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SA.E.R.E. R.7986

FORTPAU SUBRMOINES FOR, FltING ltMOLY1JCIAL ZEROS

by

K. Madsen* and J.K. Reid

ABSTRACT

In this report we present subroutines for finding all the rootsof a polynomial and bounds on their errors. To find the zeros we usethe algorithm of Madsen (1973) and to find error bounds we use the workof Peters and Wilkinson (1971) with some significant modifications. Boththe real and complex cases are treated.

* Institute for Numerical Analysis, Technical University of' Denmark,2800 Lyngby, Denmark.

A

Computer Science and Systems Division,A.E.R.E. Harwell, Didcot,Oxford, England. JU:,c•

February, 1975 By

HL.75/1172 (C.13)

Di ct

(i) --

B . ~~~slri mrp1~i nsc aea

CONTENTS

Page No.

1. Introduction 1

2. Finding the roots in the complex case 2

3. Error estimation 6

4. Changes to the algorithms in the real case 13

5. Description of Fortran subroutines 15

6. Test results and comparisons with other methods 18

7. Appendix. Specification sheets and listings 23

8. References. 65

ISBN--0-70-580215-9

(ii)

j1. Introduction

In this report we consider the problem of finding all the zeros of

the polynomial

f(z) = ao + a z +...+a nzn (1.1)

and estimating error bounds for them. To find the set of zeros we use

the algorithm of Madsen (1973), which we have found to compare favourabI

with other algorithms. For error estimation we apply Rouche's theorem

as recommended by Peters and Wilkinson (1971), but with some difference

in detail. Both of these algorithms are a little simpler in the case

where the polynomial has complex coefficients, so we describe the

algorithms for this case in sections 2 and 3 and the modifications for

real case in section 4.

We believe that our code is in accord with the ANSI standard, havir

checked it with Bell Telephone Laboratories' Fortran verifier (Ryder,

1973) and run our test programs with array subscript checking. We

describe this code in section 5 and in an appendix give specification

sheets, and listings (produced by the Bell Laboratories'verifier, so the

include cross references). For the Harwell subroutine library we have

made a small number of changes in order to shorten the argument lists,

at the expense of a departure from ANSI standard. We have decided aga7

including a single-length version in the library because the IBM 370/1E

has a very short single-length word (6 hexadecimal digits, so that

numbers just greater than unity are held to about I part in 10 6) but hE

double-length hardware which executes very little slower than the singi-

length hardware. Some results obtained with the library subroutines,

together with comparisons with other algorithms are given in section 6.

Our code (listed in section 7) contains comments suitable for machine

processing which detail the changes needed for the single-length and

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Harwell li.'rary versions.

We would like to acknowledge the help of M.J.D. Powell in carefully

checking a draft of ýhis reoort and making several valuable suggestions.

2. Finding the routs in the complex case

We use the algoritrim of Madsen (1973) to find the .-oot of minimal

or near-minimal modulus and then use forward deflation to construct a

polynomial of degree n-1 whose roots are the remaining roots of the

original polynomial. The process is repeated until approximations to

all the roots have been found. Wilkinson (1963) has shown that forward

deflation is stable provided a large root is not accepted before a much

smaller one. Our algorithm does not guarantee that the moduli of the

roots are strictly increasing but our experience has always been that they

are found in roughly increasing order. A version with the composite

deflation of Peters and Wilkinson (1971), which should be stable no matter

in which order the zeros are found, was tried but it did not give more

accurate results. In any case it is not clear how to apply the

composite deflation when two complex conjugate roots of a real polynomial

have been found so that deflation by a real quadratic factor is wanted.

It remains necessary to describe Madsen's (1973) algorithm in detail

and we will consider its application to the original polynomial. The

general strategy of the algorithm is that. given an iterate zk, a

tentative step dzk is found and the next iterate z k+l is taken at the best

point (in the sense of If(z)I) encountered in a short search of values

on the line through zk and zk+dzk. Because the search may sometimes

yield no better value than that at zk we may sometimes have z k+l =Zk and.

in this case ensure that the next tentative step is shorter and in a

different direction. The inclusion of searches ensures rapid convergence

to multiple roots and reliable convergence when difficulties are

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ATTRIBUTES: Col 1: C if in COMMON

F encountered. Such a search is, however, wasteful if we are so near a simple

root that Newton's iteration is reliable and fast. We have therefore

devised a *test (given by inequality (2.7)) which normally ensures this.

While the algorithm performs searches we say it is in stage 1; otherwise

it is in stage 2, performing straightforward Newton iteration. It begins in

stage 1, which we now describe.

