LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAICN 510.84 I&63c AUG. 51976 1 he person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN ■ fell? 1". JAN. • S APR 8 I » > < LOAN :; SCfc OCT EHO] ,-rjo PHOTO REPRODUCTION OCT 22JECU PHOTO REPRODUCTION NOV 1 6 fttf REPRODUCED* OCT ti?n w 2 7 REC'B I BBJKDBHOODC 2 3RECD L161 — O-1006 Digitized by the Internet Archive in 2012 with funding from University of Illinois Urbana-Champaign http://archive.org/details/computationalmat64belf ENGINEERING LIBRARY UNIVERSITY OF ILLINOIS UR8ANA, ILLINOIS iced Computation UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS 61801 CAC Document No. 6k COMPUTATIONAL MATHEMATICS ABSTRACTS Edited By Geneva Belford, Jonathan Lermit, George Purdy, and Ahmed Saraeh February 1973 CAC Document No. 6U Comput at ional Mathematics Abstracts Edited by Geneva Bel ford Jonathan Lermit George Purdy Ahmed Sameh Applied Mathematics Group Center for Advanced Computation University of Illinois at Urb ana-Champaign Urbana, Illinois 6l801 February 1973 This work was supported in part by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U.S. Army Research Office-Durham under Contract No. DAHC0U-72-C-0001. TABLE OF CONTENTS Pace PREFACE 1 ERROR ANALYSIS 2 FUNCTION EVALUATION AND COMPLEXITY THEORY 3 APPROXIMATION 6 LINEAR ALGEBRA 9 NONLINEAR EQUATIONS IT QUADRATURE 18 ORDINARY DIFFERENTIAL EQUATIONS 19 PARTIAL DIFFERENTIAL EQUATIONS 21 INTEGRAL EQUATIONS 23 INTERVAL AND SIGNIFICANT DIGIT ARITHMETIC 23 OPTIMIZATION 2U GRAPH ALGORITHMS 27 KEY TO REPORT SOURCES 30 PREFACE In our efforts to keep abreast of developments in computa- tional mathematics, we have found it useful to make this listing of abstracts of the many departmental reports which we have accumulated during the past year. The abstracts are largely those provided with the documents; where no abstract was given we used selected sentences from the intro- duction. Occasionally we condensed an abstract which seemed too long for our purposes. Sources of the reports are identified by letter codes prefacing the report number. A key to these codes is given at the end of the listing. We wish to thank Nancy Freece not only for her excellent and rapid job of typing but also for doing a considerable amount of the work of hunting through the documents to assemble the information listed. ERROR ANALYSIS Algorithms to Reveal Properties of Floating-Point Arithmetic by Michael A. Malcolm Two algorithms are presented in the form of FORTRAN sub- routines. Each subroutine computes the radix and number of digits of the floating-point numbers and whether rounding or chopping is done by the machine on which it is run. The methods are shown to work on any "reasonable" floating-point computer. STAN-CS-T1-211 March 1971 8 Pages Program Incl. On the Precision Attainable with Various Floating-Point Number Systems by R. P. Brent For scientific computations on a digital computer the set of real numbers is usually approximated by a finite set F of "floating- point numbers". We compare the numerical accuracy possible with different choices of F having approximately the same range and requir- ing the same wordlength. In particular, we compare different choices of base (or radix) with the usual floating-point systems. The emphasis is on the choice of F, not on the details of the number representation or the arithmetic, but both rounded and truncated arithmetic are consi- dered. Theoretical results are given, and some simulations of typical floating-point computations (forming sums, solving systems of linear equations, finding eigenvalues) are described. If the leading fraction bit of a normalized base-2 number is not stored explicitly (saving a bit), and the criterion is to minimize the mean square roundoff error, then base 2 is best. If unnormalized numbers are allowed, so the first bit must be stored explicitly, then base h (or sometimes base 8) is the best of the usual systems. YORK-RC 3751 February 1972 28 Pages Performance Statistics for the FORTRAN IV (H) and PL/1 (Version 5) Libraries in IBM OS/360 Release 18 by Kenneth E. Hills trom The computational subroutine libraries associated with the FORTRAN IV (H) and PL/I (F) (Version 5) compilers of OS Release 18 have been tested. The testing techniques are described and accuracy and timing statistics are presented; proper operation under error conditions was verified. ANL-7666 August 1970 70 Pages Program Incl. FUNCTION EVALUATION AND COMPLEXITY THEORY On The Number of Multiplications for the Evaluation of a Polynomial and Some of its Derivatives by Mary Shaw and J. F. Traub Some recent work in computational complexity has dealt with the number of arithmetic operations needed to evaluate a polynomial or a polynomial and its first derivative. Here we consider the evaluation of a polynomial and its first m derivatives and, in parti- cular, the calculation of all the derivatives. CMU - No # August 1972 18 Pages Computational Complexity of Iterative Processes by J. F. Traub The theory of optimal algorithmic processes is part of computational complexity. This paper deals with analytic computa- tional complexity . The relation between the goodness of an iteration algorithm and its new function evaluation and memory requirements are analyzed. A new conjecture is stated. CMU-CS-71-105 October 1971 25 Pages On the Additions Necessary to Compute Certain Functions by David G. Kirkpatrick We describe low level or functional complexity, fitting it into the general framework of computational complexity and presenting a number of results in arithmetic complexity. We put forward a new notion of independence, called rational independence, which applies to the terms of a computed expression. We are able to show that this notion serves as a measure for the number of additions needed to compute a function. TOR-Tech. Rpt . 