JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 65, No. 1, January- March, 1961

Computational Problems Concerning the Hilbert Matrix l

John Todd 2

(November 21, 1960)

The interaction between t heoretical mathematics and practical computational expire­m ent is illustrated by a discussion of recent work, at various centers, concerning t he Hilber t matrix.

1. Introduction

Recent work, in various centers, concerning the I-Elbert matrix shows very well the interaction between (theoretical) mathematics and practical computational experiments. Some aspects of this work will be discussed here in an expository manner. Altbough we sball be concerned mainly with the finite segments Hn of the (infinite) Hilbert matrix

H = {hi,i } = {(i + j + 1) - 1) , i,j= 0,1,2, .. . ,

many of the results can b e generalized, e.g., to matrices of the form

or

2 . Theory

Thc literature of the last half-century contains many theoretical resul ts concerning the Hilbert matrix. Of these only the b asic inequality is immediately relevant [8, p . 234]:

0>

2.1. I] 2:a! is convergent, then o

'" '" '" 2: 2: aman/(m+ n+ 1)< 1l' 2: a;n' o 0 0

We note the following consequence of this. 2.2. The equation H x= 1l'x cannot be satisfied by

any vector x in Hilbert space, 12. Suppose

H x= 1l'X

is satisfied for a vector x € 12. Then it is permissible to multiply bo th sides by x' , i. e., Lake inner products, to get

x' I-lx= 1l'X' x.

1 Tbis paper is baser! 011 an in vited address to tbe Societil Italiana peril Progresso delle Seienze, in Sicil y in 1956; it was prepared witb partial support from tbeOIDee of Naval Researcb .

2 Present address: Califo1'llia Institute of T echnology, P asadena 4, Calif.

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If X= (Xo,Xl, .. . , ... ) SO that ~x,;, is convergent this gives

~~xmxn/ (m+n+ 1) = 1l'~x;"

in contradiction with (2.1). The question as to whether there is an x, neces­

sarily not in 12, for which the equation

I-lx= 1l'x

is true, was rai ed by O. Taussky [20]. This question will be discllssed in sections 5 and 6 below.

3 . Applications

Among the areas in applied mathema tics in which Lhe I-Iilbert maLrix or related maLrices have turned up are aerodynamics (A. R. Collar [3]) , diffrac tion of electromagnetic waves (W. M agnus and F . Ober­hettin ger [12]), an d statistics (1. R. Savage and E. Lukacs [17]),

4 . Computational Problems- Inversion

Theoretically, the problem of inver ting the finiLe matrice H n i solved because of the existence of an explicit represen tation of the inverse, a result which is, essentially, due to Cauchy. The matrix is of Lhe form known as a double-al ternan t, and the inversion has been discussed e.g., by A. C. Ai tken [1, p. 134], A. R. Collar [3], and G . Palya and G. Szego [15, pp . 98- 99 , 299- 300]. The expli cit form of the inverse, and the actual numerical values for n = 1(1)10, have been given by 1. R. Savage and E. Lukacs [17].

Practically, however , the problem remains inter­esting. For instance, although the I-Iilbert matrix itself turn s up in idealized siLuations, it may be expected that slightly different matrices turn up in the practical circumstances which are approximated by these. It is, Lherefore, desirable to study the mechanical inversion of the I--Iilbert matrices H n ,

especially as they enjoyed a reputation for ill­condition i. e., "numerical instabili ty," long before the advent of high-speed computers and the basic papers of von Neumann and Goldstine [25] and Turing [24], in order to compare the observed in verse with the theoretical one.

The Hilbert matrix and related ones (e.g., Lotkin [10]) have been often used to test matrix inversion

programs prepared for high-speed computcrs. All the experiments of which we are aware [2 , 16, 23]­and in these several different schemes for inversion were used- completely confirm the bad reputation. The r esults quoted by Todd [23] are fully repre­sentative. For machines with 10 to 12 decimals, with floating poin t arithmetic to about 8 significant figures, there is rapid deterioration of the qu ali ty of the alleged inverses which are produced- and usually complete failure to produce any in verse for Hn with n about 8.