The tentative step dzk is found with the help of stored values of

Zk, f(zk)O f'(zk),zk-l and the previous tentative step dZkl. If the

last iteration was successful (zk i Zkkl) then the Newton correction

nk= f(zk)/f'(zk) (2.2)

is calculated and the next tentative step is taken as

k nk if InkI • 31zk-zk-lI (2.3a)dz 31zk-zk-lleie nk/Inki otherwise (2.3b)

where e is chosen (rather arbitrarily) as arctan(3/4). If the last

step was unsuccessful (zkNZk-l) then we take the tentative step to be

dzk = -• eie dzkl. (2.3c)

After a successful iteration we normally expect to want to take a Newton

step (2.3a), but we include the alternative (2.3b) because accidently

coming near to a stationary point of f(z) is likely to make the Newton step

ridiculously large. We include a change of direction in (2.3b) because

if a saddle point is being approached the direction nk may be a worse,

search direction than almost any other. After an unsuccessful iteration

we want to change the search direction to one likely to be successful

and reduce the step size; this leads to formula (2.3c). Its repeated

use is sure to yield a descent direction.

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F The algorithm used by PAO6BD to find the roots has already been

Once the tentative step has been found we test the inequality

!f(zk+dzk)I < 1f(zk)t . (2.4)

If this is satisfied then we calculate the numbers

ff(zk+p dzk)1, p=l,2,...,n (2.5)

continuing for as long as these are strictly decreasing. If inequality

(2.4) does not hold th;n we calculate the numbers

If(zk + dzk/2P)I, p=O,l,2, If(zk + jeie dzk)0 (2.6)

again continuing until the sequence ceases to decrease. In all cases we

take Zk+l to be the best point found. Note that if there is a true multiple

zero of multiplicity m or if we are a fair distance from a cluster of m

zeros then zk+mnk will be a very good estimate of the solution and will be

found by our search. In fact we get quadratic convergence to a multiple

zero. Note also that when an iteration fails following a search to the

end of sequence (2.6), the choice (2.3c) leads to a step likely to be in

a direction of decreasing jf(z)I.

To complete our description of stage 1 we need to specify starting

iterates. We take these to be

dz 0 (fO)/f'(O) if f'(o)=O1l otherwise (2.7)

z min (Ia Ill/k\ dz0zI = (ijmink>o IdZao

The iteration really starts from zI but we have to include z0 and dz0

because they are needed for choosing the tentative step dzI. This choice

of zI is used because its modulus is certainly less than that of any root

of f(z) and it is in the direction of steepest descent of If(z)I from the

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origin. It is therefore likely that we will converge to a root of near-

minimal modulus.

We do not make our main test for switching to stage 2 (straightforward

Newton iteration) until a stage 1 search has led to the choice

Zk+l=Z k+dz k This is obviously sensible and has the added virtue that

we should (correctly) avoid the switch when converging to a multiple root.

The test itself is based on the Kantorowitz theorem (see, for example,

Ostrowski (1966)) which states that if K. is the circle with centre

zk+nk and radius Inkl where nk is the Newton step (2.2) then the

conditions

f(zk) f'(Zk) A 0 (2.8)

21f(zk)I max If"(z)l ! If'(zk)l 2 (

zE:Ko

ensure the convergence of Newton's iteration starting from z k This

leads us to test the inequality*

2If(zk)jjf'(Zk- 1 ) - f'(zk)I 5 If'(zk)l 2 lZkl-Zkl (2.9)

Of course this is not equivalent to test (2.8) because we have replaced

max If"(z)l by a rather crude difference approximation but we

ZEo K

nevertheless expect it usually to predict correctly that straightforward

Newton iteration will be satisfactory. We check inequality (2.9) at every

step in stage 2 and switch back to stage 1 if it is violated. We also

check the inequality (2.4) and if this is violated return to stage 1

beginning by modifying the tentative step with formula (2.3c) as in stage 1.

We complete this section by describing our convergence criterion.

We terminate if a stage 1 search or a stage 2 iteration leads to a new

iterate Zk+l different from zk and yet such that the inequality

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IZk+l-ZkI < EZ:k+ll (2.10)

holds where e is the largest number such that to machine accuracy I+E = I.

We also terminate if the condition

If(zk+l)I = lf(zk)I • 16nlaolc (2.11)

holds. The expression 16nlaolE is a generous overestimate of the final

roundoff made in calculating f(z) at the root of smallest modulus and we

expect that such accuracy will be attainable. The normal convergence

pattern is that If(zk)I decreases until well below 16niao0 E and then

roundoff errors cause a new iterate Zk+l=Zk to be taken so that (2.11) is

satisfied. If such accuracy is unattainable then the step will decrease

steadily because of the application of (2.3c) until (2.10) is satisfied.

This combination of convergence criteria means that we are certain to

obtain a good solution and almost certain to obtain the best possible.

