t (6N-L)(L,+2N) J + "o K + 4M (K + 4M)2- i(L + 2N)2 (1 _ 1:2)2

0

Since [ (6N-L)(L+2N) [ <(6M+K) (K+2M) = K2+8Kllll+12M2..: (K+4M) 2 and

[L+2N [ ..:K+4M we find

1 ( c ~,e - c~) 2

3 a'+ ~ 1 + 4 ( I a k I I a~,e [2)

Since c 1n = o, we have (c },e - c~) 2 = (c U - cmm) 2 + 4[c fm'2

•

Moreover, a' + ~ 1 + 4( [a_km[2

+ [a'm,/) < 411A' ~~ = 4[AH~ •

Hence ( _ ) 2 + 4 r [ 2 1 c ,e,e cmm c fm 0 :;, - ___ __;;;____; __ ___;=--

12 ll Al!2 • E

We summarize the results in

Theorem 3.1. For each pivot-pair (i,m) there exists a unimodular

shear T im such that

(3.1.8)

The pivot strategy, which must guarantee that the Euclidean norm

decreases in a sufficient degree for convergence to normality, will

be derived from the lower bound (3.1.8) of this decrease. Therefore

we need

Lemma 3.1. There exists a pivot-pair (i,m) such that

(c 1,g- cmm)2

+ 4[cim[2

:;, n(t-1) IIC(A)II~.

72

(:~.1.9)

Proof. We have

2:: ( .)2 ifj J

n

n 2(n-1) )::: - 2 c .. c .. .

i=1 ll JJ

But since r, c .. ll

0 (i.e. tr (A*A AA*) = o)' i=1

n n 2: c.

i=1 :.i I: +

i=1

'A1lence for n ?-> 2 n

(c .. - c .. )2 = 2n 2~ ~-l J ,j i=1

c .. c .. o. ll JJ

n ?->4 2:: c~.

i=1 Jl

Fence L: {(c .. - c .. ) 2 + 4lc. ·l 2 f :;:,411CII~. i~j ll JJ lJ

•

If the pivcts ();,m) are chosen in such a way that

then (3.1.9) holds, as follows from (3.1.10). D

A consequence of theorem 3.1 and lemma 3.1 is

(3.1.10)

Theorem 3.2. Let A be ann x n matrix and C(A) = A*A -AA*. Then for each pivot-pair (t,m) for which (3.1.9) holds there exists

a unimodular shear 'I' tm such that

/!c (A)[[_} l!J (3.1.12)

3. 2. The convergence theorem

We now retvrn to the optimal norm-reducing unimodular shears intro­

duced in the chapters one and two.

Lemma 3.2. If the pivot-pair (t,m) is chosen in accordance with

(3.1:9) and T.em is an element of the class mtm(A) of optimal

norm-reducing unimodular shears constructed by the algorithm de­

scribed in the preceding chapters, then

73

Proof. The estimate (3.1.12) of the decrease of the Euclidean

nonn by transformation with an approximation of the optimal norm-

reducing shears holds a fortiori for the optimal one. 0

Theorem 3. 3. Let the sequence {~}, starting fro~ A0

= A, be

constructed recursively by

k 1,2, ...•.

where (~k'~) is the pivot-pair according to rule (3.1.9)

and T~k'~ is an element of the class ~tk'~(~_1 ) of optimal

norm-reducing unimodular shears constructed by the algorithm de­

scribed in the preceding chapters, Then the sequence {~}converges

to normality.

Proof. {lf~l!~} i~ a monotonically decreasing sequence of numbers,

bounded below by E ~~[ 2 (see theorem 0,8). Therefore, i=1

3n(n-1)

we have

I!C(~_1 )11~

IIAII~

Thus also !IC(~)IIE - 0 fork- oo, From the corollary of theorem

0.13 follows that {Ak} converges to normality. 0

Corolla;y. The sequence Df departures of {~(~)},

corresponding to the sequence {Ak} of theorem 3.3, converges to

zero.

74

Proof. Consequence of theorem 0.12. 0

Finally we prove an analogous theorem for the real norm-reducing

algorithm described in chapter one.

Theorem 3.4. Let A = A0

be a real matrix and let {~} be the

sequence of real matrices constructed according to the recursive

relation

,k=1,2, •••• ~ := T .£:~~ Ak-1 T ,ek'~ where the pivot-pair (,ek,mk) is

and T 0 is an element of the "'k'~

selected according to rule ( 3.1 .11)

class Zt,ek'~(Ak_1 ) of real opti-

mal norm-reducing unimodular shear computed with the algorithm

described in chapter one. Then {~} converges to normality.

Proof. In the case of a real matrix the approximate optimal

norm-reducing shear transformation of lemma 3.2 produces a real

matrix. From the corollary to theorem 2.9 we conclude that lemma

3.2 still holds for this real transformation. Hence the proof of

theorem 3.3 can be copied. 0

75

CHAPTER 4

JACOBI-LIKE METHODS FOR ALMOST DIAGONALIZATION OF ALMOST NORMAL MATRICES

4. 0. I ntroduction

During the numerical realization of the Jacobi-process for

Eermitean matrices the Hermitean character is usually preserved

by storing and computing only the elements in the upper trangle

and the corresponding elements in the lower triangle to

b.e thGir complex conjugates.

It is generally not pos~dble to preserve in such a way the normal

character of a normal matrix: al~eady the floating-point repre­

senti1hon of a normal matrix A differs not only from A, but gen­

erally it will even not be normal. Therefore, algorithms for ex­

actly normal matrices are mainly of theoretical interest. Their

scope of applicability has to be extended to a larger class,

namely to the class of almost normal matrices.

In section 4.1. we shall describe an algorithm by which, using

unitary shears, a complex almost normal matrix is transformed in­

to almost diagonal form. The ultimately obtainable departure of

cliagonal form ·depends on the departure of normality which is in­

variant under unitary transformation.

In section 4.2. we shall formulate a real variant of this process.

In this case our aim is to reach an almost bl?ck-diagonal matrix,

being a perturbed Murnaghan canonical form. This proves to be

possible, unless the matrix has, with a pair of complex conjugate

eigenvalues, yet other eigenvalues with the same real part.

In the sections 4.3 and 4.4 we shall indicate which element in

the class ~~m(A) of real (respectively complex) optimal norm­

reducing shears may be preferably used in a norm-reducing trans­

formation on the pivot-pair (~,m). As we have seen in the chapters

one and two, the optimal norm-reducing shears with pivot-pair(t,m)

are determined up to row congruency, i.e. if T0 E ln" (A) , ,vill ,vill

E m,1

(A), where U o is any ...,m "'m

,m) • As we have seen in chapter

gence to normality of the sequence {~} is

T n E lnv (Ak 1) that ~k'~ Nk'~ -

element

transformation

k 1. 2 ••••

shear with

the oonver­

of the

is used in the

Since usually norm-reduction is only the first of an algo-

rithm for the computation of the eigenvalues

able to make use of the degree(s) of freedom

~ g .~ (Ak_1 ) in order to promote the

of A, it is reason-

within the class

of the k k .

matr1ces in the sequence {~}. Therefore,

use the element from ~n (A 1

) that '"k'~ -K-

it seems appropriate to

minimizes the departure

of diagonal form of the transformed matrix ~·

4. l. Almost diagonalization of a complex almost normal matrix ix

We introduce the following notations

H :== t(A+A*), ,D .­

A := i(A-A*), K .-

diag(Re(a .. ) ) , JJ

,2, ••• ,n, F := H-D. (4.1.1)

diag(i Im(a .. )), j JJ

1,2, ••• ,n, G:=Z-K.(4.1.2)

If A is normal and the Hermitean part H of A is diagonal, i.e.

H = D, then DZ = ZD. Consequently, = 0 if d .. f d ..• Hence J.J. JJ

a normal matrix of which the Hermi tean part has diagonal form, is

a direct sum of diagonal blocks. Each diagonal block is the sum

of a real multiple of the unit matrix and a skew-Hermitean matrix.

In the sequel we shall

almost diagonal Hermitean parts. Of these matrices the diagonal

elements can be grouped into elements within a cluster

baNing almost equai real parts. In order to describe this phenom­

enon we introduce the notion of a ,;-partition of a complex matrix.

77

Let us permute the rows and columns of A so that the real parts of

the diagonal elements are non-increasing. Without loss of generality

we may assume this was true originally, i.e. if 1 "' j < k < n, then

Re(a .. ) ;;. Re(akk.). JJ

Let

A

A 11

A 21

A 12

be a partition of A with square diagonal blocks. Let~ ;;. 0.-

This partition is called a ~-partition of A lithe real parts of suc­

cessive diagonal elements to the same diagonal block dif-

fer not more than~, whereas the differences of the real of

diagonal elements occurring in different diagonal blocks are greater

than this value,.. Or, eg_uivale~tly: Re(a .. - a. 1

. 1

) "'~if and ll l+ ,l+

only if a .. and a. 1

. 1 belong to the same block. ll l+ ,l+

Let the dimensions of the diagonal blocks be n, n , ••• , n. Then 1 2 K k

L: j=1

n.=n. The real parts of the diagonal elements belonging to the J

j-th diagonal block do not di'ffer more than (n .-1 h. Let for this J

~-partition the g_uanti ty 1: * be defined by

: = min (Re (a .. - a . . ) ) , . . ll JJ l<J

and a .. belong to different diagonal JJ 2 2 2

Since (-r *) "'ma.:x[ Re (a . . - a .. ) ] "' 2f!AIIE' . . ll JJ l' J

where

is only one block, then we take 1:* = +=.

78

blocks. Evidently ],_

"*"'22 IIAIIE. If there

Let U be a block diagonal unitar.y matrix with the same

A, thus

u

Let

u 11

0

0

A' 11

A' 21

A1 := U*AU =

Then

A! . = Ut. A. . U .. lJ ll lJ JJ

0

u 22

0

A' 12

AI 22

i,j

0

0

A' 1k I k

1,2, .•• ,k.

tion as

(4.1.6)

Analogously to (4.1.1) and (4.1.2) we denote qy H1 , D', F 1 , Z1 , K'

and G' the corresponding matrices derived from A1 • These matrices

corresponding to A and to A' are partitioned as A.

Using these notations we start to estimate the non-diagonal blocks

of z.

Lemma 4.1. according to (4.1.3) with some

given non-negative value of ~· D, F and Z are defined according to

(4.1.1) and (4.1.2), ~*according to (4.1.4). Then for the elements

zij of Z not belonging to a diagonal block holds l

izij'"'" 2""2 [ll(A)+ 2(l!ZliE+ ll(A)) (!IF!IE+ ll(A))/ -.*] , (4.1.8)

79

where 6(A) is the departure of normality of A.

11breover,

2: !IZ n IIE2

tfm --vm

Proof. In with theorem 0.10 we can write A = N +

where the normal matrix N has the same eigenvalues as A and

!! PilE l:!. (A). Then

D + F = N + N*)+ t(P + P*), Z = t(N - N*)+ t(P - P*).

Let Q : P + P*), R := t(P- P*),

Since N is a normal matrix, the Hermitean and skew-Hermitean parts

D -+ F - Q and Z - R respectively,. commute:

(D + F - Q) (z - R) = (z - R) (D + F - Q) •

Hence

DZ - ZD = DR - RD + ( Z - R)(F - Q) - (F - Q)( Z - R). (4.1.10)

From this matrix identity we find the following identities for the

(i,j)-elements: n

(d .. -d .. )z. ·"" 2:: Yz.k- r.k)(fk.- qk.)-(f.k- q.k)(z. .- rk.)} + ll JJ lJ k=1~ l l J J l l kJ J

(4.1.11)

With the triangle inequality and that of Cauchy-Schwarz we derive

from (4.1.11) the following inequality:

2t 2t 2t 2t I d .. -d .. jjz .. j <~> {(Biz.kl ) +(Bjr.kl ) }{(~Irk./ ) +(~jqk. I ) } + ll JJ lJ k l k l k J k. J

2t 2t 2t 2t + { ( Z If. k I ) +( Z:: I q. k I ) }{ ( E I zk. I ) +( E Irk. I ) }+Id .. -d .. 11 r .. 1. k l k l k J k J ll JJ lJ

Since F and Q are Hermitean, and Z and Rare skew-Hermitean we can

majorize the right-hand part in the way : 1 ' 1

jdii-djjll zij j..;2'"'2/dii-djj I I!RIIE+2. 2'"'2(11 ZI!E+I!RIIE) (IIFIIE+[Q~).

Since ITQ~ = nt(P + P*)I!E..;;fiP!IE = l:!.(A) and similarly [RilE<~> l:!.(A},

80

we have -t .d..

ldii-djj J lzij 1,..;2 /dii-djjl 6(A)+ 22 (llZJIE+ 6(A))(IIFIIE+ 6(A)).

If (i,j) is the index-pair of an element not belonging to a diago­

nal block, Id .. -d. ·I > 1:*, hence (n .• 1.8) is obtained. ll JJ ' In order to prove (4.1.9) we start from (4.1.11):

z .. = r .. + [(z- R)(F - Q)-(F - Q)(Z - R)] .. /(d .. -d .. ). lJ lJ lJ ll JJ

Hence, if z .. does not belong to a diagonal block: lJ

lzij I ,..; lrij I+ l[(z-R)(F-Q)-(F-Q)(Z-R) ljl/-c*· (4.1.12)

We now take squares, and in the left-hand side.we sum over the non­

diagonal parts of A, whereas in the right-hand side we sum over all

i,j. Then we obtain

2: UZn ITE2 ,.;11jRI+i(Z-R)(F -Q~ /r:* + I(F-Q)(z -R)I/r*ll~ tiro "'m

,.;3liRIT~ + 6(-r*)- 2 1lZ-RI!~ ffF-Qll~

<e;;3t.2(A) + 6{'r*)-2 (nznE+ t.(A)J (\!FilE+ t.(A)) 2•

(Here I RI denotes the matrix with elements I r .. 1 etc.). D lJ

The qualitative interpretation of (4.1.9) is very simple. If !!FifE

and 6(A) are small relative~ to 1:* and t.(A) is small relatively to

nAHE' then t~m nz~mll~ is small relative~ to IIAII~ (note that

!lznE ""!!AI! E). Hence an almost normal matrix with a Hermitea~ part of almost dia­

gonal form is almost a direct sum of diagonal blocks. Each diagonal

block is the sum of a Hermitean matrix which is an almost multiple

of the unit matrix and a skew-Hermitean matrix.

We shall now prove a lemma which is useful in dealing with diagonal

blocks of the above type. To such a block we want to apply a uni­

tary _transformation to diagonalize the skew-Hermitean part of the

block. The lemma asserts that after this transformation the

Hermitean part of the block is still an almost multiple of the

unit matrix.

