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OFFICE OF NAVAL RESEARCH LONDON

,- - , TECHNICAL REPORT

' »• - ONRL-35-51

COLLOQUIUM ON LINEAR EQUATIONS

This document is issued for information purposes only. Publication, repri ting, listing, or reproduction of it in any form, or of any abstract thereof, is not authorized except by specific prior approval of the Commanding Officer, Office of Naval Research, Branch Office, tfavy No. 100, Fleet Post Office, New York, New York.

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T2C-\::LGS.L iiii"Gnr Ol3lL-35-51i

COLLOQUIU" C>: LIKEAR EQUATIONS

Summary

Ihiii report presents a summary of 3ome of the rork presented at a Colloquium on Linear Equations and inequalities, neld at Oberwolfach, Ger- many in October 1953« Hals work includes: ar application of the Stiefel- Rutishauser quotient-difference algorithm which involves a novel relation of an inverse povrer series vitu a continued fraction; a method presented by irof. Stiefel applying orthogonal polynomial sets to the approximate solution of symmetric, positive-definite systems; an elementary proof by Dr. iihlers of the inequality£ft'a.y» ^ -^/a: , where the a^'s are positive

real; a description by Dr. Linger of the Jenne-Friedrich method of evaluating determinants; and by the writer, z. variant of the classical SchlOmiloh expansion of an arbitrary function in a series of i functions.

21 June 195U Approved by: E. Eprend an Deputy Scientific Director

prepared by? Wallace D» Sayes Scientific liaison

Officer

COLLOQUIUM ON LINEAR EQUATIONS

j INTRODUCTION i

During the week $ to 10 October 1953» a Colloquium on Linear Equa- tions and Inequalities was held at the Mathematical Research Institute at Oberwolfach, in the Black Forest. This Colloquium had the same informal and pleasant nature characteristic of such meetings at Oberwolfach, and was attended by about twenty mathematicians, mostly from German and Swiss uni- versities. A list of the participants is given in Appendix I.

Since a thorough description of the Institute, together with a detailed review of its history, nas been given in Technical Report ONRL-12-52, this will not be repeated. Briefly, the Institute is housed in a building known as the Lorenzenhof, located at Oberwolfach-Walke, Schwarzwald, about forty miles from Freiburg i/Br. The history of Oberwolfach starts with the organization by Professor W. S'iss in the middle of 19Ui of a Central Mathe- matics Institute for the purposa of acting as a national research and con- sultation center. The organization was divided into two parts, pure mathematics and applied mathematics, and was housed in the Lorenzenhof. The policy of the Institute as established very early in its existence was to employ as many good people as possible for research along lines which would produce mathematics and mathematicians of lasting value irrespective of the applications of the moment. During the allied occupation, the Institute served as a refuge for mathematicians in distress. In the surroundings at Oberwolfach, their interest in mathematics had an opportunity to be re- awakened.

Ever since the fall of Germany, the finances of the Institute have been rather precarious. Tne receiving of the assignment from the mili-

\ tary government to prepare the FIAT reviews on pure mathematics helped, principally in the matter of food, and the Institute has for the last several years received a small annual allocation from the Land Baden, enough to cover rent and a few basic expenses. Tne situation of Oberwol- fach is at present rather clouded and it is not at all certain that it will continue to get even the nominal support essential for its bare existence. The reason for this lies in a recent political change in Ger- many, the consolidation of two of the Lander into one. The former Lander of Baden and Wflrttemberg have been combined into a single one, with, of course, corresponding changes in the Departments of Education. The nev< Ministry of Education has now to be convinced anew of the value of the Institute to the Land and to the Federal Republic.

During most of the week of the Colloquium three meetings a day were held, one in the morning, one in the late afternoon and one in the early evening. Some of the twelve or fifteen contributions presented at Oberwolfach are discussed below. In keeping with the informal and "work- shop" nature of the meeting, much of the work presented was tentative or unpolished in character and much of the time was spent in critical discussion. No time limit was ever imposed.

AN APPLICATION OF THE Q'uOTIENT-Dirr'ERKKCE A LGORI7HM

A quotient-difference procedure for the investigation of eigenvalue problems was presented at the 1953 meeting of Gall! in Aachen by Professor E. Stiefel of the E.T.II. in ZUrioh. This procedure was reported in detail in Technical Report ONEL-76-53. At Oberwolfach, Dr. M. Rutishauser, who col- laborated with Prof. Stiefel in the development of the algorithm, presented an application of this algorithm to the problem of obtaining solutions to linear equation systems* This application involves the use of continued f ra c ti ons.