Furthermore this result is usually obtained with only one more iteration than

is necessary to get this good accuracy.

3. Error estimation

We seek to estimate error bounds for all the roots produced by the

algorithm of the previous section. We base our algorithm on ot f

Peters and Wilkinson (1971) and will follow their notation. The most

significant difference is that they look for non-overlapping discs each of

which contains precisely the same number of exact and approximate roots,

whereas we allow overlapping discs. This 's because it can happen that

one root is well determined although it is inside the best disc

obtainable for another (ill-determined) root. We therefore look for a

separate disc for each root and take its centre to be at the calculated root

itself. This also allows a very simple form of output to the user since

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all that is required is a radius for each root.

We suppose that we have approximate roots ai, i=l,2,...,n. Then

the polynomial

nP(z) = a n H (z-ai) (3.1)

i=l

should agree with the original polynomial (1.1), but because the roots

are not exact there will be an error

Q(z) = P(z) - f(z). (3.2)

Now Rouche's theorem states that if P(z) and Q(z) are analytic

functions in and on the closed curve C and the inequality

IQ(z)I < IP(z)I (3.3)

holds on C then P(z) and P(z)-Q(z) have the same number of zeros inside C.

We apply this here by looking for circles with centres ai, i=,2 ,...,n on

which condition (3.3) holds. The following theorem shows that there is

no need to worry about the overlapping of some of these discs.

Theorem 1 If condition (3.3) holds on each of the circles centre ai'

radius ri, i=l,2,...,n, then the roots 0i of f(z) may be ordered so

that

Jai-oil :5 ri, i=I,2,.....n . (3.4)

Proof Regard the perimeters of the circles as dividing the plane into aset of non-overlapping regions Ri, each of which is the intersection of a

subset of the set of discs that the circles enclose and their complements.

A simple case is illustrated in Figure 1. Let Rki be the region

containing ci' i=l,2,...,n. Note that a region may contain more than one

ai so that there may be coincidences among the ki (e.g. klk 2 =3 in

Figure 1). The set of regions Rk. i=1,2,... ,n together contain all the

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Figure 1 Four circles centre a. dividing plane into seven regions Ri

rofts a. of P(z) and each Rk has a boundary on which condition (3.3)

holds so contains exactly the same number of roots of f(z) as of P(z).Therefore the regions Rki(i=l,2 ... ,n) contain all the roots of f(z) and

these may be ordered so that Rk contains ¢i (i=l,2,....,n). The result

now follows since each Rki is contained in the disc centre a. radius ri. UBeing able to use overlapping discs leads to simplifications in

coding and sometimes to much better error bounds. This improvement isillustrated by the example shown in Figure 1, where a1 and a2 are quite

ill-conditioned but a3 and a.4 are not. The procedure of Peters and

Wilkinson would have forced us to regard all four a. as a cluster and to

enclose them in a single disc. This would h~ve made little difference tothe bounds for a1 and a2 but would have significantly worsened the bounds

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for a, and a4. Another illustration of this point is given in §6 with

an example using our code.

We now suppose that the error polynomial is

nQ(z) = I Eiz

1 (3.5)i=o

and describe how to find a suitable circle for a typical root. For

simplicity of notation let us imagine that the roots are reordered so

that the one for which we are seeking a circle is al and [ai-ail increases

monotonically with i. Rouche's condition (3.3) is satisfied on a circle

centre al and radius r if the inequality

n .n

X lEil (r+IliV' < Ian i (T i-I -al r)I (3.6)i=0o

holds. If m is such that

I am-OiI < r < lam_l-all (3.7)

(we ignore the case r = ic-Call for some i because clearly inequality (3.6)

cannot be satisfied in this case) then we may rewrite inequality (3.6) in

the form m

k(r) < TI (r - Jai-alI) (3.8)

where k(r) is given by the equation

nI I i I (r+l~ I ali=o

k(r) =n (3.9)

lani ITl (I ai -alI-r)

Peters and Wilkinson (1971) solve the equation

(1.1) k(O) = (rI - max ai-I (3.10)

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and then check whether rI satisfies inequalities (3.7) and (3.8). Almost

always inequality (3.8) holds because

(i) -the right hand side of (3.10) underestimates the corresponding

expression in (3.8),

(ii) the safety factor (1.1) has been introduced

and (iii) k(r) is nearly constant in the usual case where the roots

ai, i=1,2,...,m are well separated from the rest.