39 February 1972 75 Pages [M.S. Thesis] Some Results in the Study of Algorithms by J. Ian Munro Functions such as polynomials, matrix products, and digraph transitive closures are in practice computed very often. In recent years, considerable attention has been turned to efficient computation of such functions, with the hope of minimizing the time needed to compute them on a general purpose digital computer as we now know it. In this thesis several new algorithms are given for the evaluation of such functions, and in some cases these are shown to be optimal under a reasonable model of computation. It is shown that if a polynomial, or set of polynomials, is to be evaluated at a large number of points, and these points are all given at the same time, then the number of arithmetic operations needed per "polynomial-point" is less than linear in the degree of the polynomial. An algorithm requiring work of the order of max (n log n,e) is given for the determination of the strongly connected components of an n-node, e-edge digraph. Attention is turned to problems of computing arithmetic functions under a model of - computation in which a large degree of parallelism is permitted. New optimal and almost optimal algorithms are presented for several such computations. TOR-Tech. Rpt . 32 October 1971 102 Pages Ihesis Application of Continued Fractions for Fast Evaluation of Certain Functions on a Digital Computer by Amnon Bracha The purpose of this paper is to develop a method for evalua- tion of certain elementary functions on a digital computer by the use of continued fractions. The time required for this evaluation is drasti- cally reduced by using "short" operations like shift and add, instead of multiplications. Functional consistency is the most important factor that allows the expansion of a function into a continued fraction. Several cases are discussed and in particular the solution of the quadra- tic equation is discussed in more detail to demonstrate the convergence of the method. UIUC-R- 72-510 March 1972 29 Pages Bounds on Polynomial Evaluation Algorithms by Larry Joseph Stockmeyer The purpose of this work is to investigate the number of arithmetic operations required by algorithms which evaluate polynomials. Previous results show that a polynomial of degree n requires at least n/2 multiplication/divisions and at least n addition/subtractions for its evaluation if the coefficients of the polynomial are suitably independent irrational numbers. However, the coefficients of any poly- nomial that would be evaluated in practice are represented only to a finite accuracy and are therefore rational numbers. The above results are extended to show that the same lower bounds hold for almost all rational polynomials if the polynomial is being evaluated efficiently. Another lower bound result is given that shows that almost all rational polynomials of degree n require at least vn multiplication/divisions for their evaluation by any algorithm, efficient or not. Several algorithms are presented which can in theory evaluate any rational polynomial using 0(/n) multiplications and many additions. While of no practical use for rational polynomials in general, these algorithms do turn out to give methods for evaluating a polynomial at a matrix argument which are more efficient than pre- vious methods. MAC TR-98 April 1972 5*+ Pages APPROXIMATION Variational Study of Nonlinear Spline Curves by E. H. Lee and G. E. Forsythe This is an exposition of the variational and differential properties of nonlinear spline curves, based on the Euler-Bernoulli theory for the bending of thin beams or elastica. For both open and closed splines through prescribed nodal points in the euclidean plane, various types of nodal constraints are considered, and the correspond- ing algebraic and differential equations relating curvature, angle, arc length, and tangential force are derived in a simple manner. The results for closed splines are apparently new, and they cannot be derived by the consideration of a constrained conservative system. There is a survey of the scanty recent literature. STAN-CS-229-71 August 1971 26 Pages An Algorithm for Fitting Related Sets of Straight-Line Data by Geneva G. Bel ford Many physical experiments give rise to sets of curves related by the requirement that, although certain of the curve parameters may vary from curve to curve, others should be the same for all of the curves. To get the "best" values of the common parameters, one would like to fit all of the curves simultaneously by the appropriate theore- tical expressions. This paper deals with this problem, presenting an algorithm for its solution, in the case that the curves are straight lines with common slope and "best" fit is defined in the uniform (or minimax) sense. CAC No. 5U December 1972 12 Pages An Implementation of the Remes Algorithm for Minimax Approximation by Roland Olofsson The ALGOL procedure for the Remes algorithm given by G. H. Golub and L. B. Smith (Comm ACM Ik (1971 ), 737-7^6 Alg klk) has been implemented and tested on a CD 3200 computer. Some examples of ill conditioned generalized polynomials have been used to test the algorithm. A comparison is also made to another program based on the Polya algorithm. UMlNF-20.72 April 1972 Ik Pages On the Existence and Characterization of Minimal Projections by P. D. Morris and E. W. Cheney A projection from a normed linear space X onto a subspace Y is a bounded linear operation P: X -*■ Y having the property that Py = y for all y e Y. Projections play an important role in numerical analysis and approximation theory. The use of projections is based upon the acceptance of Px as an approximation of x in the subspace Y. The quality of approximations produced by P depends upon ||p|| and ||I-P||, The