Solution of integral equations by hybrid computation

by G. A. BEKEY and J .. C. MALONEY

University of Southern California Los Angeles, California

and

R. TOMOVIC*

Belgrade University Belgrade, Yugoslavia

INTRODUCTION

The mathematical description of many problems of en-gineering interest contains integral equations. Typical of a large class of such problems is the Fredholm inte-gral equation of the second kind,

b

y(x)

==

f(x)

+

A

!

K(x,t) y(t) dt (1) where f(x) and the kernel K(x,t) are given functions, a and b are constants, A is a parameter and y(x) is to be found. From a computational point of view, equa-tions of this type may be considered as problems in two dimensions, where one dimension (t) is the dummy variable of integration. For digital computer solution, both variables must be discretized. For analog com-puter solution, it is possible to perform continuous integration with respect to the variable t for a fixed value of x and perform a scanning process to obtain step changes in the second variable. In either case, the solution is iterative and results in a sequence of func-tions {Yn(X)}, n==1,2,... which, under certain con-ditions, converge to the true solution y(x) as n increases.

It is evident that such a sequential solution, with a two dimensional array K(x,t), may be extremely time

* This work was initiated during 1964-65, while the author was an NSF Senior Foreign Scholar and Visiting Professor at the University of Southern California, Los Angeles. It was sup-ported in part by the Office of Scientific Research U. S. Air Force under Grant No. AF-AFOSR 1018-67.

143

consuming for pure digital solution. On the other hand, the scanning and iteration procedures, which require storage of the successive approximation to the solution, do not lend themselves to pure analog computation.

Rather, the problems require a hybrid combination of high speed, repetitive integration, memory, and flexible control logic.

The advantage of using such hybrid computational methods for the solution of integral equations was real-ized quite early. A special purpose computer for solu-tion of integral equasolu-tions was proposed by Wallman in

1950^1• Basically, the computer technique consisted of replacing the integration with respect to two independ-ent variables by scanning at two differindepend-ent rates. These original proposals for· iterative solution of integral equations were based on the classical or Neumann method.² In 1957 M. E. Fisher proposed an iteration technique for high-speed analog computers equipped with a supplementary memory capacity which resulted in considerably faster convergence than the classical technique.³ However, little practical experience with his method is available due to the complexity of the function storage and playback apparatus.⁴ One test of Fisher's method was made using a repetitive analog computer in which the computer operator manually adjusted a set of potentiometers in a special function generator at the end of each iteration cycle.^s

The purpose of this paper is to examine hybrid com-puter solution of integral equations, by both the Neu-mann and Fisher methods.

144 Fall Joint Computer Conference, 1967 The Neumann iteration method

The classical or Neumann iteration procedure for the solution of (1) is specified by

D

Yn+l(X) == f(x)

+

^A

^f .

K(x,t) Yn(t) dt (2) with Yo(t) == O. Under conditions discussed in Refer-ence 4, the process converges to a limit Yoo(x) which is the solution of (1).

From a computer point of view ~ an integration over the whole range of t must be made for each particular selected value of x. If the range of x is divided arbi-trarily into I segments of length .1x, the function is represented by the values of y(x» at the midpoint of each segment, i.e., y(xJ), i == 1,2,3, ... 1. It is clear that a total of I integrations in the t domain must be made before a single change in Yn(t) is made in equa-tion (2). Such an integration in t for a single value of x == Xi will be called a minor cycle. In order to in-crease the index n, i.e., to derive the next approximat-ing function Yn+l (t), one has to complete I minor cycles. This group of minor cycles will be called a major cycle. The complete solution theoretically re-quires an infinite number of major cycles. However, practical experience'" has demonstrated that accuracies of the order of 1 % are attainable in about 20 major cycles using Neumann's method.

It should be noted that digital computer implemen-tation of the strategy defined by (2) requires that the variable t also be discretized and that an appropriate numerical integration formula be used. For example, if Euler or rectangular integration is used, Equation (2) becomes

J

Yn+l(XI) == f(xi)

+

^{A I} K(xi,tj) Yn(tj) .1t (3)

J=l

i == 1,2, ... ,1

where .1t == constant is the integration step size and J is the total number of steps in the interval (b -a).