A detailed examination of the condition of Hn has been given by Todd [23]. Given a matrix A and a specific process of inversion, thc error in computing A - 1 can be found, in theory. In practice IV hat is wanted is a cheap estimate of this, in terms of quan ­tities associated with the matrix and e.g., the word­length of the machine. Various condition-numbers have been introduced to measure the condition of the matrix. One of these, the P -condition number is defined as P(A)=iAi/i Mi where IAI and IMi are the greatest and least among the absolute values of tbe characteristic values, of A. It has been shown [23], that ex being a certain positive constan t,

while for the n X n matrix

r- 2 1 1- 2 1

1-2 l

l 1-2 1J 1-2

associated with a second-order differential equation, we have

Using these results in the general error-estimate ob­tained by von Neumann and Goldstine [25], for an elimination process, and comparing these theoret­ical error-estimates with actual observed estimates, shows that the two are in reasonable agreement (see Todd [22]).

5. Computation Problems- Characteristic Roots and Vectors

As soon as programs for the computation of char­acter'istie roots and vectors were constructed , it was natural to test them on the H n , since input pro­grams for this h ad already been constructed. There was no reason to suppose th at it would be particu­larly difficult to handle- since it is considered that the presence of multiple roo ts, or roots close in absolu te value woulcl b e the most likely source of trouble. The theory of positive m atrices due to P erron and Frobenius (see e.g., Wielandt [26]) guar-

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an tees the cxistence of a single dominant character­istic root with a characteristic vector all components of which arc positive, so t hat the power method should be satisfactory. This is indeed the case and the dominant root, and the corresponding vector have been obtained for values of n up to 200 at vario lls cen ters [5, 18].

If more characteristic roots and vectors are needed , the Gantmakher and Krein [6] theory of completely positive matrices guarantees that the characteristic roots are all simple and posi tive and the compo­nents of the vectors exhibit a characteristic pattern : th e first has all its componen ts positive and each succeeding one has one more change of sign.

The last criteria shows up the weakness of the results obtained. For instance, a reasonably effi­cient code using floating poin t with 8 significant figures which gives the dominant root A5 of H5 cor­rect to 8S gives the smallest r oot negative! Actually,

detH5: 3.75 X I0- 12

and

so the results are not too surprising. The usual method for handling this casc is an

approxim ate diagonalization of the m atrix using the Jacobi process, or the variation on this due to W. Givens [7] in which the matrix is reduced to triple diagonal form and the roots located by a Sturmian process. So far few actual results of th e usc of these Inethods have b een published.

6 . More Theory

Following experimental computations, O. Taussky [19] showed that, if An denotes the dominant ch aracteristic root of H n , then

An = 7r + O(ljlog n )

so that " n----'7 7r . The actual approach to 7r is very slow, e.g., it is known [5,18] that: A50 : 2.08, A60 : 2.11 , A75: 2.1 4, " LOO: 2.18 , A125 : 2.21, A200: 2.27; the exact order or magnit ude of 7r - A n docs not appear to b e known .

A constructive attack on the problem, already mentioned in section 2, of the existence of a vector v satisfying

H V= 7rV

appeared difficul t and an affirmative solu tion to it was obtained by T . Kato [9], who used an indirect approach . His method, which will now be de­scribed , was suggested by observations of the mono­tonic beh avior of the coordinates of the dominant characteristic vectors of Iln, when these are normal­ized so that the first coordinate is unity.

Suppose the characteristic vector VI! of H I! cor­responding to An is

vn==(vi, V2, ... ,v:), V~= l.

Jun1erical evidence suggesLs tll at both the seq uence {An} n,nd all the sequences vl, v1+1, v1+2 , . . . (i= 1,2, ... ) ar e m.onotone. Thc fLrst fact is known (e.g. , CollaLz [4)) and the second is a special case of it

more general result regarding the domin an t vecLors a , b of tvvo positive matrices A, B where A is ex X ex and B is (3 X (3. If B domin ates A in the sense that (X~ (3 and if Cii= bij/aii is a non decreasing function of i, j (as long a.s it is signifLcan t) , then b dominates a in ~t similar sense , provided fur ther t111tt B has all its seeond-ordcr minors positive. This is proved using the fact (already noted) that in these circum­stances the dominant vectors can be obtained by the "power" method. The special case used is that ill which A = H n , B = H n+l .