81

Lemma 4.2. Let H be a Hermitean matrix and

t : max ( j h .. - h .. !) • . . n JJ ~,J

(4.1.13)

For each unitary matrix U the departures of u..l."•"'v·''"" ... form of H and

U*HU satisfy the inequality

(4.1.14)

If a:= i(max (h .. ) + min (h .. )), then I o:- h. ·I os;; it for j JJ j JJ JJ

each j. Let F be defined by F : == H - diag(h .. ) • Since JJ

H =a I·+ diag(h .. - a) + F, for each unitary U we find JJ *

U*rlli o:I +U* [diag(h .. -a) +F]U. Hence we have for S(UHU) JJ

S2 (U*HU) os;; 11U* [ diag(hjj- o:)+F]U!I~ os;; z n t 2 + !IF!!~ • D

With the preceding lemmas we are able to estimate the departure of

diagonal form s(A') of the matrix A'= U*AU (described in (4.1.6))

in terms of l'l(A), the non·-d~agonal part of the Hermitean pal.'t of

the non-diagonal parts of the skew-Hermi tean parts of the diagonal

blocks of A1 , and the quantities~ and~* belonging to the parti-

tion.

Theorem 4.1. Let A be a complex matrix for which holds

Re(a .. ;;;. Re(a .. ) if 1 os;; i < j ...:; n. Let A be partitioned acc;rding n JJ

to (4.1.3) for some given non-negative value of~, i.e. and

a. 1

. 1

belong to the same diagonal block iff Re(a .. -a. 1

.+1

) os;;-;. J.+ ,~+ J.J. J.+ ,J. Let U be a unitary block diagonal matrix having the same block

structure as A. Let A 1 := U*AU. The matrices F, G and Z are defined

according to (4.1.1) and (4.1.2). Let~* be defined according to

( 4 • 1. 4) • Then

s 2 (A')< IIF!I~+ 3l'l2 (A)+ 6h·*)~ (!!ZfiE+ l'l(A) )2 (!!FilE+ n(A) )2 +

k + ~ {I!G!.!!~ +in .(n .- 1 ) 2 ~2 }

j=1 JJ J J

82

where n. is the dimension of the j-th diagonal block A .. of A. J JJ

Proof. We start by rellll;l,I'.ICI.r.lg that S2 (A) "' S2 (H)+ (z). Hence k k

(A 1 )= 4, ff-(A!.)+ I!A! .!IE2 = ~ {S2 (H!.)+S2 (Z!.)}+ ~ !IA!.IIE2 •

j=1 JJ ::LJ j=1 JJ JJ ifj ::LJ

Since H'. . = F! . + D t . and Z! . = K~ . + G~ . , JJ JJ JJ JJ JJ JJ

k c A,) = ~ c !IF ! . nE2 + n G! . rrE2 ) + 2:; RA! .11 E~ •

j=1 JJ JJ ifj J.J

We now apply lemma 4.2 to the Hermitean part H! J

parts of the diagonal elements of A .. differ JJ

whence I!F!.I! 2E.,.;}-n.(n. -1?-r2 + [[F .. I!E2 • JJ J J JJ

Thus k ( 2 2) k .J..._ )2 2 E !IF! .HE+ I!G! .liE ..;;; ~ {411.(n.-1 -r + l!F ..

JJ JJ j=1 J J JJ

(4.1.16)

of A! .• The real JJ

more than (n.-1)-r, J

Since Ai_j = uri Aij Ujj' we have I!AJ}E

of lemma 4.1 to estimate ~ !lA .. IIE2 •

IIA .. lfE. We make now use ::LJ

J "J ' irj ~

l!A .. IIE2

= ~ (!IF .. I!E2 + I(Z •. IIE2

) ::LJ ifj 1J 1J

11~+ ~L12 (A)+ 6(-r*)-2 (I!ZIIE+Cl(A) )2 (I!FIIE + ll(A) )2

•

(4.1.18) With (4.1.17) and (4.1 .18) we find from (4.1.16)

+ ~· {l!G!.II~ +in.(n.-1)2-l}. D j=1 JJ J J

83

We conclude from (4.1.15) that S(A') will be small

HAnE if the following requirements are fulfilled:

1 • 11(A) << !lA!~

2. IIF1!E << llA!lE

3· 't" << IIA!IE

4· !IF liE + 11(A) << -.*

5. k

l!G'. -llE2 << I!AIIE 2. L!

j=1 JJ

to

If the conditions 1 and 2 are fulfilled then obviously there exists

a ,;-partition so that the requirements 3 and 4 are statisfied. This

'_ist of requirements for the smallness of s(A 1 ) /!lA liE may be used as

a lode-star for a safe (but in many cases unnecessarily pessimistic)

choice of the value of -r which deten"tlines the -r-parti tion of A. The

value of -r is to be chosen so that the terms

6(T*)- 2 (!lZIIIE + 6(A)) 2 (!!F!!E + ll(A)) 2 and t t 2 ~ n.(n.-1) 2 are both

j=1 J J small relatively to liAl!E. In the following corollary to theorem 4.1

we mention a value of -r which fulfills this requirement.

where F is defined according to (4.1.1). Using this value of 't" in

the of A and u, and using the same notations as in

theorem 4.1, we find that the departure of diagonalform of A'= U*AU

satisfies the inequality

Proof. We derive from (4.1.19)

·h*)-2< 1:-2 = (I!FIIE + ll(A))-1 KAn;1

• (4.1.21)

From (4.1 .15) and (4.1.21) we conclude that (4.1.20) holds. D

In order to

write this

a qualitative interpretation of (4.1.20) we re­

in the following manner:

k k 2.: n.(n.-1)2 +-1-

J J IIAII~ E I!G~. lli .

j=1 JJ (4.1.22)

~(A) << IIA!IE' IIF''IE << IIAII'IE and U is such that Hence, if k 2 Z !IG~ .liE

j=1 JJ << IIA'I/~, then S(A ')<< IIAIIE· This means that if A is al-

most normal, the Hermitean part of A almost diagonal, and ~ chosen

to (4.1.19), then A1 ·almost iff the skew-

Hermitean parts of the diagonal blocks of AI are almost diagonal.

We would remark that the estimates (4.1.20) and (4.1.22) for S(A')

are very crude, and that for the reasons.

Within a diagonal block of our Re(aii- ai+i,i+1 ) ~ ~' whereas I Re( a .. - a .. ) I ~ 1:* > 1: if and a .. belong to different

ll JJ JJ u~'~g,;r~'~ blocks of the partition. For the majority of (almost nor-

mal) matrices (namely, those whose have well-separated

real there exists a 1: of the order of !!FilE + ~(A), for which

-c* (of the corresponding 1:-parti tion) is of the order of I!A!fE •

Then we find from (4.1.15) that (A 1 ) is of the order of k 2

ftFjj~ +62(A) + Z IIG~ .!lE • For these matrices the value of ,. given j ==1 J J

in (4.1.19) is a severe over-estimate of the

smallest value of 6( 1:*)-2 (IIZIIE+ !.I( A) )2 (!lFII

value of 1: gjving the

( ) ) 2 1 2 k ( )·2 1". A + 4 -r 2.: n. n.-1 j=1 J J

and a severe under-estimate of the 1:* corresponding to the latter·

value of 1:. Hence for these matrices taking both for 1: and 1:* the

value given by (4.1.19) means severely overestimating of the

hand side of (4.1.15).

The estimate obtained for S(A') suggest an algorithm to transform

an arbitrary matrix into almost diagonal form.

85

Algorithm. Let A(o) be an arbitrar,y complex matrix of the order n.

This matrix has been transformed into almost diagonal form in the

following way.

(i)

(ii)

We apply the norm-reducing process of theorem 3.3 to the

matrix A(o). Thus the matrix A(o) is transformed into anal­

most normal matrix A(1 ), i.e. li(A(1 )).,.e: , where e: is some 1 1

prescribed positive number.

B,y applying the classical Jacobi-process we transform A( 1 )

unitarily into a matrix A(2 ) of which the Hermitean part is

almost diagonal, i.e. )+A(2)))E;;e: , where e: is some 2 2

prescribed positive number.

(iii)We permute the rows and columns of A( 2 ) in such a way that of

the resulting matrix A(3) the real parts of the diagonal ele­

ments are arranged in non-increas~ order.

(iv) We choose such a value of ~ that for the corresponding~­

partition the quantity

f(~)

is m1n1mum. This can be done by tr,ying the values

0 Re(a(3)- a(3 )) Re(a(3 )- a(3 )) ••• Re(a( 3 ) - a(3 )). ' 11 22' 22 33' ' e--1,n-1 nn

With the value of -r 0btained in this way, A( 3 ) is .. -partitioned.

A(3) A(3) A(3) 11 1 2 1 k

•

•

86

where 1 .;;; k ,..; n.

(v) We apply the classical Jacobi-process to the diagonal blocks

of A( 3 ) (i.e. A( 3 ) is transformed with a block dia­

gonal matrix u< 4 ) , having the same block structure as A ( 3 ) )

in such a way t~t the skew-Hermitean parts of the diagonal

blocks A~~), A~~), ••• ,A(~ of the transformed matrix

A( 4 )::: u(4 )*A( 3 ) U(4)are almost diagonal, i.e.

~ s2(t(A(4 )- A(4 )*)),..; s 2, where e: is some

·=1 3 3 ~ositive number.

The question whether and in what sense A( 4) is an almost diagonal

matrix is answered by the following theorem,

Theorem 4.2. For each e: > 0, we are able to prescribe in the al-

gorithm sketched above, such values of e: , e: , 't' and e: that

s(14))o;;;e:. () () 1 2 3 6{11 t(A 2 - A 2 *)11 + e: }2 (e + e: T !

Take 1: = [ E 1 1 2 J • ! n(n-1 )2

. * With this value of 1: we have, s~nce 1: > 1:,

6(1:*)~{!1t(A(2)_ A(2)*)!fE + e:1 }2 (e1 + e)2 =

= (~2 n)t (n-1)(8 + e: ){!lt(A(2 )- A( 2 )i1E + e L 1 2 1 '

k 2 2 For the term 2:: n.(n.- 1) 1: in (4.1.15) we find

j=1 J J

k t 2:: n. (n .-1 )2"2

,..; i n(n-1 )21:

2 = j=1 J J

• (~2 (n-1)(e + e: ){flt(A( 2)- A( 2)*)11E + e } • 1 2 1

Hence we find from (4.1.15) (since ITA(2 )11E= !!A( 1)llE"' UA(o)IIE)

ff(A(4 )).;;; 3e 2 +e:2 +e:3 +2(~2 n)t(n-1)(e +e ){IIA( 2 )11E +e:} 12 3 12 . 1

(4.1.23)

By taking appropriate values of e , e and e it is obviously pos-

sible to obtain S2 (A (4 \..; e2

1 0

2 3

Remark. From theorem 0.13 follows a criterion by which effective-

ly can be tested whether ~(A) ~ e • 1

n3 - n .1. .1. For if ( -.:r2) 4

11 c(A)IIE 2 ~ e1

, then a fortiori ~(A)~ e • 1

We shall now demonstrate that the ~-partition of an almost normal

matrix with almost diagonal Hermitean part corresponds to a par­

tition of the eigenvalues with regard to their real parts. To that

end we need

Theorem 4.3. Let i(A +A*) = D + F, where D = diag(Re(a .. )). JJ

Suppose Re(a .. )?Re(a .. ) if 1 ~ i < j ~ n. If 11 + iv, 1.12

+ iv , ••• , ~~ JJ 1 1 2

" + iv (". ? 11 • 1

) are the eigenvalues of A then ""n n ,...~ ~+ ·

n ]2 ( 2 E [Re(a .. ) - ll· ~ IIFIIE +MA)) • j=1 JJ J

Proof. As is stated in theorem 0.11, the matrix A is the sum of

a normal matrix A having the same eigenvalues as A, and a matrix P,

so that \I PilE = 1~A - N11E = ~(A)

For the Hermi tean part i(A + A*) of A we have

i(A + A*),= D + F = i (N + N*) + i(P + P*).

Hence D - i(N + N*) = i(P + P*) -F. (4.1.24)

Since, according to theorem 0.4, the real parts of the eigenvalues

of A are the eigenvalues of the Hermitean par~ + N*) of the

normal matrix N, we find from (4.1.24) with the Wielandt-Hoffman­

theorem

n :1.

( E [Re(ajj)- llj]2

)2 ~ lli(P + P*)- FIIE~I!PIIE+ !!FilE= IIFffE+~(A). 0

j=1

We conclude from theorem 4.3 that the real parts of the diagonal

elements of an almost normal matrix A, with almost diagonal

Hermitean part i(A +A*), may be considered as approximations of

the real parts of the eigenvalues of A.

88

We first consider matrices having eigenvalues with well-separated

real parts ll ~ ll ~ ll ~ •• ~ lln• Again we assume that 1 2 3

Re( a .. ) ~Re( a .. ) if i < j. n JJ

Theorem 4·4• Let A be a complex matrix with eigenvalues

~=t..tj+ivj

if i < j.

( j = 1 , 2, ••• , n). Let Re (a .. ) ~ Re (a .. ) and ll.; ~ ll . l.l. JJ ~ J

Let d : = min ( lt..ti- ll j l') ll . fll '

J. J

e :=~(A) + s(t(A +A*)).

(4.1.25)

(4.1.26)

If e <id then there exists a partition of A, which is a ~-par­

tition with~= 2e and ~* > d- 2e.

Proof. According to theorem 4.3, for the real part of the diag-

onal elements of A holds

n ( E [Re (a .. )- ll. J2

J J J (4.1.27)

We now consider in the complex plane the vertical strips

I£ = { z ; I Re ( z)- ll i I .,;;;; e:} , i = 1 , 2 , ••• , n. ( 4.1 • 28)

If f.!£ -f llm' then since e~, the strips I ,e and Im are disjoint. As a

consequence of (4.1.27) the diagonal elements of A are elements of

n U I,.e and the number of diagonal elements belonging to the strip

i=1

I , equals the multiplicity of ll 0 in the sequence f.! , ll , ••• , f.l • "' "' 1 2 n

Since e <id, the (horizontal)distance of the diagonal elements of

A belonging to different strips is greater than d - 2e: > 2e:. The

(horizontal) distance of the diagonal elements belonging to the

same strip is not greater than 2e. Hence for the ~-partition of A

with 1: = 2e holds ~* > d - 2e. [J

We can derive an analogous result for matrices with eigenvalues

clustered in vertical strips of the complex (eigenvalue) plane.

,.

Theorem 4.5. Let A be a complex matrix with eigenvalues A.=~.+iv. J J J

(j 1 ,2, ••• ,n). Let Re( a .. ) ;,Re( a .. ) and ~· :;;, ~· if i < j. Let ll J J J. J

these eigenvalues cluster around k distinct vertical lines z = mh

(t = 1,2,,,.,k), and let o be the ;vidth of these clusters, i.e.