They give no recommendation for dealing with the case where rI does not

satisfy (3.8), but presumably intend that an iteration should be set up

with 0 and rI in (3.10) being replaced by rj and rj+I. This yields a

montonically increasing sequence, so that we must eventually either

satisfy inequality (3.8) or break the right-hand inequality (3.7)

necessitating a new start with a greater value for m. Actually Peters andm

Wilkinson take the centre of their circle to be i = E cti/m rather thani=l

a and this means that the right-hand side of (3.10) is quite a good

estimate of the right-hand side of (3.8) so that the procedure gives a

realistic radius r without very much computation. Unfortunately this is

not the case with a, in use, except when m=l and we have sometimes

obtained solutions such that the right-hand side of (3.8) exceeds the left

by factors as big as 1,000. We have therefore decided to use an iteration

based directly on (3.8). Given an iterate r. we seek a new iterate such

that

m(1.05) k(rj) < I (rj~l - cai-cll) < (1.1) k(rj). (3.11)

In view of inequality (3.7) it seems sensible to start with ro = lam-all

rather than Wilkinson and Peters' ro=O. To solve (3.11) we use the method

of bisection because m is usually small so that function evaluations are

cheap, no great accuracy is required in view of the slack in (3.11),

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S (double-length standard version only) is a REAL work array of

and suitable initial upper and lower bounds are available in rj and the

Peters and Wilkinson overestimate obtained from the analogue of equation

(3.10). Typically between four and seven iterates are required.

Peters and Wilkinson suggest that rj+l should be checked directly

in (3.8) but a simple test often avoids any need for this. If the

inequality

i n q a i y( r~j + ++ l aY l am + l -al - . n 1 .0 5 (3 .1 2 )

rji + I lm+ all - rI

holds then it follows from the definition (3.9) that the inequality

k(rj+l) ! 1.05 k(rj) (3.13)

also holds. From this last inequality and the first inequality (3.11)

we deduce that rj+l satisfies inequality (3.8).

Rounding errors occur when the coefficients of P(z) are calculated,

but can be reduced by multiplying out the factors (z-ai) in order of

increasing Jail. We therefore use the same order as that in which they

were calculated. If bounds ei on the errors were available we couldn

work with the error polynomial E (lcil+ei)z in place of the polynomiali=o

(3.5) and obtain strict bounds on the errors in the calculated roots.

We hoped to use the running error analysis of Peters and Wilkinson for

this purpose, but unfortunately found that it sometimes gave gross over-

estimates, as explained in the last paragraph of this section. We

therefore have ignored this source of error, relying on such errors being

reflectedin larger coefficients ci in the polynomial Q(z). It should ber

noted that the sequence of constructed polynomials H (z-ai), r=l,2,...,ni=l

is not identical with the sequence of deflated polynomials used when

finding the roots because we perform the multiplication in the same order

as that in which the roots were found. Therefore errors produced by the

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nmultiplication are likely to show in enlarged coefficients Ei" Not

bounding these errors means that our bounds are not strict bounds, but

they are more realistic. In none of our tests did the actual errors exceed

the estimated bounds.

It is straightforward to allow for the effects of uncertainties in

the original coefficients by working with the error polynomialn

-E (Ieil+bi) where bi, i=O,l,...,n, are bounds on these errors. This

we do in our subroutine.

We complete this section by explaining why the running errorn

analysis of Peters and Wilkinson for the coefficients of H (z-ai)i=l

sometimes gives very pessimistic results. If the computed productr rIT (z-ci) is E b r)zi then they generate coefficients fir) from the

i=I 1=0

recurrences.

f(l) -f(l) = 0o 1f(r+l) = I ccljf~r)+ lb r+l)l

f~r+l) = ffr) +Jr+l 1 fr) + arl br)l +1b r+l)1, r=1,2 .... n-l (3.14)1 : -1 rJlfl r~l1l~~ +1 br1

f(r+l) = f(r+l) 0- r+l

to yield bounds 2 -tfn) for the coefficients of P(z) if floating-point

computation with t binary places is used. This can sometimes lead to

very pessimistic bounds since the recurrence (3.14) fails to distinguish

properly between H(z-ci) and H(z+IctiI). In fact the recurrence (3.14)

is obtained by bounding the corresponding recurrence

er+l) ( a er) ( b(r) E b r+l) (3.15)i~ 1 Elr+l. i 2 1 3.5

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NPl (standard versions only) is an INTEGER which must be set to n+l.