Since t is only a dummy variable, the total range in t must equal the range of x and it is possible to choose the number of steps in t and x to be equal, i.e., let I

==

J. . More sophisticated numerical procedures do not change the need for minor and major integration cycles. Equation (3) requires only the algebraic opera-tions of addition and multiplication and is well suited to digital computation.

For hybrid computer solution each minor cycle may be performed continuously (using analog integration) and the reSUlting values stored. Assignment of other specific computational functions to the analog or digital portions of a system may significantly affect overall accuracy, as discussed in a later section of the paper.

Thus, the generation of the functions f(x) and K(x,t) as well as the multiplication under the integral sign in (2) may be perfonned either in the analog or digital computer. A flow chart illustrating the programming of Neumann's method is shown in Figure 1. A stopping criterion given in the flow chart is based on reducing the difference between successive approximations to a sufficiently small value.

Figure I-Flow diagram of Neumann's method The Fisher iteration method

An examination of Equation (2) and Figure 1 reveals that Neumann's mithod requires the storage of Yn+l (XI) at the end of each minor cycle (i.e. for i

==

1,2, ... I).

At the end of I minor cycles the entire vector [Yr.(t)]

under the integral sign is replaced and a major cycle has been completed.

Fisher's method^3.'" for the solution of the same prob-lem requires that one eprob-lement of the vector [Yn(t)] be updated at the end of each minor cycle. Consequently, the Fisher version of the digital process of equation (3) becomes

J

Yn+l(XI) == f(xi)

+

^{A I K(XI,t}J) Yn.l-l(t_J) .1t (4)

J=l

i == 1,2, ... 1,

j

==

1,2, ... , J, I

==

J

Note that yet) under the summation sign now carries a double subscript. The idea is to replace at the end of each minor cycle the existing value of Yn(XI) in the memory with the newly obtained value of Yn+l(Xi)' The notation (n,i)i== 1 ,2, .. , I implies that during each major cycle the unknown function y(xl) is gradually adjusted as the index i is increased, not waiting, as in

Solution of Integral Equations by Hybrid Computation 145 the Neumann method, until all minor cycles are

com-pleted. In other words, Fisher's method is based on using each piece of new information as soon as it is available so that the adjustment from Yn(X) to Yn+1(X) proceeds gradually, rather than being performed all at once after I minor cycles.

The hybrid computer version of Fisher's method takes the form

x t

f(xl)+A

f

K(Xht) Yn+1 (t)dt

a b

i==1,2, ... I (5)

The first integral on right hand side of (5) contains the results which have been obtained during the pre-vious minor cycles of the present major cycle. A flow chart showing the hybrid computer implementation of this strategy is shown in Figure 2.

Figure 2-Flow diagram of Fisher's method Illustrative examples

Fisher has shown⁴ that for symmetric kernels his algorithm converges whenever the Neumann algorithm does, and in certain cases also when the classical method fails. (A symmetric kernel is characterized by K(x,t)

==

K(t,x». Furthermore, using a simple exam-ple, Fisher has demonstrated that his method may speed up convergence significantly. In order to obtain practical results concerning this comparison, several problems were solved using a small hybrid computer

(IBM 1620 digital computer and Beckman 2132 analog computer).

In order to facilitate the evaluation of the two meth-ods, solutions, Yn(t) were compared with known exact analytical solutions, z(t) , by means of a root-sum square criterion, defined by

I [z (Xl) - Yn(XI) ]2 (6) 1=1

Example 1. The following equation was solved:

1

y(X) 2 + 2 j'(X-t) y(t) dt (7)

3

with the initial approximation yo (t) == 2

3 The

step size was chosen as ~X == 0.1 and the multiplica-tion was performed on the analog computer. The solu-tions obtained by the Neumann and Fisher methods are compared in Figure 3 using the criterion F n defined in (6). The considerably faster convergence of Fisher's method is clearly illustrated.

~2.0

a: a:

1&.1 1&.1

~ 1.5 ::J o

(I)

~ ^1.0 ....

o o a: 0.5

~ ^{... .}

~ ... .

\ ···A .... "

\ ""'''''' ''6.