present work is devoted mainly to the characterization problem of minimal projections. CNA-37 January 1972 32 Pages Stability Properties of Trigonometric Interpolating Operators by P. D. Morris and E. W. Cheney We consider the familiar process of trigonometric interpola- tion with 2n+l equally-spaced nodes. This process is interpreted here as a linear projection operator P acting on the space C of all real, continuous, 27r-periodic functions. The range of this projection P is the subspace n of all trigonometric polynomials of order n. Many other useful projections exist which map C onto II, and attention here is focused on their extremal properties. CNA-51 August 1972 20 Pages Extremal Properties of Approximation Operators by K. H. Price and E. W. Cheney In 1910, de La Vallee Poussin published some researches on the following approximation problem: given the continuous function x = x(t) defined on the interval [-1,1], and given n+1 points t ,...,t in [-1,1], find the polynomial p of degree < n which minimizes max |x(t.) - p(t.)|. 0 ♦ V <°> ♦ v (l > (n+1) _ (n+ (n) (n-l) . . u = pGu + qu + ru + w, n = 1, 2, ...., . CNA-52 August 1972 29 Pages Error Bounds for Approximate Subspaces of Closed Linear Operators in Hilbert Space by G. W. Stewart It is well known to specialists in matrix computations that the eigenvectors of a matrix corresponding to a set of poorly separated eigenvalues are quite sensitive to perturbations in the elements of the matrix. It is also known that for Hermitian matrices the invariant sub- space corresponding to a cluster of eigenvalues is insensitive to such perturbations. It is the object of this paper to extend such results to nonnormal matrices and, more generally, to closed operators in Hilbert space CNA-1+ October 1970 25 Pages 15 On the Numerical Properties of an Iteration for Computing the Moore- Penrose Generalized Inverse by G. W. Stewart In this paper some of the numerical problems associated with computing the generalized inverse of a matrix are discussed and illustrated by a detailed analysis of an iteration of Ben-Israel and Cohen. CNA-12 March 1971 27 Pages On the Solution of Large Systems of Linear Algebraic Equations with Sparse, Positive Definite Matrices by David M. Young The object of this paper is to discuss the current status of iterative methods for solving large systems of linear algebraic equations. Primary emphasis is on those systems involving sparse matrices where iterative methods appear more attractive than direct methods. CNA-55 August 1972 U5 Pages The Solution of Large Systems of Linear Algebraic Equations by Iterative Methods by David M. Young In this paper we provide a summary of some of the available iterative methods for solving large systems of linear algebraic equa- tions. We are primarily concerned with the case where the matrix of the system is "sparse", i.e., has only a few non-zero elements in each row. Systems of this kind typically arise in the solution by finite difference methods of elliptic partial differential equations: CNA-19 May 1971 23 Pages An Algorithm for the Generalized Matrix Eigenvalue Problem Ax = ABx by C. B. Moler (Stanford University) and G. W. Stewart A new method, called the QZ algorithm, is presented for the solution of the matrix eigenvalue problem Ax = ABx with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the QR algorithm, and reduces to it when B = I. A Fortran program and some illustrative examples are included. CNA-32 October 1971 50 Pages and STAN-CS-2U2-71 Programs Incl. 16 A Generalization of the LR Algorithm to Solve Ax = ABx by Linda Kaufman In this paper, we will present and analyze an algorithm for finding x and A such that Ax = ABx where A and B are n x n matrices. The algorithm does not require matrix inversion, and may he used when either or "both matrices are singular. Our method is a generalization of Rutishauser 's LR method for the standard eigenvalue problem Ax = Ax and closely resembles the QZ algorithm given by Moler and Stewart for the generalized problem given above. Unlike the QZ algorithm, which uses orthogonal transformations, our method, the LZ algorithm, uses elementary transformations. When either A or B is complex, our method should be more efficient. STAN-CS- 72-276 April 1972 72 Pages Program Incl. Richardson's Non- Stationary Matrix Iterative Procedure by R. S. Anderssen and G. H. Golub Because of its simplicity, Richardson's non-stationary itera- tive scheme is a potentially powerful method for the solution of (linear) operator equations. However, its general application has more or less been blocked by (a) the problem of constructing polynomials, which deviate least from zero on the spectrum of the given operator, and which are required for the determination of the iteration parameters of the non-stationary method, and (b) the instability of this scheme with respect to rounding error effects. STAN-CS- 72-30U August 1972 73 Pages Bi diagonal iz at ion of Matrices and Solution of Linear Equations by C. C. Paige An algorithm given by Golub and Kahan for reducing a general matrix to bidiagonal form is shown to be very important for large sparse matrices. The singular values of the matrix are those of the bidiagonal form, and these can be easily computed. The bidiagonal iz at ion algorithm is shown to be the basis of important methods for solving the linear least squares problem for large sparse matrices. Eigenvalues of certain 2-cyclic matrices can also be efficiently computed using this bidiagonalization. STAN-CS-72-295 June 1972 28 Pages An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations by Harold S. Stone Tridiagonal linear systems of equations can be solved on conventional serial machine in a time proportional to N, where N is the number of equations. The conventional algorithms