The quotient-difference scheme starts with an original set of numbers designated by f- and follows the equations

k ''

Ak ^K ^K

If J; has the form T aj \, , then /I i = ° .If the A i are positive real, distfrTct, and arranged in order cf descending magnitude, then

Q; -» ,\ as L -*• cc

The method presented bv Dr. Rutishauser is based on a relationship afforded by the quotient-difference algorithm between a certain inverse power series and a continued fraction. This relation is given

HM - L A^« x "*" y- ^r, A t=o

_ to Op ^f Qo _Ao Q X-I-X-I-X-I

This method may in some cases be useful for the numerical evaluation of semi-convergent series.

Tne particular application is to the problem of solving the equation system

A* =

•2-

expressed here in matrix form. A formal solution is

i convergent if A has only positive real eigenvalues and if ^ is sufficiently- large. If f- represents a particular element of r- , where

| ^ r6 - r

| rc+, =• (XE-A) n

! then the corresponding element of x is given by

KM * i -pi - Y T \

The procedure is to pick a convenient value of r\ , calculate the ft , carry out the quotient difference procedure on a particular element, and to calculate the corresponding element of x through the continued fraction expression. This continued fraction may be convergent even though the equivalent inverse power series is not.

As an example, the negative of the Laplace operator is considered in the usual finite difference form, applied to a square region with homogeneous boundary conditions

o-o-o-o-o-o 0 x x x x C -1 OxxxxO A = -1 U -1 0 x x x x 0 -1 OxxxxO 0-0-0-0-0-0

The quantity \ is chosen to be equal to 5, and the operator \£-A is expressed

1 Ê - A = 1 1 1

1

The distribution r for which a solution is desired is chosen

0 0 0 0

O O 0 o r — -i — r x - oloo - 0

0 O 0 o

-3-

for which

0000 oloo 3330 oloo r 222o r . 612 6 3 1 1 1 o 2"2521 3 " 11 13 12 3 oloo. 222o 6 11 6 3

12 21 12 6 .... 32 UO 36 12 133

rk z 36 59 Uo 21 r5 = 155 211 133 . 28 36 32 12 155 .

The quotient-difference procedure is applied to the element for which r was non-zero.

13

1

5 5

5

"f -12

+ To-

21 126 ~ io ~T~

«/5

126 5^13

126 T

21 ~ 10

2-5-7-13 if ^

25 2-7

»/* 5-Ul 13-7

nv»

738 "13T9 -*

59

111 and x for that element is expressed

y 1 .J. J± _£. jo^ J5_ * - S - I - 5" - < - £" - |

-U-

The successive approximations to the particular element of interest are

It may be seen that, vîth the number of terms taken, no reasonable approximation has been reached.

An alternate method that might decrease the amount of calculation required in the quotient-difference algorithm would be to separate the first m terms, giving

! 4j'X\ * r* 1- - r -ri +- '" * A* * HX- » - * - • - J This type of variation to the method is necessary in case one or more of the elements in the partial fraction expression are zero (or infinite).

SQLuaoN OF SYMMETRIC, rosrnvE-.jBFiNi'ns SYSTEMS OF EQUATIONS USING ORTHOGONAL POUCiJOLlALS

Professor E. Stiefel of the E.T.H. in Zttrich presented a method for obtaining approximate solutions of finite systems of equations in v.hich the Matrix is symmetric and positive definite. The basic equation system is expressed

Ax = k (1)

and the eigenvalues of tiie matrix are presumed to be less than unity and arranged in order of increasing magnitude

o< \, ^ X,<A. <Xn <- \

Since an upper bound to the eigenvalues of a symmetric matrix may be found, this requirement can alv.ays be satisfied by an appropriate reduc- tion. For convenience, the case of distinct eigenvalues is considered.