Such bounds are particularly unrealistic in such a case as

f(z) = zn-l since the numbers f(r) grow with r in much the same way as the

recurrence for generating the coefficients of (z+l)n and these coefficients

are nCr,r=O,l, .... n. In fact the recurrence (3.14) has some added

nterms when compared with the recurrence for the coefficients of (z+l)

4. Changes to the algorithms in the real case

The only change made to our root-finding algorithm in the case where

f(z) has real coefficients is that once a complex root has been found it is

either perturbed into a real root or its conjugate is taken as another r'

This ensures that each deflated polynomial is also real and that work

saved for genuinely complex roots. Once a complex root ak=Xk+iYk has been

found we evaluate f(xk) and use Peters and Wilkinson's (1971) running

error analysis to bound the roundoff error made in this evaluation. The

recurrence used for evaluating f(xk) is given by the equations

sn =an

si xksi÷++ai, i=n-l,...,O (4.1)

f(xk) = so

and the corresponding running error analysis is given by the equations

gn = 0 (4.2)gi = Ixkl(gi+l+isi+I) + Isil' i=n-l, ... I 0

to yield the bound Eg0 for the error in f(xk) where e is the relative

floating-point accuracy. If the inequality

Is0 1 < 2cg0 + If(a k0 (4.3)

holds (where f(ctk) is the computed value at ak), we take v,, to be a real

root and deflate with it. Otherwise we deflate with the complex conjugate

pair xk ± iyk. It is very important that the case where we have a simple

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real root well separated from the other roots should not be taken as a

complex pair because a double deflation in such a case would be disastrous.

We therefore consider this case and suppose that hk is such a real root and

Lk is the (complex) approximation to it found by the algorithm. xk (real

part of ck) is a better approximation to h in the sense that the

inequality

1ýk -Xk1 :5 lh -O•kI (4.4)

holds since hk is real. Further the exact value f(z) behaves like a

constant times (Z-hk) if z is near 4k" It is therefore reasonable to expect

that the inequality

Jf(xk) 0 5 f((xk)l (4.5)

holds between the corresponding exact values. We therefore expect xk to be

at least as good a root as ak so that it is virtually certain that in-

equality (4.3) will hold, since eg0 is a rigorous upper bound on the round-

off error in computing f(xk) and may also be taken (slightly incorrectly)

as a bound on the error in computing Jf(lk)1.

A disadvantage of the test (4.3) is that we may decide we have a

real root when a complex conjugate pair is the true position. We feel

that this disadvantage is quite slight because it is reasonable to regard

any point for which inequality (4.3) holds as a good approximate root.

The only change to the error bounding algorithm is that once a

bound for a complex root has been found, this bound is also used for its

complex conjugate.

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5. Description of Fortran subroutines

In this section we describe the Fortran subroutines themselves. It is

a short section because we have included rather full comments in the code

itself, since we believe that this provides the most convenient form of

documentation. Specification sheets and listings of the code are given

in section 7. The specification sheets document all three versions (double-

and singe-length standard Fortran and double-length IBM Fortran). The

listings are of the standard Fortran double-length versions and contain

specially coded comments detailing changes needed for the other versions.

We have been using a simple preprocessor (written in standard Fortran) to

convert from one version to another. Where a statement differs between

versions we preceded the genuine (double-length standard) statement by the

alternative version or versions modified by the insertion of C in column 1,

/ in column 72, and the letter I or S (for IBM version or single-length

version) in column 71.

The listings used are those produced by the Bell Telephone

Laboratories' Fortran verifier because they include cross-references which are

very helpful when reading the code. They consist of lists of all the

identifiers and labels in lexographical order together with the statement

numbers of all references to them and the following coded inforwition

for each identifier:

TYPE: Col 1: E if explicitly typedCol 2: I for integer

R for realD for double precisionC for complexL for logicalH for Hollerith

USE: FA for arithmetic statement function argumentFN for functionE for external subroutine or functionGT for assigned "go to" variableIF for intrinsic functionSN for subroutineV for variable

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U

ATTRIBUTES: Col 1: C if in COMMONCol 2: E if an EQUIVALENCECol 3: A if a dummy argumentCol 4: S if value set by program unitCol 5: S if scalar

A if arrayCol 6: if array then number of dimensions

Because double-length complex facilities are not available in

standard Fortran we do not use any complex variables, although the arguments

A,R and E may be regarded as COMPLEX*16 (or COMPLEX for the single-length

version) on the IBM 360/370 series. All complex variables are written

as arrays of length two and all complex arrays are written as 2-dimensional

arrays whose first dimension is two. This leads to the code being almost

as easy to read as if the complex facilities were used. Complex division

is more complicated than can be written conveniently as in-line code so we

have taken this out of line to the subroutines PAO6ED/7ED.

We begin by describing the versions for complex polynomials, that

is PAO6AD/BD/CD/DD/ED.

The main call is to a short routine PAO6AD which calls PA06BD to find

the roots and PA06CD to find error bounds. PAO6DD and PAO6ED are short

subroutines for polynomial evaluation and complex division, respectively.