Let d ·- min [m.- m"l' 1<j<t<k J h

e := t:,(A) + S(t(A +A*)).

If c: + o < then there exists a of A which is a ,_

with 1: = 2 ( e + o) and '* > d - 2 ( c; + a).

From theorem 4.3, (4.1.31) and (4.1. ) we deduce that

the diagonal elements of A are elements of the vertical strips

I..r;={z; (z)-m..el"'e:+o}, .R=1,2, ••• ,k. Ifd>2(c:+o),

then the strips are disjoint and the number of elements

of A in the strip equals the number of eigenvalues of A in this

Consequently since b + e < the (horizontal) distance of the

diagonal elements of A belonging to different strips is greater

than d - 2(6 + e)> 2(o + e) whereas the (horizontal) distance of

the diagonal elements belonging to the same is not

than 2(o +e). Hence for the ,-partition of A with,= 2(o + e:) holds '*>d.- 2(o + e). o

4. 2. Almost block diagonalization of a real almost normal matrix

An~logously to the preceding section we now investigate a real A

of which the diagonal elements are assumed to be non-increasing.

Let this matrix be 1"-partitioned with some value of ' : aii and

ai+1 ,i+1 belong to the same diagonal block iff - ai+1 ,i+1 ..,. T.

We assume that we have in this partition of A k diagonal blocks k

Thus L: n. i=1 l

with dimensions n1

, n.

90

Let the indices of the diagonal elemenvs of the blocks run

from p. up to and including q .• If ~ ~

:==tea +a pi,pi qi,qi) (4.2.1)

then for each diagonal element a££ in the block holds

(4.2.2)

We now consider the following decomposition of A

A= M+ G + L.

In (4.2.3) M is block diagonal with diagonal blocks

(I~ the n .. x n. unit matrix) and G is block .L ~ ~

blocks

i = 1,2, ••• ,k. Gii = !(Aii - AiiT),

If for each i (1 .;;; i .;;; k) holds n . .;;; 2, then M + G is a mur.t'll:~gr:~an l

canonical form. If, moreover, lfLIIE is small relatively to llAUE ,

then A may be said to have an almost Murnaghan canonical form •

In order to estimate ITLIIE we investigate the c-parti tioned matrix.

A=

We would recall that

and a. . belong to the l~

'* = min faii- ajjl' blocks of the

of lemma 4.1 •

a a ,;;:: ~ whenever a ii- i+1,i+1-' i+1,i+1

same diagonal block, and that by definition

where and a .. belong to different diagonal JJ

Without proof we state the real analoguE

Lemma 4.3 •.

(4.2.5) with

Let the real matrix A be '1"-partitioned according to

some value of "· Then

91

i

k 2:

where

With lemrr.a 4.3 we are able to prove

Let the real matrix A of order n have uvl<-.J..acoJ.''""""J.."-4";

elements: a .. 3 a .. if 1 ,..; i < j ..:; n. Let this matrix be ll JJ

• If A is decomposed according to (4.2.3), then

k + t E

i==1

where H = t(A +AT), Z == t(A- AT) and n. (i 1,2, ••• ,k) is the J.

dimension of the diagonal block A .. of A. ll

We start by remarking that

k [[L = s 2(H) + 1:: fiZ .. [~ + 2:: 2:: (a if" ai)2 •

if'j lJ i==1

According to (4.2.2) (a 1;;- ai) 2 ..:; - 1 ) 2

• Hence the con-

tribution of the diagonal elements of 1 to 1~1~ is not greater .!';

than i=1

n . ( n. - 1 ) 2 • Thus J. J.

k nL1f~,..;s 2(H)+ L: IIZ.I[2

E+!-r2

2:: if'j J.j i=1

With lemma 4.3 we find

2': llz .. fl ~ ,..; 3 t? (A) + 6 ( 't*)- 2 (nz [~ + l'l( A) ) 2 ( S (H) + 6( A)) 2 • ( 4. 2. 9 )

if'j lJ

With '(4.2.8) and (4.2.9) we obtain (4.2.7). D

We conclude from (4.2. 7) that ffLI~ will be small relatively to

nAITE if the following requirements are fulfilled :

92

1. /1, (A)<< 11 A !lE

2. S(H) << \1 All., ]<,

3. T << 1\Al\E

4· S(H) + n(A) << -r*.

If the condition 1 and 2 are fulfilled, then obvious~y there exists

such a 't"-partition that the requirements 3 and 4 are satisfied. A

suitable value of T is mentioned in the following 1 1

Corollary. Let -. :=- (s(H) +n (A))2 ITAII; • (4.2.10)

Using this value of ,; in the -r -partition of A, we find that the

Euclidean norm of L =- A - ~1 - G satisfies the inequality

ITLIIE EO ff (H)+ 3/1,2

(A)+ 611AII; {IIZIII.J+ n(A)f (s(M)+ n(A)) +

k 2 +tllA.liE{S(H)+n(A)} ~ n.(n.-1).

j=1 J J

Proof. From (4.2.10) we derive

(4.2.11)

C..*)-2 EO T- 2 = {s(M) +/I, (A) r 1 nAr~ t. (4.2.12)

From (4.2.7) and (4.2.12) we conclude that (4,2.11) holds. o

From (4.2.11) ,we conclude that if we take for 1: the value described

in (4.2.10), the perturbation Lis small relatively to A if

l1(A)<< IIAIIE (i.e. A almost normal) and S(H)<< IIAIIE (i.e. the sym­

metric part of A almost diagonal).

Analogously to the complex case, described in the preceding section,

the estimate of I!LIIE given in theorem 4.6 suggests an algorithm by

which an arbitrary real matrix may be transformed into almost block

diagonal form.

In the Erst stage of this algorithm the origir.al matrix is trans­

formed into an almost normal matrix by means of the norm-reducing

method. Finally, the almost normal matrix is transformed by Jacobi

iterations, in order to diagonalize the symmetri.c part of the al­

most normal matrix, Let the resulting matrix be

A ( 1 ) = M( 1 ) + G ( 1 ) + 1 ( 1 ) • ( 4. 2.13)

In tcis formula we consider 1(1 ) as a perturbation which can be

estimated b,y means of (4.2.1). M( 1 )+ G(1 ) is a block diagonal nor­

mal matrix, each diagonal block M\ 1) + G ~ 1) consisting of a multi­

ple M\1.) = ex. I of the n. x n. ~it ma~~ix and a skew symmetric .JJ J n. J J .

matrix G~ 1). J JJ

If a diagonal block has dimension~ 2, its eigenvalues can be

readily computed. If all diagonal blocks of A(1 ) have dimensions

~ 2_, M( 1 ) + G(1 ) is a so-called Murnaghan canonical form.

If fo; some diagonal block A~ 1.) n. > 2 and the skew-s;ymmetric part ( ) JJ J .

G. 1. of that block is not almost zero, the situation is less favor­JJ

able. It is obvious that all real parts of the eigenvalues of the

blocks M~ t_) + G ~ t_) are ex. ( j = 1 , 2, ••• , k). From oonsidera tions sim-JJ JJ J

ilar to those which led to theorem 4. 5 it follows that this sit-

uation can occur only if the original matrix has three or more

eigenvalues with almost the same real part and among them at least

one complex conjugate pair. It is a well-known fact that a:nY skew­

symmetric matrix can be orthogonally transformed into its Murnaghan

canonical form (see theorem 0.3). However, for d.imension > 2 at

this moment no general algorithm is available to achieve this

transformation with the aid of a sequence of Jacobi rotations.

In this case we advise a transfer to complex arithmetic and use of

the algorithm for complex matrices described in the preceding sec­

tion.

4. 3. The real diagonalizing representative of7Jkrn(Al

In section 4.2 we considered the Jacobi process applied to the sym­

metric part of an almost no:r.mal matrix. In fact, this process ge­

nerates a sequence of real matrices which converges to Murnaghan 1s

canonical form, unless with a pair of complex conjugate eigen­

values A.. + iu. the matrix A has still other eigenvalues with tbe J - ... J

94

same real part A·• B~t the Jaoobi process applied to the ~JW~etric J

part H .. + of a real matrix A may be considered from an-

other point of view. As a matter of fact, a Jacobi transformation

with Q.£ minimizes not only S(Q~ H Q.£ ) but also the departure of m T m a

diagonal form of Q£mA Q~m' as is stated in

Lemma 4. 4. Let A be a real matrix and Q£m an orthogonal shear.

Then Qtm minimizes S(Q~mA Q£m) if and only if Qtm minimizes

s( QtmH Qtm) •

Proof. It is easy to d.emonstrate that

s2(QT HQ ) + s2(Q:m lm .t:m "'

T 82 (QT H Q ) + g2 (A-

2A ) •

.t:m lm 0

Remark. It is well known that Qtm minimizes S(Q~~ Qtm) if and

only if the (.t:,m)-restriction of Q~mH Qtm is diagonal.

Lemma 4.4, in conjunction with theorem 4.6, suggests which element

of the class ?n1m(A) of unimodular optimal norm-reducing shears is

suitable for a numerical norm-reducing process, if we want to com­

bine diagonalization and norm-reduction. It seems reaso:r..able to

select from~. (A) the element Tn that diagonalizes the (t,m) -,vm /Jffi

restriction of the symmetric part of A. For this T tm lntm(A)holds,

as we have seen in lemma 4·4

S(T.-1A T. ) = min S(P"m-

1A P.m) •

..vm "'m P E m, (A) "" "" tm tm

This unimodular norm-reducing shear Ttm will be called the

diagonalizing representative of nttm(A).

The diagonalizing representative Ttm of nz.t:m(A) can be obtained

from the upper-triangular representative Btm of m.,.t:m(A) by multi­

plying this shear with an appropriate orthogonal shear Qtm :

95

Let A' := T;:~1 A T.em· Then Qtm is determined from the relation

a lm + a~ 0 . (4.3.2) The (.e,m)-restriction of B:1A B0 is

.Aim Aim

a.emY- amjy-1

+(a u-amm) z )

-1 amm+ zy am.£

where y and z are computed with the algorithm of theorem 1.5. Let the (t,m)-restriction of Q.£m be

(

co. s cp - sin cp) Q£m

sJ.n cp cos cp

Using (1.2.4) and (4.3.2) we find the angle of rotation cp of Q£m

from

where w == - amtx + a tmY + (a.er amm)z.

The same diagonalizing shear T£m may also be constructed as the

product of the lower triangular representative 1£m of ~£m(A) and

an orthogonal shear R.em: T.em = L.em R.em· Then

(

)-'- 0 J (cos 4 ~ sin 4) ZX

2 X SJ.n (j; COS (j;

The rotation parameter q; of R£m has to be determined by the equa­

tion (4.3.2).

Similarly to (4.3.4) we find for the rotation parameter q; of the

orthogonal factor Rtm

(1-z 2)a + i'a - xz(a - a ) tan 2~ = ------~~~m~--~mt~_,~~£~~--~mm==­

x(a~£- amm)+ 2za£m

- X!N + 2a £m

x(au- amm)+ 2za£m ' ~ < \jJ..;; n/4

where w = - am£x + a£my + (at£- amm)z.

In the next chapter we shall explain that for reasons of numerical

stability it is advisable to derive the diagonalizing representa­

tive To of ~o (A) in the case of x..;; y from B with (4.3.4), kill kill £m

and in the case x > y from 1 in accordance with (4.3.5) and £m

(4.3.6).

4. 4. The complex diagonalizing representative ofn"Ztm(Al

For a complex matrix, too, it makes sense to perform simultaneous­

ly the normalization process (described in chapter two) and a

Jacobi-like diagonalization process. ThereTor we have to select

an appropriate element from 7Jt,£m (A), the class of unimodular op­

timal norm-reducing shears. In contrast with the analogous real

problem the unitary shear which diagonlizes the (£,m)-restriction

of the Hermitean part differs from the unitary shear that r;:ini­

mizes the departure of diagonal form.

We propose to select, as has been suggesced by Goldstine and Hor­

witz for normal matrices [10], the transformation shear T E:7n (A) £m £ID

in such a way that the departure of diagonal form is minimized~.e.

S(T:1A To ) = min S(P:1A P ).

kill kill p E: m (A) km £m £m £m

The shear T£ru will be called the diagonalizing representative of

m t~(A). In this section we describe the construction of the diagonalizing

representative of ~tm(A).

97

As .we have seen in theorem 2.2

r: Cla! 12 + ja! [2 + ra'.[ 2 + [a'.f 2

) = ccx + PY + 2Re(yz). • .n U J.m .EJ. mJ. J."'"' ,m

Hence this sum is independent of the choice of the representative

from the class of row congruent matrices with Euclidean parameters

(x,y,z). Consequently, for the determination ofT" E m (A) for "'m ;;m

which (4.4.1) holds, we may confine ourselves to minimizing "-1 ....._ A

S(P..em A P,em) with ?n,em(A). Let A be a pre-treated matrix,i.e.

am£ = 0 and let (x,y,z) be the Euclidean parameters of the row

cong!'l<ent shears of ""in ,em (A).

The representative of ~..em(A) will be considered as

the product of the upper-triangular rep~esentative B of ~ (A) ,em ..em and an appropriate unitary shear Q. • Hence, T := B. Q •

"'m ..em "'m :em The (..e ,m)-restriction of B..em is

) Let the (..e,m)-restriction of ~m be

Q = ( cos cp -e -:ie sin cp)

:em i8 e sin cp cos cp

Now the nni tary factor Q..em has to be

(4.4.1). This means that we have to

S(Qf B:1

A B0

Q0 ) = min

obtained from relation

select Q..em in such a way that

S(P* B-1 A P0 m) • .Em .em "' "'m "'m ,m "'m p E u

J:m J:m

We easily find that the (.e,m)-restriction of B;~ A BJ:m equals

( ao.U wamm) (4.4.4)

where w := y a..em + z(a;;;;- amm). (4.4.5) Then the (.e,m)-restriction of A' := Q* B-1 A B Q is :em J:m .em .em

98

a.£.£- VSlll <p 1. -ie . 2

A' = .£m (

. 2 --zve sJ..n2 cp + wco s ;:p )

(4.4.6) 1. iS .