do not lend them- IT selves directly to parallel computation on computers of the ILLIAC IV class, in the sense that they appear to be inherently serial. An efficient parallel algorithm is presented in which computation time grows as log N. The algorithm is based on recursive doubling solutions of linear recurrence relations, and can be used to solve recurrence relations of all orders. STAN-CS-71-251 December 1971 21 Pages NONLINEAR EQUATIONS Error Bounds, Based Upon Gerschgorin's Theorems for the Zeros of a Polynomial by Brian Thomas Smith Suppose L distinct points z ,z ,...z are given in the complex plane and associated with each point z is a positive integer M , the 'multiplicity' of z, ; These points z , with their multiplicities M , are supposed to approximate the zeros of a monic polynomial P(z). The object of this thesis is to measure how well a set of such points approximates the zeros of P(z); that is, we want to determine how close each point z is to its subset of M zeros. TOR Tech. Rpt . 9 J uly 1969 137 Pages [Ph.D. Thesis] An Algorithm for the Solution of a Quadratic Equation using Continued Fractions by Kishor Shridharbhai Trivedi This is an effort to investigate representations of numbers other than positional notation for computer arithmetic. Using continued fraction representation of numbers, an algorithm to solve a limited class of quadratics has been developed. This algorithm is suitable for hardware implementation and is reasonably efficient. Feasibility of constructing an arithmetic unit with continued fraction representation depends on discovery of many more such useful algorithms which can share the same hardware. UlUC-R-72-525 June 1972 62 Pages Program [M.S. Thesis] A Method for Solving Polynomial Equations by Continued Fractions by Amnon Bracha A method for the approximation of all the real roots of an n-order polynomial equation is developed. It is assumed that intervals containing the solutions are known. Bilinear transformations are used to approximate the solution. Convergence is achieved. UIUC-R-72-521 July 1972 20 Pages l8 Kublanovskayas algorithm for solving the nonlinear eigenvalue problem by Per Lindstrom In the present contribution an iterative algorithm proposed by B. N. Kublanovskaya [Dokl Akad Nauk 188, lOOU-1005 0-969) compare also SIAM Journal of Num. Analysis 7, 532-537 0-970)) for the solution of the nonlinear eigenvalue problem T(A]x = Is tested numerically. T(A) is an n*n real or complex matrix that depends analytically on the eigenvalue parameter A. It is our aim to study different possible algorithms for this problem in order to assess their practical value, and this is a part of this investigation. UMINF-5.71 August 1971 lh Pages Algorithms for the nonlinear eigenvalue problem by Axel Ruhe The following nonlinear eigenvalue problem is studied: Let T(A) be an nxn matrix, whose elements are analytical functions of the complex number A. Seek A and vectors x and y, such that T(A)x = and y^T(A) = 0. Several algorithms for the numerical solution of this problem are studied. These algorithms are extensions of algorithms for the linear eigenvalue problem such as inverse iteration and the QR algorithm, and algorithms that reduce the nonlinear problem into a sequence of linear problems. Numerical tests, performed in order to compare the different algorithms, are reported, and a few numerical examples illustrating their behaviour are given. An ALGOL program for one of the algorithms is given as an appendix. UMINF-7-71 August 1971 30 Pages Program Quadrature Multi-Dimensional Quadrature Formulae by P. Keast Two classes of formulae, for numerical quadrature in several variables, are considered. Comparisons are made on the basis of accuracy and efficiency of the various formulae used, and on the basis of ease of obtaining similar formulae in high dimension. A new technique for obtaining a particular type of quadrature rule is developed. Several numerical examples are given. TOR Tech. Rpt. 1+0 February 1972 78 Pages Program 19 ORDINARY DIFFERENTIAL EQUATIONS Studies in the Numerical Solution of Stiff Ordinary Differential Equations by Wayne En right A class of second derivative multi-step formulas is developed. The stability of these formulas is investigated and they are shown to be suitable for stiff equations. An implementation of a variable-order, variable-step method based on this class of formulas is described and some numerical results are presented. The reliability of methods suitable for stiff equations is also investigated. The condition of a-acceptability is introduced and it is shown that if a method satisfies this condition, then for any member of a particular class of stiff systems, the numerical solution can be guaran- teed to lie on the exact solution of a slightly perturbed system. Methods based on the second derivative formulas mentioned above are shown to be a-acceptable. Numerical comparisons of methods in terms of both reliability and efficiency are considered and the background is established for a comprehensive comparison. TOR Tech. Rpt . h6 October 1972 82 Pages Program [Ph.D. Thesis] On the Identification of Multi-Output Linear Time-Invariant and Periodic Dynamic Systems by Ahmed H. Sameh and Walter L. Heimerdinger In this paper we describe an efficient computational algorithm for estimating the coefficients of the characteristic polynomial of a linear time-invariant multi-output dynamic system, using only output observations, for qualitative analysis of the transition matrix or for evaluating its eigenvalues. We also give some computational results of the identification of those systems using the Ho-Kalman approach. Furthermore, an identification scheme for high-frequency periodic systems of unknown periods is described in detail. CAC No. 53 November 1972 29 Pages