Showing papers in "Mathematics of Computation in 1976"

Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization

Abstract: Numerically stable algorithms are given for updating the GramSchmidt QR factorization of an m X n matrix A (m > n) when A is modified by a matrix of rank one, or when a row or column is inserted or deleted. The algorithms require O(mn) operations per update, and are based on the use of elementary two-by-two reflection matrices and the Gram-Schmidt process with reorthogonalization. An error analysis of the reorthogonalization process provides rigorous justification for the corresponding ALGOL procedures.

Dissipative two-four methods for time-dependent problems

Abstract: A generalization of the Lax-Wendroff method is presented. This generalization bears the same relationship to the two-step Richtmyer method as the KreissOliger scheme does to the leapfrog method. Variants based on the MacCormack method are considered as well as extensions to parabolic problems. Extensions to two dimensions are analyzed, and a proof is presented for the stability of a Thommentype algorithm. Numerical results show that the phase error is considerably reduced from that of second-order methods and is similar to that of the Kreiss-Oliger method. Furthermore, the (2, 4) dissipative scheme can handle shocks without the necessity for an artificial viscosity.

Optimal ^{∞} estimates for the finite element method on irregular meshes

Abstract: Uniform estimates for the error in the finite element method are derived for a model problem on a general triangular mesh in two dimensions. These are optimal if the degree of the piecewise polynomials is greater than one. Similar estimates of the error are also derived in L/sup p/. As an intermediate step, an L/sup 1/ estimate of the gradient of the error in the finite element approximation of the Green's function is proved that is optimal for all degrees.

Computation of π Using Arithmetic-Geometric Mean

Abstract: A new formula for π is derived. It is a direct consequence of Gauss’ arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre’s relation for elliptic integrals. The error analysis shows that its rapid convergence doubles the number of significant digits after each step. The new formula is proposed for use in a numerical computation of π, but no actual computational results are reported here.

Improvement by iteration for compact operator equations

Abstract: The equation y f + Ky is considered in a separable Hilbert space H, with K assumed compact and linear. It is shown that every approximation to y of the form Yln = Enaniui (where {ui} is a given complete set in H, and the an1, 1 < i < n, are arbitrary numbers) is less accurate than the best approximation of the form Y2n = f + ynbnjKui, if n is sufficiently large. Specifically it is shown that if Yln is chosen optimally (i.e. if the coefficients ani are chosen to minimize Ily Yln 11), and if Y2n is chosen to be the first iterate of Ylnl i.e. Y2n = f + Kyln then Ily Y2n ll < an Ily Yln II, with an0. A similar result is also obtained, provided the homogeneous equation x = Kx has no nontrivial solution, if instead Yln is chosen to be the approximate solution by the Galerkin or Galerkin-Petrov method. A generalization of the first result to the approximate forms Y3n' Y4n' . . . obtained by further iteration is also shown to be valid, if the range of K is dense in H.

A bound on the _{∞}-norm of ₂-approximation by splines in terms of a global mesh ratio

Abstract: Let Lkf denote the least-squares approximation to f E L1 by splines of order k with knot sequence t = (ti)1+k. In connection with their work on Galerkin's method for solving differential equations, Douglas, Dupont and Wahlbin have shown that the norm IILklIo, of Lk as a map on Loo can be bounded as follows, IlLklloo 0}. Using their very nice idea together with some facts about B-splines, it is shown here that even JlLk lloo < constk(mt ) M(k) with the smaller global mesh ratio Mt given by A4 :=) max (ti+k-ti)(t+k ti). A mesh independent bound for L2-approximation by continuous piecewise polynomials is also given.