""2Ve Sln2cp 2iS . 2 - we s1n ;:p

• 2 1. iS ·n2 amm+ vs1n ~ - ~e Sl ;:p ,

where v := au- amm. (4.4.7) A I

If f(cp,e) := 82 (A.£m) , we have

f(<p,S) = l'-ivei8sin2cp - <pl 2 + j'-ive-i8 sin2cp + wcos2 cp[ 2

= i[vi 2 sin2 2cp + lwl 2 (cos4(jl + sin4

<p)- !Re(vwei8 )sin4q>

The parameters <p and e of Q.£m are those for which f(cp,8) is a min­

imum. The minimum value of f proves to be ij'wf. Hence we have

Let A be a pre-treated matrix, (x,y,z) the Euclid­

ean parameters of~ (A), and T0 the diagonalizing representative ,em ,.,m of ~,em(A).Let further A' = Ti~ A T,em, v = a,e,e- amm and

w a,emY + (a,e,e - amm)z. "I

Then S2 (A,em) = ijwj 2• (4.4.8)

The shear T,em is the product of the upper-triangular representative

B,em of ~,em(A) and the unitary shear defined in (4.4.3) : = B ,em Q,em .For the parameters cp 8 of the unitary shear Q.em

holds

<p = 0 if w = o, (jl n/4

s o e = o if \) = o, w ~ o, (4.4-9)

-tan2cp ie

: ( 0 < cp < n/ 4) , e I:~ I if wv ~ 0.

" ' non-diagonal elements of A,em holds alm = i wand For the

a~ - iw if v = o, while a~ = - iW'v2/f vl.-2 if v ~ o.

We would remark that the pre-treatment of the

causing the (t,m)-restriction of the

..~.~-'-!S-'-lta...t. matrix,

matrix to have

upper form, renders the construction of the diagonal-

99

representative T.em of 1?'1-.e m(A) much :nore

described by Goldstine [10].

than that

In the next chapter we shall explain tt~t it is advisable to con­

st~ot the diagonalizing representative Ttm of ~.em(A) in the

case x >y from the lower triangular representative Ltm of ~tm(A~

We then makP. use of the following matrix

(

{x

" B • tm z

{x

L.em

Hence T ,em = B tm Qtm 1 tm R tm Qtm , where Qtm and R ,em are unitary

shears.

Unlike the algorithm described in theorem 4.2., the Goldstine­

Horwitz algorithm for the diagonalization of normal matrices,based

on minimizing the departure of diagonal form, does not guarantee

convergence to diagonal form. Voyevodin and [32] have con-

structed a class of non-diagonal normal matrices for which the

Goldstine-Horwitz algorithm becomes stationary before diagonal

form is reached. For matrices of this class holds for each pivot-

(,e,m)

laPm12 + 1am,el2 = lti2(Atm).

Hence we cannot expect that for all matrices A the sequence {~},

which is constructed with the representatives of

nv1

~ (Ak_1

) and converges to normality, also converges to k' k

diagonal form.

100

CHAPTER 5

NUMERICAL STABILITY AND THE NORM-REDUCING PROCESS

5. 0. Introduction

It goes without saying that each author in the field of er-

ror analysis has to acknowledge the impact of the enormous amount

of results obtained by Wilkinson in a series of papers culminating

in the detailed expositions entitled "Rounding Errors in Algebraic

Processes" [34] and nThe Algebraic Eigenvalue Problem" [35] • In

the latter book Wilkinson gives general error analyses which apply

to almost all the numerical methods for the eigenvalue problem that

are investigated in that book~ These analyses enable Wilkinson to

assess the numerical stability of several algorithms.

For our norm-reducing algorithm it is required to ascertain that

the numerical error associated with the actual computation of an

approximation of a similarity transformation, is bounded in some

appropriate sense.

In [35] Wilkinson has given a general err6r analysis of eigenvalue

techniques based on similarity transformations.

Let A A , A , A , •.. , ~ be the successive transforms of 0 0 ' 1 2 --k

A computed with inexact arithmetic. Owing to the rounding errors 0

ih the similarity transformations, A (p = 1,2, ••• ,k)is onzyalmost p

similar to A0

• The resulting matrix ~ , obtained after k steps ,

can be considered to be a perturbed exact transform of A 0

Ak = Pk-1 Ao pk + G(k) (5.0.1)

Wilkinson shows that 'it also may be advisable to consider~ as an

exact transform of the original matrix with some perturbation :

~= (5.0.2)

Wilkinson ([34], page 125) explains that "Bounds for G(k)are like­

to be usefui a posteriori when we have computed the eigensystans

of ~ (at least approximately) and have estimates for the conditions

101

of ·its eigenvalues. A priori bounds for F(k), on the other hand,

enable us to asses the algorithm" (viz., with respect to its numer'­

ical stability).

Useful a priori error bounds for F(k) and G(k) have been found for

quite large classes of transformations based on the use of unitary

transformations. "The great numerical stability of algorithms in­

volving unitary transformations springs from the fact that , apart

from rounding errors, the 2-norm and Euclidean norm are invariant,

so that there can be no progressive increase in the size of

the elements in the successive transformed matrices. This is im­

portant because the current rounding errors at any stage are essen­

tially proportional to the size of the elements of the transformed

matrix' ([34], page 162). Therefore, our almost diagonalization(by

unitary shear transformations) of an almost normal matrix can be

performed in a numerically stable way.

For non-unitar,y transformations the situation is less favourable;

useful a priori bo1mds for F(k) and G(k) do not exist when A is a

general matrix. Wilkinson remarks that"···· when we use a trans­

formation matrix with large elements, the rounding errors made in

the transformation are equivalent to large perturbations in the

original matrix'' •

As we have seen in the chapters one and two, the elements of opti­

mal norm..: reducing shears may be very large. Hence it is not a priori

certain that our norm-reducing algorithm will be numericallystable.

In the sequel of this chapter, we shall prove a "local stability"

result, namely, that it is possible to perform the transformation

with Tk in the k-th step of the norm-reducing process in such a

way that - -1 c ) ~ = Tk ~-1 + Fk-1 Tk + Gk '

where both IJFk_1 !1E/II~-1 1lE and HGk!IE/II~IIE are small{irrespective

of the condition number of Tk).

102

On the basis of this result we would venture the following

tative argument, from which a global result would follow.

Let ak_1 be the of Ik_1, the spectrum of ~-1 + hence also of - Gk' and ak the spectrum of Ik. Then (dependent

on the condition of the eigenprcblem of ~-1 ), ok~ differs little

from ak_1 iff 11Fk_111/lf\:_1 [[ is small, and (dependent on the condi­

tion of the eigenproblem of Ik) ok differs little from ak-t iff

ftGkll/1!\:l! is small. Further we remark that, since we are normal­

izing, the condition of the k-th eigenproblem improves as k in­

creases (at least in a global sense)o Therefore, we feel that our

local result described above explains, essentially,the a

observed of our norm-reducing algorithm.

The possibility of performing the transformation by Tk in a stable

way can be sketched in the following manner.

As we have seen, the decrease of the departure of normality a

nonn-reducing shear transformation with pivot-pair Ce,m) does not

depend on the choice of a representative Ttm from the class ~m(A)

used in the transformation

A'

So it makes sense to try to find an element S"m of ~ (A) for ..v ,em

which the floating-point computation of A1 is numerically stable in

the backward sense, i.e.

A' st~1 (A +F) s..em (5.0.5)

with I!F!IE/ I!AI[Esmall. Elements from ~ ..em(A) with this

will be called stable representatives of ~tm(A). In the sectiam

4.3 and 4·4 we have argued, however, that it is advisable to use

the diagonalizing representative T..em ·Of »v..em(A). But· since the

elements of nttm(A) are row congruent, there exists a shear

U..em so that Ttm Stm Utm • Thus the computation can be performed

in a stable way as follows

-1 A11 ::o S.8m A S.8m , A 1 (5.0.6)

103

Then for

A"

and forA'

Hence

A'

we have, conformally to (5.0.5)

-1 s-tm (A + F) stm

U -:1

A" TJ + G tm tm '

(5.0.7)

where 11 FHEis small relatively to !1 1\I!Eand ff q!Eis small relatively

to I[Ai !lg

In order to determine the stable representatives of ~tm(A) we

first investigate the characteristics of the rounding errors.made

in matrix multiplications (sections 5.1 and 5.2). In section 5.3

the res~lts of this general error analysis will be applied to shear

similarity transformations. In section 5·4 we shall shOw that ei-·

ther the upper triangular or the lower triangular representative.

of JnJ £ m(A) is a stable representative of this class and that this

is independent of the largeness of the Euclidean parameters of

m 1 m(A). Finally in section 5.5 it will be shown that the diago­

nalizing representative TA; m of J/ttm(A) itself is a stable repre­

sentative if

Hence in this case (which occurs if A is almost diagonal and a ' 11

and a are well separated) the decomposition mentioned in {5.0.6) mm -1

is not necessar.y in order to compute Ttm A Ttm in a ·stable way •

5. L Input and output perturbations related to rounding errors

5.1.0. Following Wilkinson ([34], pag, 4) we use the symbol ft to

indicate that the computed results are obtained with floating-point

arithmetic.

104

If the floating-point representation of a number has a binaxy t­

mantissa, then we assume that the rounding errors in the sunr

mation and multiplication are such that

f,e(a+b)

fi(a*b)

(a+b) (1+e ) 1

(a*b) (1 )

(5.1.1)

(5.1.2)

For floating SLilllmation and multiplication of two cor:tplex numbers

we can analogous description of the rounding errors, using the

properties (5.1.1) and (5.1.2).

Let z 1

Then

a + i b , z 2

c + i d.

f1(z +z ) = fi(a+c)+ i fi(b+d) 1 2

(a+c)(1+o )+ i(b+d)(1+o ), 1 2

-t with ~ 2 • i 1,2.

'I'hus for the modulus of the error we find

Jn(z + z)- (z + z )[ 2 = (a+c) 2 o 2 + (b+d)o 2 ~2-2t Jz + J2 • 1 2' 1 2 1 2 1

So there exists a complex number e , Je J~2-t, for which holds 1 1

Le(z + z ) 1 2

(z + z ) (1 + e ). 1 2 1

(5.1.3)

E ' 2

z 1

we can derive that there exists a complex

<2 -t+ 3/2 (1+2-t-1) for which holds

z z (1 +e ) • 1 2 2'

number

So with cor:tplex arithmetic the number of significant bits is essen­

tially one and a half less than that for real arithmetic.

5.1.1. The given d~scription of the rounding errors for real and

complex floating-point arithmetic enables us to give error bounds

for floating-point matrix computations. We state the following re­

sults wi tr~ou t proof (Wi lkinson [ 34], page 8 3) •

Let fi(A,x) denote the floating-point product of the matrix A and

the vector x.

Then for

e := fi(A,x) - Ax

105

we have

(5.1.6)

The elements of the vector jej and the matrix /A/ are the moduli

of the elements of e and A respectively. For real single-length

arithmetic • -t ( ) n =n2 5.1.7

whereas for complex arithmetic

n ~ n 2-t+3/2 (5.1.8)

Here and in the sequel we shall use the symbol ~ with the follovling

meaning. If a ~ b, then

a b (1 + 0(2-t)) , (t '-+ oo),

Similarly we use the symbol ~ in the following sense: if a ~ b

then there exists a number b*, with b* ~ b, for which holds that

a .-; b*.

These symbols enable us to avoid second order terms which tend to

obscure the fundamentally simple results. With the quantity n• de­

fined in (5.1.7) and (5.1.8), for real and complex arithmetic res­

pectively, we are able to estimate the rounding errors in the

floating-point product of matrices.

Let ft(A,B) denote the floating-point product of the matrices A

and B. Then for

E := .te (A, B) - AB

we have

(5.1.10) In the rest of this chapter we consider floating-point computatiom

having the characteristics given in this subsection.

5.1.2. We now investigate the multiplication of a non-singular ma­

trix A and a vector x:

(5.1.11)

106

e := fi(A,x) - Ax. (5.1.12)

Let f and g be any vectors

A f + g = e, (5.1.13)

then we can write

U(A,x) = A(x +f) +g. (5.1.14)

The vector fin (5.1.13) will be called the input or per-

.:::..:.;;_:....;;.;..;;. or forward perturbation corres-

ponding to the error e in this process. The

ward) error analysis (Wilkinson [;5]) corresponds to the

tion of the errors for which the output (respectively input) per­

turbation is taken zero.

Definition 5.1. p(A,x) is a stability bound for the

point computation of Ax if there exist vectors f and g for which

holds

and

llgfl2

.,.; 1') p(A,x) !1Axlf2

,

and which satisfy (5.1.13).

(5.1.15)

(5.1.16)

We now consider several ways in which the error e may be distrib-

uted over a forward part g and a backward part f in accordance

with the rule e = A

5.1.3. -1 f =A e. Then (5.1.6) implies

I fl .,.; I A -1

1 I el .,.; 1'J I A -1

1 I AI I xl •

Thus

llf!l .,.; ") C 0 (A) llxll , 2 N 2

where

c1 (A) := IIIA-1

1 ]AI 1!2

(5.1.17)

(5.1.18)

(5.1.19)

107

The quantity c£(A) will be called the left-condition number of A.

(5.1.18) shows that for all x, c,e(A) is a stability bound.for the

computation of Ax.

It is important to observe that c,t(A) is invariant relatively to

row-scaling of A. In fact, for non-singular diagonal matrices, D,

holds

JA-1

1 IAI = jA-1

D-1

1 IDAJ.

Hence

C n (A) ~ inf C ( I DA j) ' x D diagonal 2

(5 .1 • 20)

where c (A) = llAII IIA-11! 2 2 2

5.1.4. Forward error analysis •. We now take f = 0, thus g e.

Then (5.1.6) implies

I gj ~ 1J I Aj I xl ~ 11 I AI I A -11 I Axj • (5 .1 • 21)

Thus

11 gll ~ 11 c (A) 11 Axll , 2 r 2

(5 .1.

where

c c A) = = 111 AI 1 A-1

1 n • r 2 (5.1.23)

The quantity cr(A) will be called the right-condition number of A.

(5.1.22) shows that for all x, c (A) is a stability bound for the r computation of Ax.

c (A) is invariant relatively to column-scaling of A, since for r

non-singular diagonal matrices, D, holds

Hence

c (A) ~ inf· c (jADj). r 2 D diagonal

Mixed forward and backward error analysis. We shall dennn-

strate tr~t there is a non-singular matrix A and a vector x for

r:hich all stability bounds p(A,x) are very large. That means that

despite an opt.imal distribution of the error e in a backward part

f and a forward part g, these backward and forward perturbations

1 08

are not small relatively to x and Ax respectively.

Let

where f and g are such that

Af + g e U(A,x)- Ax).