Tchebycheffian Multistep Methods for Ordinary Differential Equations by Tom Lyche The purpose of this paper is to extend some of the theory for linear multistep methods to include steps ize-dependent coefficients. In particular we treat the case where we demand exact integration of a given set of linearly independent functions. CNA-16 April 1971 20 Pages 20 Numerical Methods for the Identification of Differential Equations by Raymond Jonathan Lermit This dissertation considers computational methods designed to aid in mathematical model building. Specifically, it discusses methods of determining ordinary differential equations given their solution in the form of observed data. Since the problem cannot be solved in this generality, it is necessary to supply equations containing arbitrary functions. The problem is then to find these functions given the solution of the equations. In order to be amenable to computer solution, a discretization of the functions as a linear sum of a given ortho- normal set is necessary. The problem thus reduces to one of find- ing a finite number of parameters. The solution technique is to find that function which produces a solution most closely approximating the observed data. It is thus a problem in the minimization of nonlinear functionals and may be solved by iterative methods. Modifications required to ensure that the functional is convex, thus guaranteeing a global minimum solution, are discussed. Different algorithms considered for carrying out the minimization include the generalization of Newton's method and variants of it which are more economical in computer time, especially the Conjugate Gradient method. All of these methods require derivatives which are derived automatically from the original equation using formal algebraic manipulation. The different methods are compared for rates of convergence and amount of calculation required at each iteration. Two examples are included. The effect of introducing random errors into the data to simulate observational errors, and how this may alter the convergence rates, is also discussed. CAC No. h9 June 1972 8U Pages Program [Ph.D. Thesis] 21 PARTIAL DIFFERENTIAL EQUATIONS Finite-Element Galerkin Method for Mixed Initial -Boundary Value Problems in Elasticity Theory by Hiroshi Fujii The Finite-Element Galerkin method (FEG method) has been recognized as a very powerful tool for numerical solution of boundary value problems of partial differential equations. In this paper, we treat the application of the FEG method to vibration problems (i.e., mixed initial -boundary value problems) in linear elasticity theory. We introduce some special versions of FEG schemes for approximating the solution of our problem, including schemes of explicit type as well as an implicit scheme and a continuous -time scheme. CNA-3 1 * October 1971 TO Pages Automatic Solutions of Partial Differential Equations by Leonard Andrew Lars en The problem of the present paper is to look for an automatic method that can be applied to some subset of partial differential equa- tions. The primary value of a program involving an automatic method would be to provide the person who needs only a few runs with a parti- cular type of equation the opportunity to find a solution, within accepta- ble tolerances, without having to write and debug a specialized program. UIUC-R-T2-5^6 October 1972 128 Pages Program [Ph.D. Thesis] An Application of Semi- Iterative and Second-Degree Symmetric Successive Overrelaxat ion Iterative Methods by Tran Phien The goal of this paper is to study the effectiveness of semi- iterative methods and second-degree methods based on the symmetric successive overrelaxat ion method for solving a certain class of linear systems. The linear systems correspond to the solution by a finite difference method of a boundary value problem associated with a particular self-adjoint elliptic partial differential equation. We shall use a semi- iterative method based on the symmetric successive overrelaxat ion (SSOR) method. The resulting method is called the SSOR-SI method. We shall also use the SSOR-SD method, a stationary second-degree method based on the SSOR method. CNA-U2 May 1972 h& Pages [M.S. Thesis] 22 Numerical Implementation of the Schwarz Alternating Procedure for Elliptic Partial Differential Equations "by David Ross Stoutemyer This thesis describes numerical implementation of the Schwarz and Neumann alternating procedures for the solution of the Laplace-Dirichlet problem on the union of two disks, the intersection of two disks, an arbitrary quadrilateral, and the union of two spheres. All of the above examples lead to a pair of coupled Fredholm integral equations of the second kind with singular kernels and singular low-order derivatives of the solution at corners of the region. These singularities are overcome by a change of variable together with special spacing of the abscissas and extrapolation to the limit. These methods are suitable for more general boundary conditions, more general partial differential equations, and more general geometrical configurations. STAN-CS-72-283 May 1972 131 Pages Program [Thesis] Use of Fast Direct Methods for the Efficient Numerical Solution of Non separable Elliptic Equations by Paul Concus and Gene H. Golub We study an iterative technique for the numerical solution of strongly elliptic equations of divergence form in two dimensions with Dirichlet boundary conditions on a rectangle. The technique is based on the repeated solution by a fast direct method of a discrete Helmholtz equation on a uniform rectangular mesh. The problem is suitably scaled before iteration, and Chebyshev acceleration is applied to improve conver- gence. We show that convergence can be exceedingly rapid and independent of mesh size for smooth coefficients. Extensions to other boundary condi- tions, other equations, and irregular mesh spacings are discussed, and the performance of the technique is illustrated with numerical examples. STAN-CS- 72-278 April 1972 39 Pages Numerical Solution of First-Order Hyperbolic Systems of Partial Differential Equations by Sunil K. Pal This work develops two new finite-difference schemes - an explicit scheme and an implicit scheme - for numerical solution of first- order hyperbolic systems of partial differential equations in any number of space variables. TOR Tech. Rpt . 