Compori.on of Hybrid Solutions Using Classical and Fish~r Methods

ylxl' 213 + 2.[lx-tlYltldt 'olxl' 2/3. Ax' 0.1 A-Classical Method o -Fisher Method

\,"O--_____ ~ ~~~~~·_:~~·~~.~6 _::'''6 "" ...•

0.01----.----2r---~=T'3==4t=-=-=='t5>===.~~6 ----1==;~="'49 MAJOR CYCLE NUMBER

Figure 3-Comparison of hybrid solutions using classical and Fisher methods

Time histories of the kernel K(x,t) == x-t and the functions Yn(XI) for several major cycles are shown in Figure 4.

Example 2. The following equation was solved.

1

y(x)

==

1.5x - - 6 - + 7

f

(x-t)y(t) dt (8) To illustrate the effect of choice of initial conditions, the equation was solved once with yo (t) == 0 and once with Yo(t) == 2/3. The results are shown in Figure 5 for ~X == 0.01. It is evident that a poor choice of initial approximation may lengthen the convergence. process.

Example 3. Examples 1 and 2 used kernels which

146 Fall Joint Computer Conference, 1967

I

+1

Speed = 0.2 em

Isee

Time History of Fishers Solution to y(x) c 213 +

2'£CX-

^tly(t)dl

Yo (x) I: 2/3

--Figure 4-Time history of Fisher's solution

5 . 0 r - - - . - - . - - - . - - . . - - - - . - - . . - - - r - - - - r - - - - . ComparoMln of Hybrid Soluloons USIng Classical and Fisher Methods II: 4.0

o ylx)' 3x/2 - 716 +4(X-t)yltldt

Yo(I)'2/3, A.·O.OI

II:

~

.to -Classical Method o -Fisher Method ::!3.0

ct ::>

o (J) Yo(x)=O, Ax '0.01

0-Classical Method

~ 2.0 0-Fisher Method

(J) I-o o

II: I.

O.O,f---.---.:¥=-=~~~~=~""""''''''''''o===_'==o===a

o 3 4 5 8

MAJOR CYCLE NUM~R

Figure 5-Comparison of hybrid solutions using classical and Fisher methods

could ce considered functions of a single variable.

Consider now the equation

1

y(x)

+.r

^{e-xt yet) dt (9)}

yo (t) 0

The results are shown in Figure 6. Once again the superiority of Fisher's method is evident.

10.0Ir-""V---r---r--.---,----r--~--~-~

'~

~8.0 II:

II:

W

~6.0

~ o

(J)

~ 4.0 UI I-o o

···6>\

.... "

... >&,'

,~.~ ... .

" 'A.

Comparison of Hybrid Solutions Using Clatsical and Fisher Methods

yl.).'+£~-·ty(t)dt

Ya(x)'" AI' 0.01 6-Classical Method o -Fisher Method ...

'"l!I", ···.6 ..

", ···A

"''G, ··· ... 6.

II: 2.0 ^... . ... i!..

0.00~--:---:---:-..!...-..-~----"\:)...L·

_'_"_'-_--c:>-.L. _'-_'_--_-'0-..J....

_--._~_~.-_-~:;.~

^.

.::..:~.~::.-:.:--:.:5-(.

4 5 6 7 8 9

MAJOR CYCLE NUMBER

Figure 6-Comparison of hybrid solutions using classical and Fisher methods

Discuss!on of errors

The major errors which enter into the hybrid .solu-tions of integral equa.solu-tions are the following: (a) trun-cation errors (due to the fact that the functions have

been quantized), (b) AID and DI A conversion errors, (c) other analog computer errors, and (d) phase shifts due to digital exe~ution time.

Truncation errors arise from the quantization of the variables x and t in equation (1). In order to test the

Solution of Integral Equations by Hybrid Computation 147 importance of the quantization interval, Example 1

above was solved using 10 intervals (Ax=O.l) and 100 intervals (Ax=O.Ol). A comparison of the root sum squared errors for both interval sizes using the Fisher method is shown in Figure 7. It can be seen that con-vergence is speeded up by the choice of a smaller inter-val. After a sufficiently large number of major cycles,

Fi~ure 7-Comparison 'of hybrid solutions using Fisher method

The effect of quantization also enters into the inte-gration process since the approximating function Yn(t) is reconstructed from the samples Yn(Xt), i = 1,2, ... I. first-order reconstruction the errors are proportional to

(~tY. Second-order interpolation formulas will reduce the error to 0(At)3, but the additional computation re-quired to achieve it may not be justifiable.