(5.1.25)

evidently, M is a stability bound and, moreover, for all sla.­

bounds p(A,x) holds

then

'I.' he

M..: p(A,x).

if p0

(A,x) is the infimum of all stability bounds p(A,x) ,

M.;;; p .;;; M. 0

~~~r~r1g~)~n multipliers we find easily

(llxll 2 A AT + IIAxll 2 I)-1 e TJ -2

2 2

2 lf(llxll 2 A AT + I[Axl 2 I)-1 I!

2 2 2

-2 T)

(5.1.26)

(5.1.27)

(5.1.28)

in this estimate of M holds if and only if e is an

of A corresponding to the smallest, eigenvalue

nA-1!1 - 2 of A 2

On the basis of this result it is easy to construct for each cor­

arithmetic a non-singular matrix A and a vector x

so that all

of Ax are of

for the floating-point computations

• Therefore the matrix A and the vector x ha:ve

to the following conditions:

(i) c1 (A) and cr(A) is large;

(ii). xis such that 11Axll /l!xll is near IIA-1

11 -1

; 2 2 2

(iii) e is to be near an eigenvector of A AT corresponding to the

smallest ' viz. IIA - 1 n -2 ' of A AT. 2

109

Let us now consider a correctly rounding floating-point arithmetic

with a mantissa of t bits.

A possible choice of A and x is

( 1-2-k 1+2-k) ' A=

1+2-k

·ll!here k := (t+3) : 2.

Then we find

by

( 1 +2-k \ X= -k) -1-2

Ax = 2-k,..l ( 1 +2-k) (-~ ) , e = 2-.2k+l ( ~~) We observe that, indeed, e is an eige~vector of A AT, correspond­

ing to tr.e ei.genvalue 2-2k+2 of A AT. With (5 .1 .28) vm find . _2 -2 -2k-1 ( -k) -2 • -1 -k-t

lVl = Y) 2 1 +2 • Thus M = 11 2 • Hence it folJ.ows

from (5.1.26) that for all stability bounds p(A,x) holds

( ) . -1 -k-1 • -t p A,x, :;;;. 17 2 • Since 'f) = 2.2 we find

( ) • t-k-2 t ! 2 -3 p A,x :;;;. 2 = 2 • •

5.1.6. ~.~atrix factorization. In the algorithm for the corr.puta-

tion of Ax to be examined in this subsection we suppose to have at

our disposal two matrices, tl:.e exact product UT of these matrices

equals A. The algorithm consists of two steps in which the factors

T and U of the factorization are used consecutively

b := f£ (T,x) c := ft (u, b).

Let e := b - Tx, g := c - Ub.

Then, i~ conformity with (5.1.6)

For the ultimate result of the process we have

c = Ub + g = U T (x + T-1e) +g.

(5.1.29)

(5.1.30)

(5.1.31)

. -1 . We now consider f := T e as a backward perturbation ar~d the error

110

g, generated in the second step of the process as a forward pert~

bation. Thus

c = A (x + f) + g.

From (5.1.31) we find for f

I fl = j T-1ej ~ I T-11 I ej ~ 1J [T-11 JTI lx I· Thus

llfll ~ TJ c n(T) llxll • (5.1.32) 2 h 2

For the forward perturbation g we find

I gl ~ 11 /UI jb I = TJ j·Jj jT(x + f) I ~ 1'J ju I lu-11 jA (x + f) 1-

Thus l!g!l2 ~ n, cr(u) !fA(x + f)ff

2, or

ngn ,;; n c (u) IJAxH • 2 r 2

We conclude from (5.1.32) and (5.1.33) that

max (c _/T) , cju)) J.

(5.1.33)

(5.1.34)

is a stability bound fer the computation of Ax vri th the algorithm

(5.1.29). Obviously, if both c;;(A) and cr(A) are very large, it

makes sense to try to find a factorization of A = UT such that the

stability bound mentioned in (5.1.34), is not largeo In the next

subsection we shall mention a recipe for the factorization so that

for each A: max (c_/T), cr(U)) ~n.

5.1.7. Principal factorization. For each matrix A the princ~

factorization

A U 11 V

exists. In (5.1.35) U and V are unitary and 11 is a diagonal matrb4

The diagonal elements of 11 are the principal values of A.

Let T := 11 V. If we have at our disposal the matrices U and T,then

it is possible to perform the algorithm (5.1.29) for the computa­

tion of Ax.

111

For the factors U and T we have

and, since the left condition munber is ii:variant relatively to

::cow-scaling, also

) = c_e(v) .;;; n.

:ience it follows from (5.1.34) that n is a stability bound for

the comp1~tation of Ax if the principal faCtorization is used.

We observe that this factorization would only be practicable for

n .;;; 2. In section 5.4 we shall mention for n = 2 an even more sim­

ple fao~crization.

5. 2. Error analysis of similarity transformations

5.2.0. We will now (;Onsider algorithms for the computation of tne

similarity transformation A1 = P-1A P of A with P. Let be the

r-?sul t of the algorithm by which P-1 is computed. Let

, f,e(A,P))

and

If F and G be any matrices satisfying

P-1 F P + G = E

then Yre can write

A' = P-1 (A + F} P +G.

(5.2.1)

(5.2.2)

The mtrix F in (5.2.3) will be called the input or baclnvard per­

turbatio:!1, the matrix G the output or forward perturbation corre­

sponding to the error E generated in this process.

The backward (forward) error analysis corresponds to that descrip­

tion of the errors for which the output(input )perturoat:Lon is taken

zero.

11

Def:ini tion 5. 2. p (A, p) is a stahli ty bound for the floating -

point computation of P-1A P if there exjst matrices F and.G for

which holds

and

11 E.,.; 0 p (A,

and which satisfy

IIP-1 A Pll E

(5.2.;;).

(5.2.6)

In the algorithms to be considered we distinguish two parts. In tbe

first part a numerical approximation P-1 of P-1 is ccmputed,in the

last an approximation of p=1 A P. This resu.l t is called A 1 •

5.2.1. Backward error anal;ysis. -1 We now take G = 0 and F=P E P •

We may write

-1 -1 ( ) P = P I + K • 1

K equals the residual matrix P P-1 - I, Let 1

IlK 11 .,.; k Tl· 1 E 1

(5.2.7)

(5.2.8)

If P-1 equals the correctly rounded P-

1, then l!K 11 .,.;TJk n-ic (P) ,

1E r where k is independent of the condition of P.

The computation of an approximation P-1A P, in which P-1 is used,

is performed in two steps:

Let

Then

where

B := fi(A,P), C := fi(P-I, B).

E1

:= B - AP, -1 E

2 := C - P B.

= P-1 [ (I+ K1

) (A+ E1P-1) + P

= P-1 (A +F) P ,

F (E + p ) + K (A + E I-I). 1 1 1

(5.2.10)

J p (5.2.11)

(5.2.12)

113

From (5.1.10) we conclude

IE11,;;; T) IAI !PI ' !E2/ ,;;; T) /P-1

1 /B/. (5.2.14)

These bounds for !E1 r and IE2 r' together with the bound for IIK1 liE

given in (5.2.8) will be used in order to estimate IIFIIE'

From (5.2.14) we derive

lE, P-1

/ ,;;; !E11 !P-1

I ,;;; T) !AI !PI ,P-1

/ •

Hence, since 11 !PI rl'-11 liE ..; nt H IPI IP-

11112 = ntcr(P)'

(5.2.15)

With (5 .2, 7) we derive from (5.2.14)

1 E 1 ,;;; T) c1 + TJ) IP-1

1 1 r + K 1 r AI 1 PI ~ T) r P-1

r 1 AI r P/ • 2 1

Hence

I p E2 p-1, ;; T) IPI jP-11 I Aj I Pj I p-1, •

Thus

(5.2.16)

For theEuclidean norm of the term K1

(A+ E1

P-1) in (5.2.1-3) we

find

HK1 (A + E1 P-1

)liE ,;;; IIK1 liE (IIAIIE + T) !!All 11 I PI [ P-1

1 liE)

~ T) k1 IIAIIE •

Combination of (5.2.15), (5.2.16) and (5.2.17) gives 1 1 .

IIFIIE '-' T) [n2 c (P) (1 + n2 c (P)) + k] IIA!IE' r r 1

We conclude from (5.2.18) that for all matrices A l

o (P) ( 1 + rf o (P)) + k r r 1

(5.2.18)

(5.2.19)

is a stability bound for the numerical computation of P-1 A P.

5.2.2. Forward error analysis. In this case F = 0 and G = E.

We may write

(5.2.20)

114

-1 K2 .equals the residual matrix P P - I. Let

[K HE k D. (5.2.21) 2 2

--=1 -1 ( If P equals the correctly rounded P , then I!K2

11E.,. l]k c)!

where k is independent of the condition of P. In this subsection

we use the notation introduced in (5.2.9) and (5.2.10). Then we

may write

-1 ( -1) -1 C P A + E P P + E = P A P + G, 1 2

(5.2.22) where

G = P-1E + E + K (P-1A P + P-1 E ) • 1 2 2 1

(5.2.23)

In order to estimate !!Gl!E' we derive upperbounds for liE , -1 -1 -1 . -1

trP '~lE and rnp A P + P E1

jfE relatively to [[P A P!IE.We start

to relate these norms to IIC'~E'

Since I .,.. fJ [P-11 IB[ and B = (P-

1)-1 (c-E ) ~ )we have 2

[E),;; n LP-1

1 [PI [c-E2

[ ~ 11 I I [PI [c[. Hence

~

lfE2

UE.,.. 11 112 c£(P) ilC!!E • (5.2,24)

For [P-1[ [E [ we obtain from (5.2.14)

1

I' P-1 r 1 E

1 r ... 7) r P-

1 r 1 AI !PI. Since A (P-1 )-1 (C-E ) P-1 - E , we derive

2 1

,p-1riE r ... l],p-1 rpr·rcr+K )-111 " +7JIP-111E IIP-111PI 1 . 2 1

~7J[P-1 r-1Pr rei [P-1

[ :P[.

Thus

(5.2.25)

Finally estimating I[P-1 A P + P-1 we find with (5.2.24) :

-1 -1 '( ~p A ? + P E 11'!= = r. I + !I 1 .]:!;

115

(5.2.24), (5.2.25) and (5.2.26) we find the following upper­

bouEd of !IGIIE : . ],_

frGI!E ~ YJ [n2 c .t(P) (1 + c .t(P)) + ] !!CifE • (5.2.27)

Since C P-1

A P + G, IIP-1 A P!!E ~ I!Cl!E • Hence from (5.2.27) fol­

lows that

(5.2.28)

We conclude from (5.2.28) that

(5.2.29)

i:s' a stability bound for. the numerical computation of P-1 A P •

We would remark that in the estimate (5.2.18), of the hypothetical

backward perturbation F, the right condition number cr(P) appears,

whereas the left condition number c.t(P) occurs in the estimate

(5.2.28) of the hypothetical forward perturbation G.

This difference in the estimates of !fFifE and IIGI!E suggest the type

of error analysis to be chosen in order to obtain a sharp estimate

of the errors generated in the algorithm for the numerical com­

putation of P-1 A P.If c..e(P) << cr(P), then it is advisable to use

backward error analysis; if cr(P) << c /P), then forward error

analysis is preferable. In the next subsection we describe an al­

gorithm for t~e computation of P-1 A P which is of interest if

both c..e(?) and cr(P) are large.

5.2.3. Mixed error ana~sis by means of factorization. In the al­

gorithm for the comp~tation of P-1A P, which will be considered in

this subsection, we suppose that P is the exact product of given

matrices T and U :

P T U.

-1 -1 Uoreover we suppose to have at our disposal T and U , being

the approximate inverses of T and U respectively.

116

-1 The algorithm for the computation of P A P now consists of four

steps :

A := fe (A,T) 1

A := f-l(A , 3 2

E :=A -AT '

A := f£ (T-1

, A ) 2 1

A := f;;(u-1, A).

4 3 .

E := A - A 1 1 2 2 1

--=-1 (5.3.32) := A -A u E :=A A E ' - u

3 3 2 4 4 3 Let

T-1 -1 (I + K ) :;""1 (I ) -1 = T ' = + u 1

We shall assign the rounding errors of the computation of A to 2

the backward perturbation F, whereas the errors of the second half

of the process will be assigned to the forward perturbation G.

Then

A := P-1 (A +F) P + G, 4

where, similarly to (5.2.13) and (5.2. ),

F = (E + )T-1 + K (A + E T-1 ) 1 1 1

and

G = u-1 + E + K (u-\ U+ u-1 4 2 2

) . From ( 5.1 .1 0) we know

1 E1 r .s; rr 1 AI 1 Tl. , IT-11 lA I

1

[ E31 .s; 1'] IA21 I u I ' I I I A I .• 3

Let further

(5.2. )

The estimate (5.2.18) may be applied here to the backward pertur­

bation F of (5.2.35). Then we obtain • .l.. .l..

!!FilE .s; 11 [ ].12 c (T)(1 + n2 c (T) )+ k] UAIIE • (5.2.38)

r r 1

The estimate (5.2.28) may be applied here to the forward pertur~

117

bation G of (5.2.36). Then we obtain • _;!,_

!IGIIE,;;;; 1 [ n2 c_e(u)(1 + (u))+ k] 2

(5 .2. 39)

We conclud.e from (5.2. ) and (5.2.39) that _;!,_ -~ _;!,_

ma:x:{n2 c (T)(1 +n2 c (T))+k, n2 c (u)(1 + r r 1 £

(u))+ k } 2.

is a stability bound for the computation of P-1A P with

rithm (5.2.31).

(5.2.40) algo-

If both c;; (P) and cr(P) are very large, then it makes sense to try

to ffnd such factorizations of P = TU that the stability bound

given in (5.2.40) is not

As in subsection 5.1.7 we now apply the result obtained in(5.2.40)

to the particular case that the decomposition P = TU is derived

from the principal factorization P = V /1. U, thus T = V/\.

Then c0(U),;;;; nand c (T) = c (v)...::n.

"' r r Hence, in this case

max (n3/2(1 + n3/2)+ k1' n3/2(1 + n3/2)+ k2)

is a stability bound for the computation of P-1A P with the algo­

rithm (5.2.31 ).

It is appropriate to remark here that the principal factorization

is rather impracticable unless n,;;;; 2. In the next section, however,

we apply the above results to shear similarity transformations.

The error bounds (5.2.38) and (5.2.39) will appear to be of prac­

tical value for these transformations.

5. 3. The general error analysis applied to shear transformation

5.3.0. In this section we apply the results of the preceding sec-

tions to the similarity transformations by unimodular shears :

-1 A 1 :=T AT•m' . ..em "'

118

We assume to have at our disposal the Jacobi-parameters p, q, r

and s of t~e shear Ttm. In order to simplify the notations we de­

fine P := T • .£m Thus

p = c :J Although generally ps-qr is not exactly equal to one, we take