13 1969 9h Pages [Ph.D. Thesis] 23 INTEGRAL EQUATIONS Methods for the Numerical Solution of Integral Equations of the Second Kind by David Blair Coldrick A detailed analysis of the quadrature method is given from several points of view, viz. those of Kantorovich, Buckner, and the Mysovskih-Brakhage-Anselone development. It is shown that the technique of deferred approach to the limit is valid under fairly general circum- stances. An error analysis of projection (and "shifted" project ion )methods is given. A class of degenerate kernel methods is proposed, and is compared to closely related projection methods in terms of error bounds and ease of implementation. Finally, a discussion of the problem of (weak) singularities is presented. The application of projection methods to such equations is mentioned. The technique of product integration is applied to this problem, and a fairly general convergence statement is established. For singularities the kernel k(s, t) along the line s = t, a generalization of Kantorovich's method with a higher order of conver- gence is proposed, and a theorem to this effect is proved. TOR Tech. Rpt . h5 October 1972 152 Pages [Ph.D. Thesis] INTERVAL AND SIGNIFICANT DIGIT ARITHMETIC Implementation of Basic Software for Significant Digit Arithmetic by Steven See Sun Lai This report is concerned with the implementation of significant digit arithmetic using the unnormalized arithmetic of Metropolis and Ashenhurst. One of the primary objectives was to design the basic soft- ware modules for inclusion into the OL/2 array language; however, these modules are written in assembly language and therefore are adaptable to other software systems for the IBM 360/370 machines. A discussion of significance arithmetic, including the nontrivial problem of input/output, is presented. Examples are provided in Appendix A. UIUC-R- 72-530 June 1972 l6 Pages [Thesis] Program Incl A Univac 1108 Program for Obtaining Rigorous Error Estimates for Approxi- mate Solutions of Systems of Equations by Dennis Kuba and L. B. Rail A UNIVAC 1108 computer program which obtains rigorous interval error bounds for approximate solutions of finite systems of nonlinear equations is described in this report. Since the coefficients of the original system may take on interval values, the error bounds obtained include the contributions of truncation error for Newton's method, round-off error, and possible errors in the coefficients of the given system of equations MRC Tech. Rpt. 1168 January 1972 155 Pages Program Incl. 2k OPTIMIZATION Large-Scale Linear Programming Using the Cholesky Factorization "by M. A. Saunders A variation of the revised simplex method is proposed for solving a standard linear programming problem. The method is derived from an algorithm recently proposed by Gill and Murray, and is based upon the orthogonal factorization B = LQ or, equivalently, upon the Cholesky factorization BB = LL where B is the usual square basis, L is the lower triangular and Q is orthogonal. We wish to retain the favorable numerical properties of the orthogonal factorization, while extending the work of Gill and Murray to the case of linear programs which are both large and sparse. The principal property exploited is that the Cholesky factor L depends only on which variables are in the basis, and not upon the order in which they happen to enter. A preliminary ordering of the rows of the full data matrix therefore promises to ensure that L will remain sparse throughout the iterations of the simplex method. An initial (in-core) version of the algorithm has been imple- mented in Algol ¥ on the IBM 360/91 and tested on several medium-scale problems from industry (up to 930 constraints). While performance has not been especially good on problems of high density, the method does appear to be efficient on problems which are very sparse, and on structured problems which have either generalized upper bounding, block-angular, or staircase form. STAN-CS-72-252 January 1972 60 Pages Product Form of the Cholesky Factorization for Large-Scale Linear Programming by Michael A. Saunders A variation of Gill and Murray's version of the revised sim- plex algorithm is proposed, using the Cholesky factorization BB 1 = LDL where B is the usual basis, D is diagonal and L is unit lower triangular. It is shown that during change of basis L may be updated in product form. As with standard methods using the product form of inverse, this allows use of sequential storage devices for accumulating updates to L. In addition the favorable numerical properties of Gill and Murray's algorithm are retained. Close attention is given to efficient out-of-core implementation. In the case of large-scale block-angular problems, the updates to L will remain very sparse for all iterations. STAN-CS-72-301 August 1972 38 Pages 25 The Differentiation of Pseudoin verses and Nonlinear Least Squares Problems Whose Variables Separate by G. H. Golub and V. Pereyra For given data (t . , y.), i = 1, . . . , m, we consider the least squares fit- of nonlinear models of the form n F(a, a; t) = I g.(a) 0.(a; t), a eflf, a eR, . ~ j=l J ~ J For this purpose we study the minimization of the nonlinear functional m 2 r(a, a.) = I (y. - F(a, a, t.)) . i=l 1 1 It is shown that by defining the matrix{$(a) }. . = .