An error functional expansion for $N$-dimensional quadrature with an integrand function singular at a point

Abstract: Let If be the integral of f(x) over an N-dimensional hypercube and Q/sup (m)/f be the approximation to If obtained by subdividing the hypercube into m/sup N/ equal subhypercubes and applying the same quadrature rule Q to each. In order to extrapolate efficiently for If on the basis of several different approximations Q/sup (m/sub i/)/f, it is necessary to know the form of the error functional Q/sup (m)/f-If as an expansion in m. When f(x) has a singularity, the conventional form (with inverse even powers of m) is not usually valid. The expansion in the case in which f(x) has the form f(x) = r/sup ..cap alpha../phi(theta)h(r)g(x), ..cap alpha..>-N, the only singularity being at the origin, a vertex of the unit hypercube of integration is derived. Here (r, theta) represents the hyperspherical coordinates of (x). It is shown that for this integrand the error function expansion includes only terms A/sub ..cap alpha..+N+t//m/sup ..cap alpha..+N+t/,B/sub r//m/sup t/, C/sub ..cap alpha..+N+t/ln m/m/sup ..cap alpha..+N+t/, t = 1,2,.... The coefficients depend only on the integrand function f(x) and the quadrature rule Q. For several easily recognizable classes of integrand function and for most familiar quadrature rules some of these coefficients are zero. Anmore » analogous expansion for the error functional with integrand function F(x) = ln rf(x) is also derived.« less

Rational Chebyshev approximations for the inverse of the error function

Abstract: This report presents near-minimax rational approximations for the inverse of the error function inverf x, for 0 S x S 1-1

Factoring multivariate polynomials over algebraic number fields

Abstract: The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented in the algebraic manipulation system MACSYMA.** Some machine examples with timing are included.

A note on extended Gaussian quadrature rules

Abstract: Extended Gaussian quadrature rules of the type first considered by Kronrod are examined. For a general nonnegative weight function, simple formulas for the computation of the weights are given, together with a condition for the positivity of the weights associ- ated with the new nodes. Examples of nonexistence of these rules are exhibited for the weight functions (1 - x2)X-/2, e-X2 and e-X. Finally, two examples are given of quad- rature rules which can be extended repeatedly.

Factorized variable metric methods for unconstrained optimization

Abstract: Several effilcient methods are given for updating the Cholesky factors of a symmetric positive definite matrix when it is modified by a rank-two correction which maintains symmetry and positive definiteness. These ideas are applied to variable metric (quasi-Newton) methods to produce numerically stable algorithms.

A nearest point algorithm for convex polyhedral cones and applications to positive linear approximation

Abstract: Suppose K is a convex polyhedral cone in En and is defined in terms of some generating set {ej, e2, , eN} A procedure is devised so that, given any point q E En, the nearest point p in K to q can be found as a positive linear sum of N* 0, fE G4, i-1 for a positive linear functional L acting on a suitable finite-dimensional function space (D

Linear Chebyshev approximation in the complex plane using Lawson’s algorithm

Abstract: In this paper we discuss the application of Lawson's algorithm for computing best linear Chebyshev approximations to complex-valued functions. Some numerical examples are also presented.

A stability analysis for perturbed nonlinear iterative methods

Abstract: This paper applies the asymptotic stability theory for ordinary differential equations to Gavurin's continuous analogue of several well-known nonlinear iterative methods. In particular, a general theory is developed which extends the OrtegaRheinboldt concept of consistency to include the widely used finite-difference approximations to the gradient as well as the finite-difference approximations to the Jacobian in Newton's method. The theory is also shown to be applicable to the LevenbergMarquardt and finite-difference Levenberg-Marquardt methods.

Asymptotic formulas related to free products of cyclic groups

Abstract: Asymptotic formulas for the number of subgroups of a given index of the free product of finitely many cyclic groups are given. The classical modular group r is discussed in detail, and a table of the number of subgroups of r of index n is given for 1 6 n 1. k= 1 Define the formal power series f (x), g(x) by 00 00 f(X) an OX n g(x) =, M nXn n=o n1 Then (1) is equivalent to the identity (2) g(x) = xf'(x)/f(x). Formula (2) implies that (3) ~~~~00 Mn ( Z xA = log Ax)5 n= 1 so that (4) E nA = log(l + f(x)-M1) = ( n Comparing coefficients of corresponding powers of x in (4), we find the following result, which we state as a theorem: THEOREM 1. The numbers Mn are given explicitly as functions of the numbers an by the formula Received March 18, 1976. AMS (MOS) subject classifications (1970). Primary 2OE35, 10DO5, 10-04.