p:-1:= (s -q) -r P

(ps-qr)P-1 (5.3.3)

as an approximation of P-1 • With this (..e,m)-restriction of T..em-1we

compute an approximation of T.£m-1A T..em in two

C := f.£(T.£m-1, B).

The errors -1

in the computation of this approximation of

T. AT. may be ..vm ,;r,m assigned to a hypothetical backward pe:!:'-

turbation F; we may also assign them

bation G :

to a forward pertur-

5.3.1. We start to examine the errors made in the computation of

the affected elements of A1 , i.e., the elements in the .£-th and

m-th column and row but not

The affected elements of A1

"' . ' belong1ng to A..em·

on the i-th row, where i I .£, i I m, T are obtained from the product of a row vector, say, and the

matrix P :

(a:I_..e a! ) "' (au a. ) (p q) T J.m J.m r. P i l..e,m. J. r s

(5.3.6)

Let T Tp + T T T) P. f.£(r. ,P) = e. (r. + J. J. J.

119

T T That means, e. and f. are the l l

bation, respectively, caused by

Then, since I ei Tl ..,; TJ I r/1 I Pi ,

the backward perturbation f.T :

forvrard and the backward pertur­

the numeric~l computation of r.TP. l

we find, similarly to (5.1.18),for

l

!I T T I! ..,; 11 c (P)Ifr. 11

2 r l 2

and, similarly

T lie. 'I ..,; r; l 2

to (5.1.22), for the forward perturbation

c ( P) 11 r. T P 1! • £ l 2

(5.3.8)

T

We now consider the affected elements on the i-th column, i I t ,

U m. These elements of A1 are obtained by multiplying the matrix -1 P and a column vector, c. say, Thus

l

(ali) ( s -q) (aJ:i)

a 1 • -r p a . ml m1

P-1 _j ci , i i ,e ,m.

Let

(c. +h.). l. J..

(5.3.11)

That means, g. and h. are the forward and the backward perturba-1 l -

tion, respectively, caused by the numerical computation of P-1 c .• l

'l'hen, since 1 gil .,. n 1 F-11 1 oil, we find directly from c5.1.1s)for

the backward perturbation h. l

[lh.!l ..,;YJ· c0

(P-1) l!c.l! ~YJ c (P) Rc.t[ , · 12 "" 12 r 12

and from (5.1.22) for the forward perturbation

( -1) • ( ) -1 ITg.!l ... Y) 0 p 11 c.\! = YJCn p liP c.rr • 12 r 12 "" 12

As we see from (5.3.8) and (5.3.12), the estimate of the error

which is entirely'assigned to a backward perturbation depends on

cr(P), whereas ct(P) determines the estimate of the error which

is entirely assigned to a forward perturbation,

In this subsection we consider the errors made in the nu-

merical computation of the (t,m)-restriction of At, For the back­

ward(forward) error analysis we investigate and estimate the back-

120

ward perturbation F n (forward perturbation G0 ) corresponding to hill hill

the errors generated in the numerical computation of the (£,m) -

restriction of A1 • These perturbations Ftm and G}i;m are the (t,m) -

restrictions of the perturbations F and G respective~ which are

introduced in (5.3.5). Consequently

Let -1 -1

liP P - IIIE ..,;;; k1

TJ and liP P - IITE < k2

TJ • j,_

from (5.3.2) and (5.3.3) we have 22 jps-qr-1 I jps-qr-11..,;;; k TJ,

2

<kTJ 1

and

The results of the preceding section now may be used in order to

obtain estimates for lrF1miTE and 'llG1m!fE.

Evidently in this case we have to take n 2.

From (5.2.18) we conclude

I!F.£miTE.;,; 2tcr(P) (1 + 2tc)P)) + k] ITAJ;mlfE (5.3.15)

5.3.3. We shall now take together the results obtained with the

backward and forward ana~sis applied to the affected elements and

to the (t,m)-restriction. For the backward perturbation F holds

where f.T and h. are defined according to (5.3.7) and (5•3.11) re-~ ~

spectively. From (5.3.8), (5.3.12) and (5.3.15) follows 1 1 A

OFIIE2 ;;n2 [22 c (P)(1 + 22 c (P))+ k ] 2 11An llE2 +TJ2 c 2(P)(a + ~), ' r r 1 hm r

wher~, with the notation of (2.2.10),

121

Co:nsequently A • .1.. 1

IIFIIE,..; 1'J[22 cr(P)(1 + 22 cr( )+ k1

] (IIA.£m!IE2+ a + ~ • (5.3.17)

Similarly we find for the farward perturbation G

I!GIIE 2 = !IG.-em!IE2+ . L: (!lei Tn 2+ I! gill 2)' l;(e,m 2 2

where e. T and g. are defined according to (5.3. 7) and (5.3.11) l l

spectively. From (5.3.9), (5.3.13) and (5.3.16) we may easily

where

ex'+ r' = ._J~ (!a'i..el2+ !a'iml2 + la'..eil2 + [a'mi12 ). lr..v,m

re-

de-

The interpretation of the estimates (5.3.17) and (5.3.18) for the

backward and forward perturbation is very simple. With the back­

ward analysis namely we find that

(5.3.19)

is a stability bound for the computation of T.£m-1A T.£m and with

the forward analysis we find that also :1 1

22 c.£(P) (1 + 22 c.£(P)) + k2

is a stability bound for the computation of T.£m-1A Ttm'

5.3.4. Left and right condition numbers of unimodular shears.

Let P be the (t,m)-restriction of Ttm' where T.£m is a unimodular

shear with pivot-pair (t,m) and Jacobi-parameters p, q, r and s.

In order to compute c.£(T.£m) = ct(P), we have to find the largest

eigenvalue of [P-11 I Pj (j P-11 I PI)*. Thus ct(P) is the square

root of the largest zero or· the polynomial

122

Hence

o_/(P) = (lpsj-jqrl f+ 2(!prl+jqsj )2 +

+ 2(jprj+jqsj) [(/PI2+1qj2)(irl2+fsl2)

to (2.1.4)

x = I PI 2 +j ql 2 ' y =/ r/2 +j sj2 ' z = pr + qs • ..l.

Consequently jprl + jqsl ~ (xy) 2 • Since we assume that Pis the

(t,m)-restriotion of the unimodular T , tm

xy - I zj2 = I ps - qrl2 = 1 •

Moreover (lpsl-lqrj )2 ~ lps-qrj 2 = 1. With these estimates

we have

C 2 (T 0 ) ~ 1 + 4xy = 5 + 4/ zl 2

• ..e ..vm (5.3.21)

Analogously we find

(5.3.22)

where

;;) (5.3.23)

y'

We would remark that (5.3.21) an upper bound of c (T ) t ..em

which is the same for all matrices row-congruent to T • Since ..em the value of z may be very if T Em (A), the forward ..em ..em error analysis is not appropriate to describe the errors in the

norm-reducing process.

This result contrasts with the conclusion that can be drawn from

the upper bound (5.3.22) for c (T ). In the next section we ~11 r ..em

show that for each A, t and m the class ~..em(A) contains a shear

S 0 for which holds c (sn }<3. This shows that generally it is .-vm r ""m

necessary to assign the -1

putation of Stm A Stm to a

r..e(s..e~l , r..e(A, s..em)) =

backward perturbation F

-1 Stm (A +F) S..em'

123

5.3.5. We now compute the right condition number of the tri-

representatives and 1£m of mi,m(A) (see section 2.1).

For the upper triangular B£m we have 1

ey~t) :-1) ~ (y:~ c (Yo :t). B £m

2 y

With (5.3.22) we find

c/ (B..em) ,.;; 5 + 41 zj2 jy2. (5.3.24)

For the lower triangular representative 1,em of 7lV (A) tm we have

( :) c_, ") c X~ A

1£m - -t zx ·X zx 1

Hence, with (5.3.22), we find

c/(1£m),.;; 5 + 4jzj 2 /~. (5.3.25)

By direct calculation we can derive that

c (B,e ) r m = ~zj2y-2- + I zj/y (5.3.26)

and

cr(B ,em) f1+T~!2x-2 + jz!/x. (5.3.27)

.!. .!. Since I z! = (:xy-1) 2 < (:xy) 2

.,;;; t(x+y).,;;; max (x,y), we have

min (I z/ x~\ I zjy- 1) < 1.

Consequently

min ( c ( B ) , c ( 1 ) ) < 1 + 2i < 2. 42 • r ,em r .em

The triangular shear in ~ (A) with smallest right condition ,em number will be called the stable representative of ~ (A). This ,em representative will be denoted by S • So we have found that if . . _em X~ y then s B ' if X > y then s n 1 and c (s ) <1 +{2. _em _em Nm _em r _em

124

5. 4. A numerically stable transformation by the diagonalizing representative of m,tmiA)

For the real norm-reducing algorithm we have found in

section 4.3 that the diagonalizing representative of IJl., (A) ,em may be obtained from the stable representative S by multiplying

tm this shear with an appropriate orthogonal shear V ,em :

- c; V - ~£m ,em •

If x.,.; y then the rotational parameter of V,em can be computed

from formula (4.3.4); if x > y then the angle of rotation can be

obtained from (4.3.6). these matrices S and V

0 , the transformed matrix can be

,em "'m computed with the following algorithm :

A := f£(A,S,e ) A ( -1 A ) := f£ s ' 1 m 2 ,em 1 (5.4.2)

u (v£~1 A := f£(A ,V ) A ' A ) . 3 2 Alm 4 3

A A

Let s := s ,em ' V := V £m and

rr (5·4·3)

We now the rounding errors made in the computation of

to tl':e backward perturbation F, whereas the errors generated in

the second half of the process are assigned to the forward per­

turbation G :

A4

T£~1 (A +F) T£m + G.

Similarly to (5.3.17) and (5.3.18) we find

+a +

and

I' . .,.

E (Vo ) )+ k ] (I! A'\ fE +a'+ p'

.vm 2 "'m

For the triangular shear S£m we easily find that l!ss-1- I!IE.,.;

125

As· c (s. )<10; 1 + {2, we have : r .vm

Since the transformation with an orthogonal shear Vtm is well

known to be a numerically stable process, algorithm 5.4.2 can be

performed in a stable way if in the factorization T,€m = Stm Vtm

the stable representative Sim of m,im(A) is used.

5.4.2. For the complex norm-reducing algorithm the situation is

a little more complicated. As a matter of fact, in that case the

stable representative stm is derived from the pre-treated matrix

U,e~1A Uim for which holds (u,e;1A Utm)m£ = 0, Hence the diagonal-

i norm-reduction on the pivot-p~ir (t,m) corresponds to the

transformation

A' (Utm 8tm Vim)-1

A (U,em 8tm V.Em),

where V,e is an appropriate unitary shear (described in section m 1

4.4) which "diagonalizes" (U,em S.Em)- A (UJlm Stm).

Since cr(S,em) < 1 + {2 and [S,em s~1 - Ill2

<10;! rJ' A• can be com-

puted in a stable way by the following algorithm

A ( -1 , f ,e(A, u,em)) := f£ u 1 ,em

A ( -1 , f,e(A, S )) (5.4-4) := f,e s 2 ,em 1 ,em

A I= fi(V - 1 , f,e(A , vim)). 3 £m 2

We shall now show that, independent of the condition of S£m' A1

can be computed in a stable way with an algorithm in which only

two numeri.oal similarity transformations are performed.

Let PJlm: UJlm Sim and Q£m = f,e(Uim' S,em). Hence

126

-1 An approximation of T£m A T£m can be obtained in the following

way

A 1

A := f£(A , V ) , 2 2 £m

A := f£( 2

A ) 1

A := f£(V - 1 , A ) 4 ,em 3

General~ A is only approximately equal to T - 1 A T • We shall 4 ,em ,em

be content, however, if the transformation by V is computed ,em in a stable way.

Let x, y and z be the Euclidean parameters of S E 711., (u - 1 A U \ ,em ,em ,em ,em:' which can be computed with the algorithm described in theorem 2.5.

We now assume x~ y (if x > y, a similar argument can be used).

Then

m (

cos q>

e -ie sincp

p

Since jzjy-1 < 1, we easily find from (5.3.22)

cr2(p,em).;;; 5 +4[zj2Y-2< 9·

Finally,

tiple of

we shall show that !!Q Q ;; ;em ;em

-t+ .!...

11(~ 2.2 2). We have to take

Il' is a moderate mul-12

into account that

(i) the matrix U$m which is actually used for the pre-treatment,

is not exactly unitary;

the determinant of the matrix S..em actually used is not exact­

ly equal to unity.

The Jacobi-parameters of the matrix U£m' actually used for the

computation of ~m' will be denoted by (c,-s,s,c). As we see, c -ie . is an approximation of cos (p, s an approximation of e s~n cp.

The Jacobi-parameters of Sn will be denoted by 1 1 1 .~m

(y -z, zy -z, o, ;i2 ( 1 + e: 1

) ) , where I e: 1

f .;;; i 11 •

With these notations .we find for the (..e,m)- restriction of ~m:

127

1 1

c c zy -2 c 1 +e: ) - sy2 c 1 +e: ) c 1 +e: ) ) c 1 +e:- ) ) 3 1 4 5

1 1

(szy-2 (1+e: )+ cy2 (1+e: )(1+e: ))(1+e:) , 7 1 8 9

where I e: .1 ..;; 1J t . Hence 1.

det(Q)= cszy-1 [(1+e: )(1+e: )(1+e: )-(1+e: )(1+e: )(1+e: )] + 2 7 9 3 5 6

+ c 1 +e: ) c c2 c 1 +e: ) c 1 +e: ) c 1 +e: ) + I s 12 c 1 +e: ) c 1 +e: H 1 +e: ) J

1 2 8 9 4 56

~ 61 cszY-1Ie:+ (c2 +1 sl 2 )(1+4e:),

where I e: I ,.; t11.

(5.4.7)

-1 Consequently, the upper bound k

111_ of ITQ Q - I!!E will be a ~oder-

ate multiple of 1J if c2 + lsl 2 is very close to unity, so that

the shear u£m is almost unitary.

From the foregoing we conclude that in the complex case algorithm

(5.4.6) enables us to perform in a numerically stable way thenorm­

reducing shear transformation by the diagonalizing shear T£m if,

as in the real case, the stable representative is used in the fac­

torization of T • £m

5. 5. Diqgonal dominance and shear transformations

-1 5.5.0. In this section we shall show that for computing T0

AT "'m £m

in a stable way'· the factorization of the diagonalizing represen-

tative of nL£m(A) is not necessary whenever the off-diagonal ele­

ments of A in the £-th and m-throws and columns are small relati­