(a; t.) and the + ~ i,J \] ~ i modified functional r (a) = ||y - $(a)$ (a) y||^ , it is possible to optimize first with respect to the Parameters a, and then to obtain, a posteriori, the optimal parameters a. The matrix $ + (g) is the Moore- Penrose generalized inverse of $(ot), and we develop formulas for its Frechet derivative under the hypothesis that $(a) is of constant (though not necessarily full) rank. From these formulas we readily obtain the derivatives of the orthogonal projectors associated with $(a), and also that of the functional r 9 (a). Detailed algorithms are presented which make extensive use of well-known reliable linear least squares techniques, and numerical results and comparisons are given. STAN-CS- 72-261 February 1972 k9 Pages Program Incl The method of conjugate gradients used in iverse iteration by Axel Ruhe and Torbjorn Wiberg An algorithm is devised that improves an eigenvector approxi- mation corresponding to the largest (or smallest) eigenvalue of a large and sparse svmmetric matrix. It solves the linear systems that arise in inverse iteration by means of the c-g algorithm. Stopping criteria are developed which insure an accurate result, and in many cases give convergence after a small number of c-g steps. UMINF-22.72 April 1972 21 Pages Partial Analysis of an Algorithm for Minimizing Functions Based on a Homogenous Model by James ¥. Daniel Jacobson and Oksman recently presented a method which they claimed minimized a homogeneous function of arbitrary degree in finitely many steps and which was globally convergent for a large class of general functions. We point out here errors in the proofs of these results, suggest some modifications in the method so that the global convergence result is valid, and we give some examples to show the computational effects of these changes. CNA-15 April 1971 lU Pages 26 Convergent Step Sizes for Curvilinear-Path Methods of Minimization by James W. Daniel When one seeks a point minimizing a function f over a set C — which may be the whole space — one often moves from one approximate solution x to another x ., = x +tp by searching along a ray n n+1 n n n x + tp , where the step-size t must be chosen. More generally one n n n can consider moving along a curvilinear path x + p (x ,t ) . The n n n present paper shows how the usual step-size algorithms can be used with this general approach. CNA-29 July 1971 2k Pages Nonlinear Least Squares Without Derivatives: An Application of the QR Matrix Decomposition by Richard H. Bartels A new algorithm is proposed for minimizing sums of squares of nonlinear functions of several variables without the use of deriva- tives. The algorithm is constructed using a multivariate secant technique (Broyden rank-one method) to approximate the Jacobian of the summands, and this Jacobian is used to drive the Levengerg-Marquardt iteration. The Jacobian is kept and updated in QR decomposition, and the linear least squares subproglems which arise at each iteration cycle are solved by Golub's method. A FORTRAN program and test results are offered. It is shown that the algorithm accomodates linear equality con- straints with ease. Remarks are made concerning the treatment of rank deficiencies in Golub's method with respect to algorithms of the type Presented. CNA-UU April 1972 hi Pages Program Incl. Global Convergence for Newton Methods in Mathematical Programming by James W. Daniel In constrained optimization problems in mathematical program- ming one wants to minimize a functional f(x) over a given set C. If, at an approximate solution x , one replaces f(x) by its Taylor series expansion through quadratic terms at x and denotes by x the minimiz- ing point for this over C, one has a direct analogue of Newton's method, The local convergence of this has been previously analyzed; here we give global convergence results for this and the similar algorithm in which the constraint set C is also linearized at each step. CNA-i+8 June 1972 10 Pages 27 Newton's Method for Nonlinear Inequalities by James W. Daniel A Newton-type algorithm has "been presented elsewhere for solving non-linear inequalities of the form f(x) ^ 0, and quadratic convergence has been proved under very strong hypotheses. In this paper we show that the same results hold under a considerable weaken- ing of the hypotheses. CNA-53 August 1972 10 Pages GRAPH ALGORITHMS Graph Isomorphism by Derek Gordon Corneil A procedure for determining whether two graphs are isomorphic is described. During the procedure, from any given graph two graphs, the representative graph and the reordered graph, are derived. The time required to determine both derived graphs depends on a power of n, the order of the given graph. This power is a function of an adjacency property known as the strong regularity of the given graph. For graphs that do not contain a strongly regular transitive subgraph, the power is, at worst, five. All given timing estimates for graphs that do not contain a strongly regular transitive subgraph are confirmed. The algorithm has been programmed and in the implemented version one of the following four messages will come out: (l) The graphs are isomorphic. (2) The graphs are not isomorphic. (3) The representative graphs are identical; the reordered graphs are not identical; hence, these graphs form a counter- example to the conjecture. (k) The graphs contain a 3-strongly regular subgraph. TOR Tech. Rpt . 