Analog-to-digital and digital-to-analog conversion errors were extremely important' in the solution of the example problems. However, it should be noted that.

both the Fisher and Neumann techniques are stable processes as long as the solution to the problem is ana-lytically convergent (see Reference 4). Thus, random errors (which may enter the problem from the con-verter or multiplier inaccuracies) during anyone itera-tion (major cycle) will be corrected in subsequent iter-ations. Consequently, random errors delay the converg.;.

ence process and may also cause a final indeterminacy

region in the solution, as evidenced in the oscillation of Figure 7.

Alternate mechanization of kernel generation and multiplication were also investigated. The resulting ef-fects on solution accuracy are clearly a function of the quality and precision of available analog multipli-ers and function generators, as well as convmultipli-ersion given solution clearly depend on the particular digi-tal computer being utilized, comparisons between hybrid and all digital solutions are of interest. A speed up factor of 300 to 1 was obtained by using hybrid computation over digital computation, with comparable final accuracy.

Extension to other types of integral equations

The above discussion has been devoted entirely to equations of the Fredholm type. However, extension of the technique to many other types of equations is pos-sible. Consider, for example, the Volterra equation:

y(x) = f(x)

+

^A

.f

K(x,t) y(t) dt (11) This equation differs from the Fredholm equation in that the upper limit of integration is variable. The algo-rithms of Figure 1 and 2 are still applicable if the ker-nel is redefined such that

K(x, t) = 0 for t

>

x (12) A simple digitally controlled switch (needed to imple-ment (12» can ce used in conjunction with the Fred-holm equation program to solve Volterra type equa-tion ....

Wallman ^I and Fisher³ have also indicated possible extension of hybrid techniques to the solution of multi-dimensional integral equations, integrodifferential equations and certain more general functional equa-tions. Such extensions have yet to be proved in prac-tice.

CONCLUSION

Hybrid computation techniques, involving fast repeti-tive analog integration and function generation and digital storage 'and control, are well suited to the solu-tion of integral equasolu-tions. Hybrid techniques lead to a substantial reduct jon in solution time when compared to all-digital methods. Further, the examples solved in this study substantiate the faster convergence of Fish-er's iteration scheme, when contrasted with the classical

148 Fall Joint Computer Conference, 1967 Neumann technique. Extensions to other areas of ap-plication appear promising but remain to be tested.

ACKNOWLEDGMENT

The writers express their appreciation to Messrs. L. J.

Gaspard and T. Deklyen for their assistance in ob-taining a number of the computer solutions discussed in this paper.

REFERENCES H WALLMAN

An electronic integral transform computer and the prac-tical solution of integral equations

J. Franklin Inst. 250:45-61 July 1960

2 F B HILDEBRAND

Methods of applied mathematics

Prentice-Hall Inc Englewood Cliffs N.J. 1952 3 M E FISHER

On the continuous solution of integral equations by an electronic analogue

Proc Cambridge Phil Soc 53:162-173 1957 4 D B McKAY M E FISHER

A nalogue computing at ultra-high speed John Wiley & Sons Inc New York 1962 5 R TOMOVIC N PAREZANOVIC

Solving integral equations on a repetitive differential analyzer

IRE Trans on Elec Computers EC-9 503-506 Decem-ber 1960

Graphic CRT terminals-characteristics of commercially available equipment

by CARL MACHOVER Information Displays, Incorporated Mount Kisco, New York

INTRODUCTION

"Who needs another review of Graphic CRT Termi-nals when so many good ones have recently been published?"

A reasonable

question-After all, Adams Associates is now offering the

"Computer Display Review"! - in which they're trying to do for the display field what they have been do-ing so effectively for the Computer field (with their

"Computer Characteristics Quarterly"). The Air Force has just published their "Compendium of Visual Dis-playsH2. Harry Poole's recent book, "Fundamentals of Display Systems^Ha admirably meets the author's stated objective " . . . to provide fundamental data and illus-trate basic ~echniques used in the design and develop-ment of display systems". Excellent books covering the use of displays in specialized application areas, like J ames Howard's new book "Electronic Information Displays for Management"4, are now available. Several survey articles⁵'6 have recently appeared. Reasonably complete bibliographies of computer graphics litera-ture have been published, such as the selected bibilog-raphy appearing in Ronald L. Wiginton's paper

"Graphics and Speech Computer Input . and Output for Communications with Humans" inculded in "Com-puter Graphics/Utility - Production - Art". ⁷ These examples by no means exhaust the list-but they do emphasize the lead question "Why another one?"