vely to a -a .In numerical experiments we have observed that norm-££ mm

reduction with the diagonalizing representative of ~k'~ (~_1 )

(k ~ 0,1, ••••• ) generates a sequence {~}which converges to

a diagonal form. Consequently from some stage in the norm-reducing

process the condition mentioned above was fulfilled, provided the

eigenvalues of A(= A ) were well separated. The above assertion 0

128

then implies that from that -1 T A_ T can be computffi ,ek'~--k-1 £k,~

in a stable way one shear similarity transformation.

.5.1. First we consider the real optimal norm-reducing trans­

formation by the diagonalizing representative T £m of JJt.,t:m(A). Let

x,,y and z be the real Euclidean parameters of the shears in ~m (A).

We assume that y ~ x (if x > y then the argument is similar). In

this subsection we make use of the notations introduced in (1.2.10)

(1.2.11), (1.2.13) and (1.3.6). As we have seen in section 4 • .3,the

Jacobi-parameters of the diagonalizing representative of ~m(A)are

where

tan2<p yw + 2.:\ vy - 2\z

zy ) c:: -sin<p)

COS(j)

1t 1t --<<p"""-· 4 4

For the estimate of c (T.) we need, as we see from (5.3.22), an r .-vm expression for (pq+rs) 2

• With (5.5.1) and (5.5.2) we find

( \2 [(cos<p + z sin<p)(-sin<p + z cos<p) + y cosm]2 pq+rs; = y sin<p T

[ (y2 + z2- 1) sin2::p + 2zcos2p]2

4l [g - A. - t w(x+y) ]

2

E +vi

We derive from (5 • .3.22) and (5.5.3)

c (T n ) 2 ,.- 5 + 4 [ !l - A - ivv(x+y) )2 r x,m E + w2

Since x+y, i.e. the square of the ~~clidean norm of the (~,m)­

rest.riction of T £m' may be very large, cr(T tm) which is the essen­

tial factor in the given bound for the backward perturbation (see

(5.3.17)), may also be very large. We shall now prove that the

129

norm-reducing and diagonalizing character of T 0 prevents this dan-. ~

ger on condition that

2 2 ( 2 2 2 2 ) 1 ( )2 ( atm+ am£+ ·..1: ai£+ aim+ a£i+ ami "'"16 a;,[ anun • 5.5.5)

~r"'•m

In terms of the notation of chapter 1 this condition reads

,2 + 112 (.( ... 1 " ,.. +CX+l-' 16 2 \) .

Lemma 5.1. If

then

(1 - 2k2 )v2 "'"E .;;;(1 + 2k

2)

lrl ...:~2 i and

obvious.

2 \) '

As concerns y, since F ~ cxp- y2 ~ 0 (see lemma 1.2.i), we have ~

frr"'" (cx~Y2 .;;;! (ex+~).;;;! k2 v2

•

Finally D2 ~ (cx!l-f3A.-rv?.;;; (cx2 +~ 2 +/)(r..2 +tl+v2 ).;;; (a:+p) 2 (1+k2)}.

1. .d.. Thus I D I .;;; ( cx+p) ( 1+k2

)'"2 [ v j .;;; ( 1 +k2

) 2

1 v 13 • 0

Lemma 5.2. Let w = -A:x + llY + vz, where x, y and z are the

Euclidean parameters of the class ~£m(A) of unimodular optimal

norm-reducing shears on the pivot-pair (£,m). If

then I wz I .;;; 4 kZiv I • (5.5.10) . 2 2

Proof. If v = o, then A: + /J. + a + ~ = 0 and consequently

m ,em(A) = "";,m • In this case we have z = 0, hence (5.5.10)is satisfied. We now

130

assume v I 0. Then E ~ (1-2~ )i > 0, as follows from leinma 5.1.

From (1.3.10) and (1.3. 20) follows

I wzl = 1- vD + y(g-E)[ (E-p)2

where p is the negative root of the quartic equation

1.

Since F:;;. 0, certainly \D[/Ip[ < (E-p)(E-2p)""'2, Using this result

in (5.5.11) we find, with lemma 5.1

I wz I < [ vD I I + \r I I < I vI (E-p)(E-2p)2 (E-2p)2

< 4 k2 [vi• 0

Lemma 5.3. Let x, y and z be the Euclidean parameters of ~m(A),

where A is a real matrix. Let T£m be the diagonalizing representa­

tive of ~£m(A) and let T,em = S,em V£m' where S,em is the stable re­

pre"Sentative of m,em(A) and Vtm is an orthogonal shear on the pivot­

pair (,e,m). If <:p is the angle of rotation of V,em and if

A. 2 + ~ 2 + a + p < ~ v2 , 0 < k < !, then

I z Sill(p COBqll < 3k.

Proof, If v = 0 then z = 0 and consequently (5.5.12) holds.

We now assume v I 0. We consider only the case in which y ~ x.Then

the (i,m)-restriction of Ttm is given by (5.5.1) and (5.5.2).Hence

1 z sill(p eo S<:p I < i I z tan2<p I = i I ~ _ + 2 \~/? I· Since y ~ x implies \z[ < y, we find with lemma 5.2

I ~ 4 k2 iv\ + 2 k\vl _ 1 + 2 k lz sin<p cos<:p < 2 ( 1 _ 2k)lv[ - 1 _ 2 k k < 3 k. 0

These lemmas will be used in the proof of

Theorem 5.1. Let o.: + p + A.2 + ~2 < v2 , 0,.;;;; k <

If T,em is the diagonalizing representative of ~£m(A), described in

131

section 4.3, then the right condition number c (T 0 ) satisfies the r "'m

If v = O, then a+~+ A2 + ~2 = 0; so Tf,m I and(5.5.13)

holds in this case.

We now assume vI 0. Then E ~ (1-2 k2 )v2 > 0, as follows from lem­

ma 5.1.

From (5.5.4), (1.3.10), (1.3.18) and (1.3.19) we derive

c 2 (T ) "' 5 + 4 (u- A - Hx + y)wf r tm E+~

"' 5 + 4 [~-t _ A + D (~-A )D +l(a+§) (E -p) ]2(E + w2 )-1,

p( p - E)

where p is the negative root of the

(p- E) 2 (p2 -F) + D2 (2p -E) = 0.

Thus j:)j/lel "'(E- p )(E- 2p)-i. Consequently

2 (T ) "' 5 + 4 [ I u-.}1 + I u-AIJDI + i(a + s)(E1 - p) J2 im E2 E2 (E - p )(E - 2p)2

~ ~ ~

Since Ill - AI ...;; 22 (A2 + /)2 ...;; 21i'kl vi, we find with lemma 5.1

Finally we estimate the norm of the residual matrix T T - 1 - I. tm £m

This estimate gives a bound for the quantity k(see(5.3.14)),occur-1

ring with c (To ) in the upper bound (5.3.17) for the backward r ,.,m perturbation F (corresponding to the errors generated during the

:-::::; ) computation or T,em A T,em •

If p, q, rand s are the Jaoobi-parameters of Tf,m' then s, -q, ~r

132

and pare the Jacobi-parameters of T$~ 1 and TtmT~~1 -I= (ps-qr-1)1.

The Jaco bi-parameters of T ~m are computed (by floating-point ari th­

metic) from the Jacobi-parameters of the stable representative

and from those of the orthogonal shear Vtm mentioned

in ( 5. 5 • 1 ) and ( 5 • 5 • 2) • 1 1 1

We assume that the Jacobi-parameters of s. are f~zy-~0 and?(1+e), hill 1

and that those of are c, -s, s and c (being the computed ap-

of and cos~). Then, according to (5.5.1)

(: J where I c: i I ,.; , 1 ,.; i ,.; 9. We find from (5.5.14) that

ps-qr = (1+<::1 ){c 2(1+~)(1 )(1+<::9)+ s2(1+e5)(1+e7)(1+e:8)} +

+ zsc(1 ) { (1 )(1+e::4)(1+e9)-(1+c:6)(1+e:7)(1+e::a)}.

It is now clear that lps-qr-11 will be a moderate multiple of 2-t

on condition that

(i) I z sin~ cos~ I is not

(ii) - 2 - 2 c + s - 1

-t a moderate multiple of 2

(iii) le'S zl is not than. I z siTI(p coscp I • Condition (i) is satisfied if the

longing to the i-th and m-th rows and columns are small relatively

to jaco- a I as we have seen in lemma 5.3. -V-" mm

Condition (ii) is fulfilled if c and s are computed in a stable

mariller ([35], page 276) from the for tan2cp which is de-

scr~bed in formula (5.5.2)

Condition (iii) is satisfied if the

parameter z is calculated with moderate care.

z of the Euclidean

133

We. conclude from theorem 5.1. and lemma 5.3 that the algorithm

A': ft(T ..e:i, ft(A,T £m)

for the computation of and approximation of T..e:1A T..em' where T..em

is the diagonali representative of ~m(A), can be performed

in a numerically stable way if

71.2 + + a + ~ .,.; If v2

, 0.,.; k.,.; :f. For in that case neither c (T. ) nor the upper bound of

:-::'1 r -vm ~T T"- - Ill /n are large, and consequently the backward pertur-, ..em "'m 2

bation F,estimated in (5.3.17), is small relatively to A.

If c (T. ) or lfTn Tn-1

- Ill /TJ is large (this is only possible r ,vill "'m »ill 2

if the off-diagonal elements ~f A in the ..e-th and m-th rows and

columns are not small relatively to a,e1

- amm)' then the danger of

numerical instability arises. In that case we have to reorganize

the computation in order to guarantee numerical stability. Then,

in fact, the factorization of T..em' described in 5.4.1 has to be

used.

5.5.2. Finally we consider the transformation by the diagonaliz­

ing representative T £m of the class ~m (A) of complex minimizing

shears on the pivot-pair (t,m). T,em is the product of three shears:

T,em = U,em S,em V..em •

The unitar.y shear U,e is used for the pre-treatment (annihilates

( ) , m -1 ) ' , -1

the m,,e -element of U,em A Utm • Let A = Utm A U,em• Stm is

the stable representative of "l'lv" (A') and V n is the uni tarJ shear ,vill -1 ! ,vm

that "diagonalizes" the matrix Stm A S,em'

In this subsection we use the notations introduced in chapter 2 :

(5.5.15)

a :=

y (a. 0 a:. -a". a.) ].,v J.ill ,v]. mJ.

,m

134

The corresponding functions of A1 = u.-1 A u.

_,vffi hiD (a~ = 0) are indi-

cated by A. 1 )

! I f I ! o,ll,v,a,f3 andy.

with (2.2.13)

' ' I f I ! 2 D := rx ll - y v , E := v I , F :

With these notations we formulate lemma 5.4. The proof of this

lemma is vexy similar to that of lemma 5.1.

Lemma 5.4. If

Let A1 be the pre-treated matrix, thus A.1

0.

(5.5.18)

then 1

( 1-2k2) I v 12 ~ E ~ ( 1 +2k2) I v 12 ' ill r I ~ 22k I vI ' I I 1 2 2 lr ~2k lvl and

1

IDI ~ (1+4k2)2klvl3 •

Analogously to lemma 5.2 we can prove

Lemma 5.5. Let x, y and z be the Euclidean parameters of ~£m(A') I I

and w ll y + v z.

if

then I A 1

2 + ill r 2

+ a + f3 ~ k 21 \) 1

2 ' 0 ~ k ~ i '

Jwl ~ 23/2

k 2 Jvl , jzjy-1 ~~ k, lwzj ~ 4 k 3 jvj and

I w I ( x+y) ~ 8 k 21 vI •

We are now able to formulate the complex analogue of theorem 5.1.

Let p, q, rand s be the Jacobi-parameters of Ttm' where

Tim = Utm Stm Vtm • We take

where (x,y,z) are the Euclidean parameters of ~fm(A').

Since S 0 v. is the "'m .,m

135

have, according to (4.4.3) and theorem 4.6

A

V X: m

(cos~

I \ ie . \ e SlUrp

-e ) j cos~

where tan2rp = jwj/jv'l , 0 ~ rp ~ n/4 and

w v'/lwv'l if w v' I o, e 0 ~ w v' = o.

Since c: (TX:m) .;;; 5 + 4 lpq + rsl 2 , we first compute lpq + rsl 2 •

From the unitarity of Utm follows that pq + rs only depends on the

Jacobi-parameters of SX:m VX:m' We can easily derive that

I - -, ~ I (coscp + z ei

9sinp)(z cosp - ei

9sincp) + Y i9 . 1

2 pq+rs ~ = y e SlnrpCOSrp

f{;y:2+1 zl2- 1}sin2p + 2Re(z ie) }2 e . co s2p • + 4 Im2(z eiel

4Y2

With the expressions for tan2rp and eie, given above, we find

Re2 {w w-1 ~(x+y) w- ~')} + ~ E + !wl2 y2

Using (5.5.18) and the lemmas 5.4 and 5.5 it is easy to prove

Theorem 5. 2. Let T~m be the diagonalizing representative of

11b" (A). If '"m

I A.j2 + I fll2 + a + ~ .;;; k21 v 12 ' 0 .;;; k ~ t, then c 2 (T ) .;;;. 5 + 1

r ,em

As in the real case we see that, despite the possible largeness of

x+y, the norm-reducing and the diagonalizing character of T.£m pre­

vent. c (Tn) to be very large, provided that the off-diagonal ele-r "'m

ments in the X:-th and m-th columns and rows of A are small rela-

tively to a.££- amm.

136

, we estimate the norm of the residual matrix T,emT~:- I.

This estimate will give us a bound for the quantity k (s.ee 1

(5.3.14)). With the bounds fork and c (T ) we are able to esti-1 r _em

mate the norm of the backward perturbation corresponding to the

errors made in the computation of an ofT - 1 A Tnm £m ,r,w

(see .3.17)). Let (p, q, r, s) be the Jaoobi-parameters of T_ern, These parameters

will be computed, in floating point from the Jaoobi-

simple, but tedious calculations we find -t be a moderate multiple of 2 if the conditions are

satisfied

(i) I c !2 + Is !2 - 1 and I 12 + I 12

- 1 are both a moderate 1 1 -t

multiple of 2 ;

(ii) ~~~y- 1 not large;

(iii) le s I (~+y) and I s I lzl not 2 2 2

Condition (i) is satisfied if Jacobi-paxameters of U0

and V "'m ,em

are computed in a stable way from (2.2.7b) and (4.4.9) respectively.

If a+~+ lit.l 2 + !11l 2 ..;k2 !vl 2, O..;k..;1z., then the conditions(ii)

and (iii) are fulfilled as follows from the lemmas 5.4 and 5.5.

I I ~~ I w! l·vrl-1• For sin~ cos~ ..;z tan2~

Hence !sin~ cos~! (x+y).,.; (x+y)!v1!-1 < 8 k2 and

!sin~ cos~! !z! .,.; 2{2 k3• if le s I (x+y) and

2 2 I c s I !z! are not essentially than I cos~ si~ I (x+y) and

2 2 ,~, 1"'1 -1 !c2

s2

1 z respectively and moreover z. y is not essentially

larger than !z! y-1, then the conditions (ii) and (iii) are satis­

fied.

We now summarize the results obtained in this subsection.

Let T be the _em representative of ~n (A). If c (Tn ) "'m r "'m

137

is not and the conditions (i), (ii) and (iii) are satisfied,

then we can -1 '

compute Ttm A Ttm in a numerically stable way with

the algorithm :

Ttm := ft(ft(Utm' Stm)' Vtm)

A1

i= ft('I'£~1 , ft(A, T,em) •

(5.5.19)

(5.5.20)

In (5.5.19) Utm is the unitary shear which A; the