The time taken to find error bounds is usually about 25% of that taken to

find the roots themselves but can rise to as much as 100% when there are

many multiple roots. Because of this overhead we felt it was desirable

for the user to be allowed to call PAO6BD directly and instructions about how

to do this are included in the specification sheet. When calling PA06CD

(instruction 9 of PAO6AD) the work-space provided by the user is divided

into the four areas required by PAO6CD. Also PAO6AD checks for zero

leading coefficients and sets dummy error bounds to correspond, since

PAO6CD assumes that the leading coefficient is non-zero.

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U The algorithm used by PAO6BD to find the roots has already been

explained in section 2 with the minor exceptions of the inclusion of

tests for zero leading coefficients (roots at infinity), tests for zero

trailing coefficients (roots at zero) and the scaling of all deflated

polynomials so that the largest coefficient has modulus approximately equal

to the reciprocal of the modulus of the smallest non-zero coefficient.

We implicitly assumed in section 2 that leading and trailing coefficients

were non-zero and the inclusion of scaling minimizes the likelihood of

underflow or overflow. To avoid additional roundoff we scale by a power

of the floating-point base. For speed of execution (in function calls

rather than basic arithmetic) we use single-length working for scaling and

for finding the initial iterate. Also we avoid time-wasting evaluations

of the moduli of complex numbers by working with the squares of the

moduli or the sums of the absolute values of the real and imaginary parts.

Subroutines PAO6DD and PAO6ED, called from several places in PAO6BD, should

be regarded as part of PAO6BD. PAO6DD evaluates a complex polynomial at a

complex point and finds the square of the modulus of the result. PAO6ED

is a simple subroutine for complex division. The arguments of PAO6DD and

PAO6ED are explained in comments at the heads of each subroutine. Other

coding details of PAO6BD are explained in comments.

Subroutine PAO6CD, which finds the error bounds using the algorithm

of section 3, again makes use of single-length arithmetic whenever possible.

In particular we found that CABS executes significantly faster than its

double-precision equivalent (which in any case is not standard FORTRAN).

Therefore the error polynomial is held in single-length, the function

k(r) given by equation (3.9) is calculated in single-length and all the

Rouche tests are performed in single-length. The details of the code are

explained in comments.

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V

We have tried to make the code for the case with real coefficients,

namely subroutines PAO7AD-PAO7ED, resemble that for the complex case as

much as possible, using the same labels and the same comments wherever this

is appropriate. The only significant change in PAO7AD lies in the way

the work-array W is subdivided when PAO7CD is called. The only

significant changes in PAO7BD lie in the code for deflation (instructions

138-166), which is much more complicated because we need to test for a

real root and need to include code for deflation with a pair of complex

conjugate roots. The code for polynomial evaluation (in PAO7DD) is

longer because we make use of the fact that the coefficients are real and

treat separately the case where the point of evaluation is real. The

main changes in PAO7CD are

(i) Recognising complex roots when forming the polynomial IE(z-ai)

and multiplying them in as a complex conjugate pair to preserve real

coefficients (instructions 18-28). It is assumed that conjugate pairs

are adjacent in array R.

(ii) Much more complicated code (instructions 39-59) for finding

distances from the Ith root to all the rest. This code avoids calling

CABS to find the distances between two real roots and so is much

faster where most roots are real.

(iii) Using the same error bound for a complex root and its

conjugate (instructions 114-115).

6. Test results and comparisons with other methods

Our original reason for providing a new routine for the Harwell

library was that the existing routine PAOI sometimes gave incorrect answers.

The new routine (PA07) is able to get answers all of which have good accuracy

in about half the time. We did not pursue the error in PAOI.

-18-

We have compared our algorithm with that of Jenkins and Traub

(1970) in two ways. First we compared the Algol-W code given by Madsen

(1973) with an Algol-W version of the code of Jenkins (1969) and found

Madsen's to be 4 to 5 times faster. The required changes from Algol-60

to Algol-W are very minor. Next we compared our Fortran code with a rather

free translation into Fortran of Jenkins'code. We did not try to make

a literal translation but rather to use his ideas to produce efficient

Fortran code. The resulting program executed between 2 and 4 times slower

than ours. A further advantage of our algorithm is that it is simpler and

so the code is less bulky (the object code being about 2/3 as long). The

accuracy of the roots produced by the two algorithms was very comparable

except in the last test shown in Table 1 (Jenkins' 6th example) where we

obtained much smaller errors (all less than 2xlO- 14 ) because the roots of

modulus 0.9 were not all found before those of modulus 1 and therefore ill-

conditioned deflated polynomials were not generated. To compare the

error bounds with those of Jenkins we ran the examples documented in his

report. In his 6th example our bounds were much better simply because the

roots had been found so much more accurately. In his 5th example (a

complex polynomial of degree 21 having roots of multiplicities 1,2 and 3)

several of his bounds were quite unrealistic and our bounds were all better,

most of them by factors over 100 and three (or seven if multiplicities are

included) by factors over 1,000. In the remaining examples the

differences between the bounds were not severe.