Class groups of quadratic fields II

Abstract: A computation has been made of the noncyclic class groups of imaginary quadratic fields Q(\J - D ) for even and odd discriminants - D from 0 to - 25000000. Among the results are that 95% of the class groups are cyclic, and that -11203620 and -18397407 are the first discriminants of imaginary quadratic fields for which the class group has rank three in the 5-Sylow subgroup. The latter was known to be of rank three; this computation demonstrates that it is the first odd discriminant of 5-rank three or more.

Zeros of Hurwitz zeta functions

Abstract: All complex zeros of each Hurwitz zeta function are shown to lie in a vertical strip. Trivial real zeros analogous to those for the Riemann zeta function are found. Zeros of two particular Hurwitz zeta functions are calculated.

A computational technique for evaluating (1,) and the class number of a real quadratic field

Abstract: A description is given of a method for estimating L(1, X) to sufficient accuracy to determine the class number of a real quadratic field. This algorithm was implemented on an IBM/370-158 computer and the class number, regulator, and value of L(1, X) were obtained for each real quadratic field Q(vIDU) (D 2, 3, . . ., 149999). Several tables, summarizing various results of these computations, are also presented.

Maximal binary matrices and sum of two squares

Abstract: A maximal (+1, — l)-matrix of order 66 is constructed by a method of matching two finite sequences. This method also produced many new designs for maximal (+1, -l)-matrices of order 42 and new designs for a family of //-matrices of order 26.2". A nonexistence proof for a («)-type //-matrix of order 36, con- sequently for Golay complementary sequences of length 18, is also given. Let M be a 2« x 2« (+1, - l)-matrix, then the absolute value of det M is equal to or less than p2n, where p2n = (2n)n, if « is even; and p2n = 2"(2« - 1)(« - 1)"_1,

The largest degrees of irreducible characters of the symmetric group

Abstract: The largest irreducible degrees and the partitions associated with them are tabulated for the symmetric group En for n up to 75. Analytic upper and lower bounds are derived for the largest degree. Introduction. A question has been raised by Bivins and others [2] -namely: For what irreducible representations of the symmetric group En does the degree attain its maximal value and how does this maximum behave for large n? This was apparently motivated by the practical considerations of number overflow in the computer but the same question arises in connection with sorting [1]. Each irreducible representation is associated with a partition a = (a1, a2, . .. ak), al > a2 > * * * > ak > O, of n. (We shall use a E n to mean that a is one of the Pn partitions of n.) Its degree is given by [6, p. 61]: da= n! [11i n2 > * * > nk, where n, is the current number of votes for candidate i (i = 1, . . ., k) with finally n, = a (i = I1, . . .,~ k). Computation of the Maximal Degree. The calculations were made at Edinburgh University on a 4K 12-bit word length PDP8 computer using a multi-length routine for expansile integer multiplication. The strategy is straightforward. For increasing n, partitions of n are generated in natural order (n first and In last) as described in [11]. If a partition, a, precedes or coincides with its conjugate then the degree da is computed as in the procedure degree of [9] but exponent arithmetic is used retaining integers throughout and avoiding unnecessary overflow. A description of exponent arithmetic appears in [10] but this description is slightly different from that used in this application, and the algorithm given there is a little garbled. Three arrays are declared, ex, hfac, Ifac [2: N] , where N is the largest integer occurring as a natural factor (here N is at most 75); ex [n] contains the exponent of n in the result and for all n < N, hfac [n] contains the largest prime factor of n and Ifac[n] contains the other factor. After initialization, the expression is evaluated by modifying the exponents in ex. For example, to divide by k!: for i := 2 step I until k do ex [i] := ex [i]-1; Received June 5, 1975; revised September 30, 1975. AMS (MOS) subject classifications (1970). Primary 20-04, 20C30; Secondary 05A15.