upper triangular representative of~tm(u..e~1 A U,em) and V..em is such

that S,em V,em is the diagonalizing representative of ~m (U£~1 A u1m).

If c (T. ) is large or if the condition )or the condition ) r "'m

is not fulfilled·(this is only possible if the elements

of A in the ..e-th and m-th rows and columns are not small

to a,e,e- amm)' then the factorization of Ttm' mentioned in section

5.4, necessary in order to perform the transfor-

mation in a numerically stable way.

138

REFERENCES

1. Causey, R.L. 1958, Computing eigenvalues of non-Hermitean ma­

.trices by methods of Jacobi type, J.S.I.A.M, ~,172-181.

2. Dimsdale, B. 1958, The non-convergence of a characteristic

root method, J.S.I.A.M., ~' 23-25.

3. Durand, E. 1960, Solutions n'.lllleriques des equations alge­

briques,Tome II, Masson & Cie, VIII + 447 p.

4. Eberlein, P.J, 1962, A Jacobi-like Method for the Automatic

Computation of Eigenvalues and Eigenvectors of an Ar­

bitrary Matrix, J.S.I.A.M., lQ, 74-88.

5· Eberlein, P.J., 1965, On Measures of non-normality of Ma­

trices, Arn,Wath. Monthly, 72, 995-996.

6, Eberlein, P.J. and Boothroyd, John, 1968, Solution to the

eigenproblem by a norm-reducing Jacobi type method,

Nurn, Math., 11, 1-12.

7, Eberlein, P.J ., 1968, Solution to the Complex Eigenproblem

by a Norm-Reducing Jacobi type Method, unpublished,

8, Forsyth, G.E. and Henrici, P., 1960, The cyclic Jacobi method

for computing the principal values of a complex matrix,

Trans. Am. M.s., 2!• 1-23.

9. Goldstine, H.H., Murray, F,J, and von Neurnan.11., J., 1959, The Jacobi method for real symmetric matrices,J.Ass.

Comp.Mach., ~' 59-96.

10, Goldstine, H. H. and Horwi tz, L.P., 1959, A procedure for the

diagonalization of normal matrices, J.Ass.Comp. Mach.,

§_, 176-195.

11. Greenstadt, J., 1955, A method for finding roots of arbi­

t~:cy matrices, lVfath. Tab. and other aids Comput., 2_,

47-52.

1 39

12. Greenstadt, J., 1960, ~he determination of the characteris­

tic roots of a matrix by the Ja.cobi II'Tethod, l\IJa.themati-

ca.l Methods for Digital Computers

Wilf), Wiley, New York, 84-91.

ed. Ra.lston and

13. Gregory, R.T., 1953, Computing eigenvalues and eigenvectors

of a. symmetric matrix on the I.L.L.I.A.C ., Math. Tab.

and other aids comput., l• 215-220.

14. Henrici, P., 1962, Bounds for iterates, inverses, spectral

variation and fields of values of non-normal matrices,

Num. ::VIath., 4_, 24-40.

15. Hoff'man, A.J. and Wielandt, H.W., 1953, The variation of the

spectrum of a normal Duke Math.J., 20, 37-39·

16. Jacobi, C.G.J., 1846, Doer ein leichtes Verfahren die in der

Theorie der Secularstorungen vorkommenden Gleichungen

numerisch aufzulosen,Crelle's J., 30, 51-94.

17. Kempen, E.P.M.van, 1966, Cn the convergence of the classical

Jacobi method for real symmetric matrices with non­

distinct eigenva.lues, Num. Math., 2J 11-18.

18. Kempen, II.:?.M.van, 1966, On the quadratic convergence of the

serial Jacobi method, Num.Math., 2, 19-22.

19. Lotkin, M., 1956, Characteristic values of arbitrary matrice~

Quart. Appl. llfa.th., 14_, 267-275.

20. },'[arcus, M. and N'Jinc, H., 1964, A survey of Matrix Theory and

Matrix Inequalities, Allyn and Bacon, Boston, XVIII +

180 p.

21. IvJarcus, M. and Mine, H., 1965, Introduction to Linear Algebra,

The Macmillan Compa~, New York, X + 261 p.

22. Mirsky, L., 1958, On the minimization of Matrix Norms,A.m.Math.

Monthly, ~ 1 06-1 07.

140

23. Osborne, E.E., On preconditioning of matrices, J. Ass.Comp.

Mach., l• 338-345·

24. Pope, D.A. and Tompkins, c., 1957, Maximizing functions of ro­

tation-experiments concerning speed of diagonalisation

of symmetric matrices using Jacobi 1s method, J.Ass.Comp.

Mach., !, 459-466.

25. Ruhe, A., 1967, On the quadratic convergence of the Jacobi

method for normal matrices, B.I.T., l' 305-313.

26, Ruhe, A., 1968, On the quadratic convergence of a generalisa­

tion of the Jacobi method to arbitrary matrices, B.I.T.,

§_, 21 0-232.

27. Ruhe, A., 1969, The norm of a matrix after a similarity'Trans­

fomation, B.I.T., 2_, 53-58.

28. Rutishauser, H., 1964, Une methode pour le calcul des valeurs

propres des matrices non symetriques, C.R. Acad.Sc.,

t.259, 2758.

29. Rutishauser, H., 1966, The Jacobi-Method for Real Symmetric

Matrices, Num. Math., 2_, 1-10.

30. Smith, R.A., 1967, The condition number of the matrix eigen­

value problem, Num.Math., 1.Q_, 232-240.

31. Voyevodin, V .V., The solution of the complete problem of eigen­

values by a generalised method of rotations, Comp.Meth.

and Progr.III (Comp.Centre of Moscow Univ.) 89-105.

(Russian, transl. by A.Korlaar)

32. Voyevodin, V.V., An extension of the method of Jacobi, Comp.

Meth.and Progr. VIII(Comp.Centre of Moscow Univ.)216-228.

(Russian, transl. by A.Korlaar)

33 •. Wilkinson, J .H., 1962, Note on the quadratic convergence of

the cyclic Jacobi process, Num.Math., i• 296-300.

141

34.• Wilkinson, J .H., 1963, Rounding Errors in Algebraic Processes,

H.M.s.o., London, VI + 161 p.

35. Wilkinson, J.II., 1965, The Algebraic Eigenvalue Problem,

Clarendon Press, Oxford, XVIII + 662 p.

142

SAMENVATTING

In dit proefschrift wordt een algorithme beschreven voor de bere­

kening van de eigenwaarden van een matrix. De gebruikte methode is

van het Jacobi-type: in de algorithme wordt iteratief een rij van

matrices geconstrueerd en elke iteratie-stap is een gelijkvormig­

heidstransformatie met een shear (d.i. een matrix die slechts in

een 2 x 2 submatrix verschilt van de eenheidsmatrix).

In de eerste fase van de algorithme wordt de oorspronkelijke ma­

trix met niet-unitaire norm-reducerende shears getransformeerd in

een "bijna normale" matrix.

Als gevolg van de invariantie van de Euclidische norm van een ma­

trix onder unitaire transformatie brengt elke shear in een klasse

van z.g. rij-congruente shears dezelfde norm-reductie teweeg. Zo'n

klasse wordt bepaa.ld door ha.ar Euclidische parameters. De Eucli­

dische norm van de met een shear getransformeerde matrix is een

eenvoudige uitdrukking van deze parameters.Voor unimodulaire shear

transformaties is de functie, die het kwadraat van de Euclidische

norm van de getransformeerde matrix beschrijft in termen van de

Euclidische parameters van de transformatie shear, kwadratisch en

het defini tiegebied er van is een ble"d. van een hyperboloide.

In de hoofdstukken 1 en 2 wordt een methode beschreven ter bereke­

ning van de Euclidische parameters van de klasse van rij-congruen­

te unimodulaire optimaal norm-reducerende shears.Daartoe wordt be­

rekend waar zich op de hyperboloide het infimum bevindt van de bo­

ven vermelde kwadratische functie.

In hoofdstuk 3 wordt het effect van successief toegepaste norm­

reducerende shear transformaties onderzocht. Bij een welgekozen

pivot-strategie zal de rij van matrices, die ontstaat door bij

elke iteratie de norm optimaal te reduceren, convergeren naar nor­

maliteit, d.w.z. convergeren naar de klasse van normale matrices

met hetzelfde ~pectrum als de matrices uit de rij.

143

In de tweede fase van de algorith~e wordt de door norm-reductie

"bijna normale" matrix A met unitaire shears getransfor-t

meerd in een "bijna diagonaal11 matrix.

Allereerst wordt daartoe, met Jacobi de bijna normale

matrix A zodanig getransformeerd dat het Hermitische gedeelte van

de getransformeerde matrix A bijna 1

is. De matrix A 1

blijkt een bijna blok diagonale structuur te hebben. Van elk dia-

gonaal blok is het Hermitische gedeelte bijna een veelvoud van de

eenheidsmatrix. Daarom kan aangetoond warden dat bij de vervolgens

uit te voeren diagonalisatie van de scheef-Hermitische gedeelten

van deze blokken (met Jacobi rotaties), het bijna diagonaal karak­

ter van het Hermitische gedeelte niet verloren gaat.

De uiteindelijk verkregen matrix

deelte waarvan de norm van boven

heeft een niet diagonaal ge­

wordt door een continue

functie die nul is als A normaal is, het Hermitische gedeelte van

A een diagonaal matrix is en de scheef-Hermitische gedeelten van 1

de blokken van A ook diagonaal 2

Voor reele matrices is de matrix A bijna een kanonieke vorm van 1 .

l\furnaghan mi ts de matrix, indien er een paar complex geconjugeerde

eigenwaarden ~ + i v bestaat, geen andere eigenwaarden heeft met

reeel gedeelte ~·

In hoofdstuk '5 wordt de numerieke stabiliteit van het norm-reduce­

rende proces onderzocht. De resultaten dragen bij tot de

verklaring van de nauwkeurigheid van berekende eigenwaarden in nu­

merieke experimenten met procedures die gebaseerd waren op de in

dit proefschrift beschreven algorithms.

144

CURRICULUM VITAE

De schrijver van dit proefschrift werd geboren te Zoeterwoude op

16 september 193-1 •

In 1950 behaalde hij het diploma H.B.S.-b aan het Titus Brandsma

College te Oss. Vervolgens studeerde bij wis- en natuurkunde aan

de Rijksuniversitei t te Leiden, waar hij in 1961 het doctoraal .ex­

amen wiskunde met natuurkunde en mechanica behaalde.Tot september

1961 was leraar aan achtereenvolgens het St.Antonius College

te Gouda, de Gemeentelijke M.M.S en H.B.S. voor Meisjes te Leiden

en het R.K. Llfceum St. Bonaventura te Leiden.

Vanaf September 1961 is hij als wetenschappelijk medewerker ver­

bonden aan de Onderafdeling der Wiskunde van de Technische Hoge­

school te Eindhoven.

STELLINGEN

I Zij D, E en F de volgens (1.2.13) van dit proefschrift ge­

definieerde functies van een reele matrix A en de pivots (t,m). Dan

geldt voor iedere reele .unimodulaire shear T ,em met pivots (,e,m):

D(A; t,m) = D(T - 1 tm A Ttm; t,m) , E(A; t,m) = E(T£~1

A T,em; ,e,m)en

F(A; t,m) = F(T - 1 tm A Ttni; £,m).

II Zij bij een reele matrix A en een pivot-paar (t,m) a• ~· Y•

~. ~' v, e en a gedefinieerd als in hoofdstuk 1 van dit proefschrift.

Zij (x,y,z) de Euclidische parameters van een shear op het pivot­

paar (,e,m).

Zij <p := o:x + ~y + 2yz en w := -A.x + ~y + vz.

Dan is

en

inf S2 (T,e-1

AT,e)= inf (<p+W+a), T£mE t£m m m (x,y,z)Eq:

waarbij ttm' S en t!( zijn gedefinieerd volgens de defini ties 0. 7 ,0.8

en .1.3 van dit proefschrift.

III Als ~. + iv ., a. en h". (j = 1 ,2, ••• , n) de eigenwaarden J J J J

zijn van respectievelijk de matrices A, i(A +A*) en i(A - A*) ~n

tJ. (A) de afwijking van de normali tei t van A is, dan bestaan er per­

mutaties p en q z6 dat

n 2:: (~ • - a ( . ) f .;; tJ.

2 (A) j=1 J p J

IV Zij T een niet singuliere matrix.

Als voor iedere matrix A geldt IIT-1 A TilE ijA[[Edan is Teen veel­

voud van een unitaire matrix.

V Zij f : R ..,. R , f E C2 • n 1

Als grad f Min en slechts een nulpunt X E R heeft dan geldt o n f(x ) extreem impliceert f(x ) globaal extreem.

0 0

VI Zij voor p > 0. 00

n(log n) 1 + p

1 --:r:tp ) • n

S(p) := Z n=2

Dan bestaat lim S(p) en deze limiet is positief. plO

VII Rutisha.user, Schwarz en Stiefel gaan in hun beschouw:i.ng over

de onnauwkeurigheid in de numerieke oplossing van het stelsel verge­

lijkingen Ax = b ten onrechte uit van wat zij noemen de onnauwkeurig­

heid van de berekening van Ax - b voor veotoren x in de omgeving van

de oplossingsvector xt.

H.Rutisbauser, E,Stiefel, H.R.Schwarz, NUmerik symmetrischer

Matrizen, Stuttgart, 1968.

VIII De door Dekker in de formule van Newton gebruikte benadering

van de multipliciteit van een nulpunt bij eindige arithmetiek,

niet geschikt om de bepaling van een meervoudig nulpunt te versnel­

len. ~.J.Dekker, Newton-Laguerre iteration, Report MR 82,

Mathematisch Centrum, Amsterdam, 1966.

IX In het door Bonset geschetste projektonderwijs wordt onvol­

doende aandacht geschonken aan kennisoverdraoht en vaardigheidstrai­

ning. '· H.Bonset,"Nooit met je rug naar de klas!"

Amsterdam, 1969.

X Het verdient aanbeveling Westerlaken 1s advies inzake het be-

rijden van verkeerspleinen niet op te volgen.

H.M.W.Westerlaken, "Eisen voor rijvaardigheid"

Uitgave ANWB- KNAC - KNMV, Se dxuk.

XI Het is wenselijk de procedure ter beoordeling van weten-

schappelijke medewerkers aan Universiteiten en Hogescholen ex­

pliciet te formuleren.

Eindhoven, 2 december 1969 M.H.C. Paardekooper