Special purpose subroutines exist in the Harwell library for solving

real cubic and quadratic equations (PAO3A/AD and PAO5A/AD, respectively).

They use direct methods involving only the extraction of square and cube

roots but sometimes they lose accuracy through unnecessary cancellation.

We had hoped that a direct call of PAO7BD might be nearly as fast so that

-19-

we could withdraw the special purpose subroutines, but the speed difference

is by a factor of about 7. It is hoped that better versions of PAO3A/AD

and PAO5A/AD will be written for the library in due course. Extra tests

for ensuring reliability are likely to slow down the programs a little, but

it seems likely that good programs significantly faster than PAO7BD can be

written for these special cases.

To test our subrcutines we read in sets of roots and a relative error

level. We used extended precision arithmetic to c.nstruct the corresponding

polynomial and then made pseudo-random perturbations to its coefficients at

the required relative level. The polynomial and its errors were then

handed to our subroutines. This enables us to check the actual errors

against the computed error discs. We began by using all the polynomials of

Jenkins (1969) and about as many others from local sources, but eventually

reduced our test set to those shown in Table 1 (and a few trivial ones

designed to explore corners in our code) because the remainder showed no

useful additional information. We used the same test data for the real

case by adding an additional root consisting of the complex conjugate of any

complex root.

Our test results are summarised in Table 1. For each example we show

the errors and error bounds obtained for the first and last root found

and one intermediate one. This gives the reader an indication of our

success in finding the roots in order and we have been able to choose the

intermediate root displayed so that the three roots together indicate the

full range of conditioning. We also show the times (370/168 secs) for a

call of PAO6BD/7BD (roots only) and PAO6AD/7AD (roots and bounds). It can

be seen that finding the bounds usually involves quite a small overhead.

The last two cases are exceptional because of the large number of roots which

required discs containing many other roots. Such cases are relatively slow

-20-

because we try to find a disc with only one root, then try with two, and

so on.

It was the complex version of the last example which led us to

abandon finding disjoint discs, because some of the roots (e.g. (0,0.9))

are so ill-conditioned that the best disc obtainable covers all the roots.

Therefore if we insist on distinct discs all that can be obtained is one

disc for all the roots. By using overlapping discs,however, we obtained

12 good error bounds of which two are displayed in Table 1.

The real versions of the third and the last examples led us to

abandon Peters' and Wilkinsons' running error analysis on the computed

polynomial P(z) = T(z-cr). In the last example all the roots are well

conditioned but the bounds for the errors in computing P(z) were so

pessimistic that each disc contained all the roots. Example 3 was less

dramatic but all the error bounds came to about 10-8 instead of about

lO-14.

-21-

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-9-

S.. .... . .. ... . . . .. .. .. .. ... . .l~ .• . .,. ... . .... . .. . . . . . . . . . .

* 7. Appendix. Specification sheets and listings

Harwell Subroutine Library PAO6AD/BD

1. Purpose

To find all the roots of a complex polynomial

a1 + a2x+...+a n+lanx

and error bounds for these roots.

2. Argument List

CALL PAO6AD(A,N,R,E,W,S,NPI,LW) (double-length standard)CALL PAO6AD(A,N,R,E,W,NPI,W) (single-length standard)

CALL PAO6AD(A,N,R,E,W) (IBM)

A is a DOUBLE PRECISION (REAL for the single-length version) array

of dimensions (2,n+l) which must be set by the user so that the

real and imaginary parts of ai are held in A(l,i), A(2,i). It

is unaltered by the subroutine. For the IBM version A may be a

COMPLEX*16 array of length n+l.

N is an INTEGER variable containing the degree n of the polynomial.

Its value must be positive.

R is a DOUBLE PRECISION (REAL for the single-length version) array

of dimensions (2,n) used to return the roots. These are held with

real and imaginary parts in R(l,i), R(2,i), i=l,2,..:,n. The dummy

value (1D70,1D70) is returned for each infinite root (corresponding

to a zero leading coefficient). For the IBM version R may be a

COMPLEX*16 array of length n.

E. is a REAL array of dimension at least (n+l) which must be set by

the user to error bounds on the coefficients, or to zero if these

are accurate to machine precision. On exit the first n locations

contain approximate bounds on the moduli of the errors in the roots.

W is a DOUBLE PRECISION (REAL for the single-length version) work

array of dimensions (2,LW).

-23-

-10-

S (double-length standard version only) is a REAL. work array of

dimensions (4,LW), which may be equivalenced with W.

NPl (standard versions only) is an INTEGER which must be set to n+l.

LW (standard versions only) is an INTEGER which must be set to

at least 5n/4+2 (or 3n/2+2 for the single-length version).