Now assume that An is bounded- tlmt th is is the case when we n,re dealing with FIn, follows from (2 .1 ) . Then lim An= A, where A= 7r in t1lC Hilbert case. H follows that the sequences vl, V;H, vj+2 , ... are bounded, for comparing the fLrsL coordinates in the equation

we have ho. i v7 ~ "22ho. j v~= An v7 = An ~ A,

which gives , for all n ,

v7 ~ A(ho. i) - 1.

In the Hilbert CftSe this is

If we write vi=lim v1, it can Lhen be s110wn tha t n

the infLn ite vector V = (V l, L'2, . . . ) sa tisfies

IIv= 7rv.

The result obtained by TCato [9], while answer ing the original question , itself suggests Ill any 1110re. For in sta,nce, is the vecLor v Lhe only (lin early in­dependenL) vecto r correspo nding to A= 7r? Arc these characteristic vn]ues which exceed 7r?

7 . Recent Developments

1. During the last few years the theon- described in secLioll 6 lws been developed considerably by Tosio K ltLo and Marvin Rosen blum, and some of tile questions raised there have been answered. Among the relevan t results are the following:

M. Rosenblum [28] has shown that every complex number with a positive real part is a characteristic root of H (8) for 8 fLxed, 8> 0. This was accomplished by using the t heory of special fUllctions to obtain a characteristic vector explicitly . I n [29J Rosenblum comnletely determin ed the spectrum. of H (8) , for any real 8 ~ O , - 1, - 2, ... ; this was accomplished using the Titchmarsh-Kodaira theory of singular differ en tial opera tors.

T . Kato [30J shows that if M (8) is the Hilbert bound of 1[(8) , i.e. M (8)= 7r cosec 7r8, O< 8 ~!;

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Nf(8) = 7r , 8~t, then every A ~lv[(8) is a charac­terist ic value of H CO), with a positive characteristic vector and t here is no characteristic value A<111(O) which has a positive characteristic vector.

2. A reference Lo Hilber t's own work about Hn is [27] where he evaluates det Hn. For some related results see R . B . Smith [32], W. W. Sawyer [33]. The problem of matrix inversion is discussed from the experimen tal point of view in [3 1], which includes further results on the Hilbert matrix.

8. References

[1] A. C. Aitken, D eter minants and matrices (Edinburgh, 1951) .

[2] S. E. Atta, Effect of propagated error on the inverse of Hilbert matrix. J . Assoc. Computing Maehiuery, 4, 36- 40 (1957) .

[3] A. R. Co ll ar, On t he reciprocal of a segment of a gen­eralized Hi lber t matrix. Pl'Oe. Cambridge Phil. Soc ., 47, 11- 17 (1951) .

[4·] L. Coliat z, Eill s chl i essung~sat z fLlr d ie charakter ist ischen Zahlen von MatJ' izen. Math. Z. 48, 221- 226 (1942) .

[5] G. J~. Fon;y t he an d ;\1. Ascher , Unp ubl ished computa­t ions on S WAC at N umeri cal Analysis R esearch, University of Californi a , Los Angeles.

[6] 1". Gant ma kher and M. Krein , Sur leR mat ri ces comple­Lement non-negatives et osci ll atoires . Co mpo Math. 4, H5- 476 (1937) .

[71 'vV. Givells, A met hod of co mput ing eigenvalues and ei genvectors sugges ted by classical results on sym­metric maLr ices, K BS Appl. Math. Series 29, pp. 117-] 22 (J 953) .

[8] G. H . LI a rd.v, J. E. Littlewood, and G. Po lya, Inequalities (Ca mbridae, 193 ·1·).

[9] T osio J\:ato, On Lhe Hi lber t matrix, Proc. American Math. Soc. 58, 78- S l ( ]!-157) .

[10] M. Lotkin, A set of test matrices. Math. T ab les Aids CompuL 8, 153- J61 (1954) .