An approximation scheme is envisioned., in which x'm' represents the mtn approximation to the solution, and the m^n residual vector is defined

r(;i) r k - Ax(«0 (2)

Trie general equation for improving the solution is taken to be of the form

x(m+l) _ x(m) - g^ (x(m) _ x(m-i)) + amr(~) (3)

-5-

from which the equation for r(m) is obtained

r(m+l) s (l4.c _ a A) r(;n) _ c r(m-l) U)

The process is begun by setting

Ihe vector rv") may be expressed in the form

riW' = Rm(A) k (5)

•..'here R. is r. polynomial of decree i in A. It nay b IL(0)xl'« If the vector k is expressed

k - k «, •«- M«. * *k*G«

e noted that

f6)

in terms of the normalized eigenvectors e*, the vector r(m) is expressed

r(rn) =. K*A)M» T - - + Rw0OVc«en (7)

As an example trie simple iteration scheme corresponding to the expansion

A"' r ICS-flf *v\=0

is considered. This fits into the ger.errl scheme with cm-0, a;a- 1. The polynouial R_ is given si*nply

It is clear by considering a plot of R~(A) that this gives rather poor approximations for a given order of iteration unless all the eigenvalues are close to unity.

-6-

Tm(

It is clear that an approach using one of the well-known sets of polynomials related by recurrence relations and giving good representations of other functions within a fixed interval may be used to devise an intera-

j tion scheme. As an example, with a lower bound "^0 to tne eigenvalues, I the Ifchebycheff polynomials may be used, with

»-Ao ) RmM - -_ / , + ^N

These functions are orthogonal over the interval ^ to 1 with the weighting function

IfNthe eigenvalues were known, a procedure which led to a residual function R\n) .vhich had zeros at the eigenvalues would yi«ld an exact solution. This would be best done for a given k by constructing a set of orthogonal polynomials for R^-' 7.1 th the weighting function

yM * k?S(*-Vi«. ^^(Û (3)

where o denotes the Dirac delta function, by means of which R(n'(/\) would automatically possess the desired properties. But the orthogonal property

(R;ltiRiM8W«** = 0= (rc"0 fcri+j (9) 'o

is equivalent in this case to the corresponding orthogonal properties for the residual vectors themselves. Thus the equations (u), together with the orthogonality conditions

<.m*i\ ^t*«-»n . (10) (rtm%,\ r^",J = o

from which the values of am and cm may be determined, are able to accomplish this construction with no necessity for specific information about the actual eigenvalues. This is the principal result of Stiefel's contribution. The double orthogonality condition, together with Eq. (u) ensures orthogonality in general. The actual solution is obtained through Eq. (3).

-7-

It may be desired to use approximate information about the eigenvalue distribution and about the vector k in place of the foregoing specialized procedure. If the density of the eigenvalue distribution is given by p(A)and the component distribution of the vector k is given by k(?v), the proper choice of density function in the orthogonal polynomial procedure is

?(V) - rk* (11)

With suitable choices for f(A/, several of the well-known systems of orthogonal polynomials are obtained.

CCAPARISON OF EQUATION-SOLVING METHODS

Dr. H. Unger of the Institut fflr Praktische Mathematik in the T.H. at Darmstadt discussed a number of standard methods for solving the equation system

A* = a in matrix notation. The general conclusion presented was that the Gauss algorithm, by which A is factored into two triangular matrices

A = Q& Q of form ^q G of form t=»

requires the least number of numerical operations.

Unger continued with a discussion of some of the variants of the Gauss algorithm.

(a) If the solution obtained is approximate, an improvement may be obtained by obtaining the residual

<io - Ax0 - a solving the incremental equation

using the approximate triangular matrices, correcting x, and repeating as necessary.

(b) The splitting of A may be accomplished using diagonal matrices between the triangular matrices, in order to decrease the number of divi- sions which need to be carried out in a numerical inversion.

-8-

(c) The Gauss algorithm may be carried out with submair-.lces in place of the usual elements; this requires the inverses of the submatrices lying along the diagonal, themselves obtainable by the Gauss algorithm.

(d) If a number of solutions involving the same matrix are to be obtained, it i: easier to carry out the calculation in terms of the inverses of the triangular matrices

than to calculate A-1 and obtain x. directly.

ELgiEKTARY PROOF OF AH INEQUALITY

Dr. G. Ehlers of the University of Kiel pave a simple proof of the inequality

I i

â.cu ah *£ ~(a, t-cvr •• • <-aY%>)

•where the a's are all positive real. Noting that the expression is homogeneous, it may be required that

a,av an = \ and it is then necessary to prove that

<XS rCkx. *• •- <\* 5. n

Ihe proof follow., the method of induction.