18 April 1970 [Ph.D. Thesis] Program An Algorithm to Determine the Chromatic Number of a Graph by Barry Graham A heuristic algorithm for the determination of the chromatic number of a finite graph is presented. This algorithm is based on Zykov's theorem for chromatic polynomials and extensive empirical tests show that it is the best algorithm available. Christofides ' algorithm for the determination of chromatic number is described and is used in the compari- son tests. An Algol- W coding of both algorithms is included in the appendix. TOR Tech. Rpt. ^7 November 1972 82 Pages [M.S. Thesis] Program 28 Algorithms for Finding Cliques of a Graph by Gordon D. Mulligan Various methods for determining cliques in undirected graphs are presented and analyzed. Testing schemes to compare the methods on graphs with a maximum number of cliques and on graphs that attempt to represent some applications are described. A theorem states that the maximum number of cliques in a graph increases exponentially as the number of vertices increases. For this reason heuristics are employed in clique finding algorithms to decrease the amount of search and an attempt is made to measure the algorithms as a function of the number of cliques. Three algorithms are formally defined and tested: the Bier- stone algorithm, the Bron-Kerbosch algorithm, and the Corneil algorithm. The Bron-Kerbosch algorithm is judged to be the best. TOR Tech. Rpt . Ul May 1972 89 Pages [M.S. Thesis] Program Spectra of Finite Graphs by L. Coll at z, Hamburg and U. Sinogowitz, Darmstadt Various problems encountered in practical applications, such as approximations of characteristic frequencies of a membrane with fixed perimeter and given area, or calculations of air vibrations in a bounded volume using difference methods, gave rise to the consideration of graphs in general. In this paper finite (in a subsequent paper certain types of infinite) graphs are examined and in particular some conclusions drawn with regard to their relationship to the theory of indecomposable, non- negative matrices. UAE No. 10 February 1968 28 Pages Graphs, Groups and Matrices by Abbe Mowshowitz Our object here is to exploit the connection between the adjacency matrix of a graph and its automorphism group in order to determine the latter. In what follows, we will take advantage of the fact that the adjacency matrix of a graph is a (0, l)-matrix and thus can be regarded as a matrix over GF(2). UBC Tech. Rpt. 71-OU October 1971 1^ Pages 29 Combinatorial Solutions to Partitioning Problems by J. A. Lukes In this dissertation we describe algorithms that use graph properties and dynamic programming techniques to generate the optimal partition of an arbitrary graph. In particular, let G be a graph with weighted nodes and weighted edges. We consider algorithms that solve the problem of partitioning G into clusters of nodes such that the sum of the node weights in any cluster does not exceed a given maxi- mum W and the weights of the intercluster edges are minimized. An interesting application of such an algorithm is the assignment of a program's subroutines and data to pages in a paged memory system so as to minimize paging faults. A very efficient variation of the general algorithm results if the graph to be partitioned is a tree. We show that trees can be partitioned in a time proportional to the number of nodes in the graph. STAN-CS-72-293 May 1972 120 Pages [Ph.D. Thesis] Chromatic Automorphisms of Graphs by V. Chvatal and J. Sichler The coloring group and the full automorphism group of an n-chromatic graph are independent if and only if n is an integer > 3< STAN-CS-72-273 March 1972 11 Pages 30 KEY TO REPORT SOURCES ANL Argonne National Laboratory Argonne, Illinois 60^+39 BTL CAC Bell Telephone Laboratories, Inc. Murray Hill, New Jersey 07971 Center for Advanced Computation University of Illinois, Urbana, Illinois 6l801 CMU Carnegie-Mellon University Computer Science Department Pittsburgh, Pennsylvania 15213 CNA Center for Numerical Analysis University of Texas, Austin, Texas 78712 MAC Massachusetts Institute of Technology Project MAC Cambridge, Massachusetts 02139 MRC Mathematics Research Center University of Wisconsin, Madison, Wisconsin 53706 STAN Computer Science Department Stanford University, Stanford, California 9^305 TOR Department of Computer Science University of Toronto, Ontario, Canada UAE Department of Computing Science University of Alberta, Edmonton, Alberta, Canada UBC Department of Computer Science University of British Columbia, Vancouver, B. C, Canada UMINF - Department of Information Processing University of Umea, Umea, Sweden UIUC Department of Computer Science University of Illinois, Urbana, Illinois 6l801 YORK - T. J. Watson Research Center Yorktown Heights, New York 10598 UNCLASSIFIED SecurityClassification DOCUMENT CONTROL DATA R&D (S»cuHlr elaaallleatlon el till; boa> of iticcl and Irtdmmtng amtotatlan muat ha tnnwl whan tha ovatall rapott la elaaaHlad) I. ORIGINATING ACTIVITY (Corpora/* author) Center for Advanced Computation University of Illinois at Urbana- Champaign Urbana, Illinois 2a. »EPO«T ICCURI T Y CLASSIFIC ATIO*' UNCLASSIFIED 26 CROU^ S. «CPCHr TITLE Computational Mathematics Abstracts «• OKSCNiptivk NOTII (Typo ol rtmtrt and htelualva dmtaa) Bibliographical Abstracts B- AUTHOMIS) (Firmi naata. mlddla Initial, laat nama) Edited by: Geneva Belford, Jonathan Lermit, George Purdy, and Ahmed Sameh a report oats February 1973 ?a. TOTAL NO. OF PACKS 31 7b. MO. OF REFS •a. CONTRACT OR GRANT NO. DAHC01+-72-C-0001 b. PROJECT NO ARPA Order No. 1899 •a. ORIOINATOR'S REPORT NuulCRlSI CAC Document No. 6k OTHER REPORT NOI1I (Any othat nuntwn that may ba aaalgnad thlt rapert) 10. DISTRIBUTION STATEMENT Copies may be requested from the address given in (l) above. II. SUPPLEMENT ART NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Research Office-Durham Duke Station, Durham, North Carolina 13. ABSTRACT NONE DD ,?.ir..l473 MCT.AHRTFTKT1 Security Classification UNCLASSIFIED Security Classification KEY KOKOI HOLE WT Numerical Analysis Graph Theory Mathematical Programming UNCLASSIFIED Security Classification ^"""«,.l