[ll ] W. J\Iagnus, Ueber e inige beschJ'ii, nk te Matrize n. Archiv d . Math. 2, 405- 412 (1949- 1950) .

[11 a] \ V. Magnus, On t he spectrum of Hilbert' s matrix. Am. J . Math. 72, 699- 704 (1950) .

[12] v\' . Magnus and F. Oberhet t in ger , On s.vst ems of linear equationR in the t heory of guided waveR. Comm. Pure and A ppl. Math. 3, :)93-,>1 0 (1950) .

[1.3] J . C. P. 'Mill er a nd R. A. F'airthorne, Hilber t' s double seri es theorem and principal latent roots of the res ult­ing matr ix. Math. T "bles Aid:; Com put. 3, 399- 400 (19-1S) .

[14] L . J . Pa ige and O. Tau~sky, Ed itors, Simultan eous linear equations a nd the determ inat ion of eigenvalu es . NBS Appl. Math. Ser ies 29 (1953).

[15] G. Poly a and G. Szego, Aufgaben und Lehrsatze aus del' Analysis, II (Berlin , 1925).. .

[161 H . Rubens tein and J. D. Rut ledge, Hi gh order matnx calcu lations on t he UNIVAC. Proc. Assoc. Compu.t. Mach. , pp. 181- 186 (Pittsburgh :\tIeet ing, 1952) .

[17] 1. R. Savage a nd E. Lukacs, Tables of in verses of finite segments of the Hilbert matrix, NBS Appl. Math. Series 39, pp. 105- 108 (J 954) . .

[I S] S. Schecter etal. , Unpublished computations on UNIVAC at t he Inst itute for Mathemat ica l Sciences, New York Univ., New York , N.Y. . .

[19] O. T auss ky, A remark concernin g t he e haracten stlC roots of t he fin ite segments of the HIlbert matn x, Quarterly J. Math. (Oxford) 20, 80- S3 (1949) .

[20] O. T a ussky, R esearch problem 12, Bull. Am. Math. Soc. GO , 290 (1954) .

[21] O. Taussky, editor , Cont ribut ions to t he solut i.on of systems of li near equat ions and t he deter mm atlOll of eigenvalues, NBS Appl. Math. Ser ies 39 (1954). .

[22] J . Todd, The condi t ion of certain mat rices, II . Archlv d. Math. 5 , 249- 257 (1954).

[23] J . Todd, The condit ion of the fin ite segments of the Hilber t matrix, NBS Appl. Math. Series 39, pp. 109-116 (1954).

[24] A. M. Turing, Rounding-off errors in m atrix processes. Quart. J . Mech. Appl. Math. 1, 287- 308 (1948).

[25] J . von Neumann and H. H . Goldstine, Numeri cal invert­ing of matrices of high order . Bull. Am. Math. Soc. 53,1021- 1099 (1947) .

[26] H . vViolandt, Un zerlegbare, nicht negative Matrizen. Math. Z. 52, 642-648 (1950).

[27] D. H ilbert, E in Beitrag zur TheOl'ie des LegendJ'eschen Polynoms, Acta Mat h. 18, 155- 159 (1894) ; reprinted in Ges. Abh. 2, 367- 370 (1933).

[28] M. Rosenblum, On t he Hilber t matri x, I , Proc. Am. :'I1at h. Soc. , 9,137- 140 (1958).

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[29] M. Rosenblum , On the Hilbert matrix, II, Proc. Am. Math. Soc., 9, 581- 585 (1958).

[30] T . Kato, On positive eigenvectors of positive infinite matrices, Comm. Pure Appl. Ylath. , 11, 573- 586 (1958) .

[31] M . Newman and J . Todd, all t h e evaluatioll of matrix inversion programs, J . Soc. Illdust. AppL Math. , 6, 466- 476 (1958).

[32] R . B. Smith, Two t heore ms on in verses of fini te segments of t he generalized Hilbert matrix, Math. T ables Aids Comp ut. 13 , 41- 43 (1959).

[33] W . W. Sawyer, On the illtegral eq uatioll kf(x)

=J~n (x+y) - lf(y)dy, J . London Math . Soc. 340, 451-

453 (1959) . (Paper 65Bl--44)