Assuming the inequality holds for n and that

a Ax a****** - \

then

tt,(U +• <** +• * ' + ft*» * ftw*» — n

•where a, and a~ are any two out of the set of a's. Choosing one of these greater than one and the other less than one (a^= 1 is a trivial cas«^.

combining the two inequalities yields

-y-

SOLUTION OF LINEAR EQUATIONS BY COMPLEX CONTOUR INTEGRALS

Dr, G, Ehlers presented the solution of the equation system r

2>.XBt =Y 1

1S| where the elements involved are square matrices and the matrices B are conmutable. The solution is

Y -L <f (- • • f - ! Y — ili- ^~

where y goes around the entire spectrum of B, but not around any singularity of fct

THE JENNE-FRIEDRICH METHOD OF EVALUATING DETERMINANTS

Dr. H. Unger of Darmstadt described a method for the evaluation of determinants which, existent in less available literature, is relatively unknown. The original reference is a paper by K. Friedrich, Stuttgart, 1930, or in a Sonderabdruck aus Zeit. f. Vermess.-Wesen, 1930, A second reference is by W. Jenne, "Zur Auflttsung linearer Oleichungs-systeme", Astronomische Nachrichten 273, 73 (19u9). The method is based upon a lattice-like graphical representation of the terms of a determinant.

Each diagonal term of the matrix or determinant is represented by a point, for example the term a22 by the point 2. Each conjugate pair of off-diagonal terms is represented by a line joining the corresponding points, with each term associated with a direction along this linej for example,

Additional nutation is needed. Parentheses are used to denote a complete diagonal subdeterminant (one whose diagonal terms are diagonal terms of the original determinant) or a single diagonal term. The expression

|MV«"- ~M - <Vav^ <WV is defined also. Note that |pj - tX.,^ = (pO t

-10-

The determinant ic considered to be evaluated with respect to a certain point, y. , and is given by the expression

where r is the number of terms in the I | expression, and R is the diagonal sub-determinant with terms in J*,*V -^ missing. The summation is taken over all permutations of }*• with any number of other indices (including none).

Some clarification may be brought about by an example. The complete lj x ii determinant

D =

all a12

a21 a22

a31 a32

aui aU2

l13

l23

l33

llk

l2U

l3U

lU3 %U

is denoted by its lattice representation

I

Evaluating with respect to the point 2,

- M(*-») - l~l(—') - |2U2t(-—I)

•4- |2132| (U) -f- \2312| (It) 4-\211*2) (3)

-f- )2itl2| (3) -f- \23h2] CD -h\2U32\ (1)

~\213U2^ -)21U32\ ~\231U2] — \23U12J

- |2lil32| - \2U312 )

-11-

•/

It is evident that this procedure presents no attraction over conventional methods in the general .-.ase. Its only claim to usefulness is for cases in which certain of the off-diagonal pairs are zero, in which certain of the connecting lines of the lattice are broken. In the example above, if a.. = ^31- a. ,= a, -.s-O, the results quoted above take the form

D = { i^L^f

+ US421 (0 + \XH*2\ CO. If the expansion is made relative to the point 1, the expression is simply

D = (o(fed^») -(m)(1—&) In the particular case of the determinant of a Jacobi matrix the lattice has the simple form

1 * "5 4 5" n-t n *• • • —* • - . . . _• •

In a comparison of the Jenne-Friedrich Lattice Method applied to the treatment of matrix and determinant problems with the Gauss algorithm, Dr. Unger pointed out that even in the case of a Jacobi matrix the Gauss procedure is better. Only in the case for which the sub- and super-diagonal terms are all 1 or -i does the Lattice Method show any numerical advantage. Interest in the method must probably depend on the conceptual picture it gives of a determinant and the fact that it furnishes a numerical expansion symmetric with respect to the diagonal.

aEO-asHLflmicu SERIES

Dr, W. Hayes of ONR London presented a variant of the classical Sch&lmilch expansion of an arbitrary function as a series of Bessel functions. In place of the classical variation of the argument pro- portional to an integer m the variation is as -V kl*-m* where k is constant. Series of this type appear in the theory of solutions of the equation $l%+ <V^ - $xx - k*<b -o which are periodic (of period 2TT , for example) in the variable x.