The convergence of the Ben-Israel iteration for nonlinear least squares problems

Abstract: . Ben-Israel [ 1 ] proposed a method for the solution of the nonlinear leastsquares problem m'mx^j^\\F(x)\\2 where F: D C R —► R . This procedure takes theform xk,x = xk — F'(xk) F(xk) where F'(xk) denotes the Moore-Penrose generalizedinverse of the Fre'chet derivative of F. We give a general convergence theorem for themethod based on Lyapunov stability theory for ordinary difference equations. In thecase where there is a connected set of solution points, it is often of interest to determinethe minimum norm least squares solution. We show that the Ben-Israel iteration has nopredisposition toward the minimum norm solution, but that any limit point of thesequence generated by the Ben-Israel iteration is a least squares solution. I. Introduction. The use of least squares solutions to systems of equations is an important and practical tool in many applications. Given a function F: D C R" —>Rm where D is an open convex set, the nonlinear least squares problem is expressed asrnmxeDIIZr(x)||, where || ■ || here and henceforth denotes the l2 norm. Equivalently, iffiix) is the z'th component of F, then the problem can be stated as rnin^^^fjc), where$ = &££Lj/?(x). If, as we shall assume, Zms continuously Fre'chet differentiable, then

A Computational Technique for Determining the Class Number of a Pure Cubic Field

Abstract: Two different computational techniques for determining the class number of a pure cubic field are discussed These techniques were implemented on an IBM/370-158 computer, and the class number for each pure cubic field Q(D ' ) for D = 2, 39999 was obtained Several tables are presented which summarize the results of these computations Some theorems concerning the class group struc- ture of pure cubic fields are also given The paper closes with some conjectures which were inspired by the computer results 1 Introduction The theory of pure cubic fields QiDxl3),D rational, was founded in 1892 by Markov (12) ; in his paper he gives some class numbers and fundamental units, not always in an explicit form In (7) Dedekind describes a method for deter- mining the class number of a pure cubic field Q{DX^3) He also gives a short table of class numbers for some small values of D Cohn (6) implemented Dedekind's method on a computer and obtained class numbers for some fields in which he could easily determine the regulator Cohn's technique was modified somewhat by Beach, Williams, and Zarnke in (4), and class numbers were obtained for QXP1 /3) for D = 2, 3, , 999 Other tables of class numbers have been calculated by hand by Cassels (5) and Selmer (14) It should also be mentioned that Angeli (1) has recently given a list of class numbers for all cubic fields with negative discriminant greater than -20, 000 The purpose of this paper is to present a new technique for determining the class number of Q{D1'3) This method is much faster than the computational technique of (6) and (4) The algorithm was implemented on a computer and the class numbers for Q{Dl/3) obtained for D = 2, 39999 The total number of these fields is 8122 We also describe here some of the results of these calculations 2 Some Properties of Pure Cubic Fields Let K be any cubic field with discrim-

Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions

Abstract: Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions and tables for the first fifteen leading terms are given. Relations of these coefficients with the randomization distributions are shown. Little is known about the Taylor series expansion coefficients of the Jacobian elliptic functions sn(u, k), cn(u, k) and dn(u, k). No recurrence formula exists for these coefficients. Only four to five leading terms of the series are given in literature ([1], [2]). We present in this paper properties of these coefficients, show relations between them and randomization distributions [3, p. 51], and give tables for the first fifteen leading terms. We consider the differential equations d Y1(U) C1y2(u)y3(U) = 0, du (1) d Y2 ( -) C2y1(u)y3(U) = 0, du d Y3(U) C3y1(u)y2(u) = 0du Solution functions of (1) for C1 = 1, C2 = 1, C3 = are the Jacobian elliptic functions v1 = sn(u, k), Y2 = cn(u, k), Y3 = dn(u, k) ([1], [2]). The formal Taylor series of the functions y1, Y2, Y3 read co(u u0)n ...CJ'21 l? ' y(u) = E ! ? ajj2j3C1 2 3 Y10Y20Y30 n=0 (2) w (U o n! [ b hi 2 (-3 SI "2 s3r n n!=LE n !hh2h3 1 C22C3jY10Y20y330J