3. Alternative Entry

The error analysis part of a call to PAO6AD takes typically about

20% of its time. If speed is important and error bounds are not wanted

then a call of the form

CALL PAO6BD(A,N,R,W,NPI) (standard versions)or CALL PAO5BD(A,N,R,W) (IBM version)

should be made. The arguments are the same as those of the main call,

but W need have length only (2,n+l).

4. Method

The roots are found by the algorithm of Madsen (BIT(1973) 13, 71-75),

the principal features of which are Newton iteration followed by

deflation. The error bounds are found by the application of Rouche's

theorem as recommended by Wilkinson (J.Inst.Maths Applics.(1971) 8,

16-35) except that discs are always taken with centres on the approximate

roots and errors in multiplying out the polynomial II(x-R(I)) are ignored.

The disc for each root is such that it contains exactly the same number

of approximate roots R(I) as exact roots of the true polynomial. Note

that in the case of true multiple roots the corresponding approximate

roots may be quite well separated but each will lie in the disc of all

the others and their mean will be a good estimate of the true multiple

root.

-24-

a

PAO7AD/BD

1. Purpose

To find all the roots of a real polynomial

aI+a 2 x+...+an+l xn

and error bounds for these roots.

2. Argument List

CALL PAO7AD(A,N,R,E,W,S,NPI,LW) (double-length standard)

CALL PAO7AD(A,N,R,E,W,NPI,LW) (single-length standard)

CALL PAO7AD(A,N,R,E,W) (IBM)

A is a DOUBLE PRECISION (REAL for the single-length version) array of

length at least (n+]) which must be set by the user. to contain the

coefficients and is unaltered by the subroutine.

N is an INTEGER variable containing the degree n of the polynomial.

Its value must be positive.

R is a DOUBLE PRECISION (REAL for the single-length version) array

of dimensions (2,n) used to return the roots. These are held with

real and imaginary parts in R(l,i),R(2,i),i=l,2,...,n. The dummy

value (lD70,ODO) is returned for each infinite root (corresponding

to a zero leading coefficient). For the IBM version R may be a

COMPLEX*16 array of length n.

E is a REAL array of dimension at least (n+l) which must be set by

the user to error bounds on the coefficients, or to zero if these

are accurate to machine precision. On exit the first n locations

contain approximate bounds on the moduli of the errors in the roots.

W is a DOUBLE PRECISION (REAL for the single-length version) work

array of length LW.

S (double-length standard version only) is a REAL work array of

dimensions (2,LW), which may be equivalenced with W.

-25-

-12-

NPI (standard versions only) is an INTEGER which must be set to n+l.

LW (standard versions only) is an INTEGER which must be set to

3n/2+2 (or 3n+2 for the single-length version).

3. Alternative Entry

The error analysis part of a call to PAO7AD takes typically about 20%

of its time. If speed is important and error bounds are not wanted then

a call of the form

CALL PAO7BD(A,N,R,W,NPI) (standard versions)CALL PAO7BD(A,N,RW) (IBM version)

should be made. The arguments are the same as those of the main call,

but W need have length only n+l.

4. Method

The roots are found by the algorithm of Madsen (BIT(1973) 13,71-75),

the principal features of which are Newton iteration followed by

deflation. The error bounds are found by the application of Rouche's

theorem as recommended by Wilkinson (J.Inst.Maths Applics.(1971) 8,

16-35) except that discs are always taken with centres on the approximate

roots and errors in multiplying out the polynomial fl(x-R(I)) are ignored.

The disc for each root is such that it contains exactly the same number

of approximate roots R(I) as exact roots of the true polynomial. Note

that in the case of true multiple roots the corresponding approximate

roots may be quite well separated but each will lie in the disc of all

the others and their mean will be a good estimate of the true multiple

root.

-26-

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References

M.A. Jenkins (1969). Three-stage variable-shift iterations forthe solution of polynomial equations with a posteriori errorbounds for the zeros. Stanford Report CS 138.

M.A. Jenkins and J.F. Traub (1970). A three-stage variable-shiftiteration for polynomial zeros and its relation to generalizedRayleigh iteration. Numer. Math. 14, 252-263.

K. Madsen (1973). A root-finding algorithm based on Newton's method.BIT 13, 71-75.

A.M. Ostrowski (1966). Solution of equations and systems of equations.Academic Press.

G. Peters and J.H. Wilkinsun (1971). Practical problems arising in thesolution of polynomial equations. J. Inst. Maths. Applics.,8, 16-35.

B.G. Ryder (1973). The FORTRAN verifier: user's guide. Computingscience technical report # 12, Bell Telephone Laboratories.

J.H. Wilkinson (1963). Rounding errors in algebraic processes. HMSO.

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