The function to be expanded is

-12-

1 ! I i

i '

I

m the principal problem being to evaluate the coefficients am in terras of the function f(x). If the function

gCO - £Q.O + 2L °-** W4 tr»A

i8 known, the coefficients are then given directly by

Jo o

A modification of parseval's integral shows that the functions f and g are related by the equation

r

(cf. Watson, G. N., "Theory of Bessel Functions", Cambridge, 19U;> Sects. 2,2, 19.1, and 19,ill), and the solution of this integral eauation is

(ft

IVds determination of g in terms of f completes the problem. Re- quirements for convergence are the same as for classical SchlBmilch series.

It may be noted that k may be imaginary, whereby in essence the cos and cosh functions in the above analysis are exchanged. In the Bessel function series a finite number of terms are then expressible with the modified Bessel function I in place of J„. c o

As with the classical SchlBmilch series, a null series may be obtained. Ifcis series is

OP

VNHil

It may be again noted that k need not be real in this serias.

-13-

Appendix I

Leaders of the Colloquium}

Gstroviski, Prof. A., Basel bttss, Irof, W., Freibarg

Participantsj

Bauer, Dr. F., LiMnchen Beyer, Frl, G., Oberwolfach Ceschino, Dr. F., Paris Collatz, Prof, L,, Hamburg Ehlers, Dr. G., Kiel Gericke, Prof, II,, Freiburg Gortler, prof. II.. Freiburg HMmmerlin, Dipl. Math, G,, Freiburg Hayes, Dr, W., ONR London von Hoerner, Dr. S., Gttttingen Hornecker, Dr. G., Paris Kuntzmann, Prof. F., Grenoble Ostmann, Prof. H. K., Berlin Pisula, Dr. K., GJJttingen Rutishauser, Dr. H., Zflrich Stiefel, Prof. E., Zdrich iitucken, Dr. II., Neukirchen Thtlring, Frof. B., Neukirchen Unger, Dr. H., Darmstadt Witting, Dr, H., Freiburg Wundt, Dr. K., Freiburg

-11*-

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College Park, Maryland Prof. II. a. Martin Prof. A. '.Veinstein

Massachusetts Institute of Technology

Cambridge 39, Massachusetts Dr. C. W. Adams Prof. S. Bergman Dr. J. Forrester Prof. C. C. Lin Frof. p. M. Morse prof. G, Springer

University of Minnesota Minneapolis, Minnesota

Prof. R. H. Cameron Frof. S. E. V/arschawski

New York New York

Frof. Dr. A Prof. Prof. Dr. S Prof.

University City, N. Y. R. Courant

. Douglis F. John E. Isaacson

. C. Lowell H. L. Ludloff

University of Notre Dame South Bend, Indiana

Prof. K. Fan Frof. A. E. Ross

University of Pennsylvania Philadelphia, Fa.

Prof, N. J. Fine Prof. R. J. Kline

Princeton University Frinceton, New Jersey

Frof. S. Bochner Prof. S. Lefschetz Prof. D. C. Spencer

Institute for Advanced Study Princeton, New Jersey

Prof. H. H. Goldstine Prof. li> Morse Prof., id. Sniff man Prof. H. Weyl

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Purdue University Lnfayette, Indiana

prof. Milton Clause

liensselaer polytechnic Institute Troy, New York

Dr, R, p. Harrinpton

Stevens Institute of Technology I'.oboken, liev: Jersey

Dr. K. S. LI, Davidson

Stanford University Stanford, California

prof, F, R, Garabedian prof. K. Klotter Frof, H. Leny Prof. G. Tolya Prof, G. SaegB

3wai»thnvore Collage Swarthmore, Pa.

Prcf. :I. W. Brinkiaann

Yale university Rev Haven, Conn,

Prof, N, Uunford Frof, K. Hille

University of Pennsylvania Philadelphia, Pa,

Frof, H, nadeiTiacher Prof, I. il. Sohoenberg

Dr. W. J. iSckert Watson Computing Laboratories 612 West 116th Street New York 27, New York

Dean Ulna Reee Hunter College New York, New York

University of Texas Austin, Ttexas

ii, J,. EUlinger prof.

TUlane University New Orleans, La.

Prof. W, L. Duron Prof. J. L. Kelley

university of Utah Salt Lake City, Utah

prof. H. N, Thomas

nl ay ne U n i ve r s ity Detroit, Michigan

Dr. il, iluskey

A'ashin^ton Universi ty St. Louis, liis30uri

Prof, «. Lei^hton Frof, Z. Nehari prof, W» J. Thron

University of Y/isconsin V.adison, Wisconsin

prof. A. C. Schaeffer

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