The justification depends on two factors, I think.

First, there appears to be a need for user oriented-hardware based information. (There is, to be sure, also a need for user oriented-software based information-but that will have to be the subject of someone else's

paper.) .

Second, there is also the need, I believe, to define terms. The user encounters words and phrases like

"jitter" and "flicker-free" which are used to describe the performance and quality of a display system . . . but the exact meaning of the terms are frequently left undefined. This lack of definitions is not the result

of a mass conspiracy by display manufacturers - but instead reflects the absence of standards and defini-tions in the field. Incidentally, one of the Society for Information Displays goals over the next few years is to establish a concensus on standards and defini-tions. This paper will attempt to clarify some commonly used display terms, as a guide to user understanding.

Generally, the discussion will be oriented to com-mercially available equipment.

*

Table I is a repre-sentative list of manufacturers of commercial graphic CRT terminals.

TABLE I

Manufacturers of Commercially Available CRT Graphic Terminals

Bolt, Beranek & Newman, Inc. (BBN) Bunker-Ramo Corporation (BR) Control Data Corporation (CDC) Digital' Equipment Corporation (DEC) Ferranti, Limited

Information Displays, Inc. (IDI) Information International, Inc. (III)

International Business Machines Corporation (ffiM) International Telephone & Telegraph Corporation

(ITT)

Philco-Ford Corporation.

Sanders Associates

Scientific Data Sy'stems, Inc. (SDS) Stromberg-Carlson Corporation (SC)

Systems Engineering Laboratories, Inc. (SEL) Tasker Instruments Corporation

UNIVAC

*This is a good place to start defining terms. By "commercially available" I mean relatively standard products regularly of-fered for sale - and meant to be used in an industrial rather than military environment. Admittedly, this is arbitrary (but, I hope, not capricious) and eliminates from considera-tion the numerous militarized command and control systems - and the fine display systems developed in University and Industrial Laboratories.

149

150 Fall Joint Computer Conference, 1967 Photographs of representative terminals are included in Figure 1.

Section 2 of this paper briefly reviews the con-stituent elements of a typical Graphic CRT terminal.

Prices of these terminals range from approximately

$20,000 to $280,000. Generally, as discussed in an earlier article,⁸ price and performance are related.

However, because there are many factors that affect performance, and the importance. of each factor de-pends upon the application, the cost-performance re-lationship is not a simple one. Section 3 of this paper is a discussion of various factors which affect perform-ance. analog deflection and .video (intensification) circuitry, plus . . .

A Display Generator which contains several types of function generators to produce graphic elements, plus (occasionally) . . .

The CRT may be deflected electrostatically, or elec-tromagnetically or by" a combination of both. Com-monly used configurations are illustrated in Figure 3.

Table II lists the deflection system used by various manufacturers.

Most commercially available Graphic CRT Con-soles use random positioning to produce a picture. terminal. Figure 4 illustrates the difference between random positioning and raster scan.

Tubes are available with a variety of conventional phosphors offering a choice of color as well as per-sistence. Bolt Beraneck and Newman Teleputer Sys-tem uses a CRT with storage screen.

So that an optically projected image can be com-bined with the electronically generated image, CRT's in terminals supplied by Bunker-Ramo and Strom-berg-Carlson have a window through which a picture

TABLE

n

Deflection Systems Used by Various Manufacturers Dual Deflection

Electro-Manu- Electro- static

+

^Magnetic

+

facturer static Magnetic Magnetic Magnetic

BBN x

The Display Generator contains the circuitry which interprets the computer digital data word and trans-lates it into analog signals to generate the graphic elements on the CRT. The Display Generator is es-sentially just a digital-to-analog converter (0/ A). Typi-cally, the Display Generator will include such function generators as a character generator, a vector (line) generator, a circle generator, position generators (con-ventional D / A converters) and a dot generator.

All of the· graphic elements produced by the

All of the· graphic elements produced by the

Im Dokument FALL JOINT (Seite 129-138)