Showing papers in "Mathematics of Computation in 1975"
TL;DR: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool as discussed by the authors, and it can be found in many libraries.
Abstract: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.
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TL;DR: The convergence rate for difference approximations to mixed initial boundary value problems has been shown to be linear in the convergence rate of the difference approximation as mentioned in this paper, which is the best known convergence rate.
Abstract: The convergence rate for difference approximations to mixed initial boundary value problems
386 citations
TL;DR: In this article, a general spectral approximation theory for compact operators on a Banach space is developed, and results are obtained on the approximation of eigenvalues and generalized eigenvectors.
Abstract: In this paper a general spectral approximation theory is developed for compact operators on a Banach space. Results are obtained on the approximation of eigenvalues and generalized eigenvectors. These results are applied in a variety of situations.
332 citations
294 citations
TL;DR: The continued fraction method for factoring integers, introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation in this paper, and the power of the method is demonstrated by the factorization of the seventh Fermat number F7 and other large numbers of interest.
Abstract: The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is demonstrated by the factorization of the seventh Fermat number F7 and other large numbers of interest. "Quand on a a' etudier un grand nombre, il faut commencer par en determiner quelques residus quadratiques." M. Kraitchik
TL;DR: A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary valud problems for almost all (quasiuniform) piecewise-polynomial spaces used in practice.
Abstract: A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary valud problems. The class of Galerkin spaces considered includes almost all (quasiuniform) piecewise-polynomial spaces that are used in practice. The estimates are optimal in the sense that no better rate of approximation is possible in general in the spaces employed.
TL;DR: In this paper, it is shown that the individual matrices are simultaneously symmetrized by the similarity transformation and their norms can be applied to the well-posedness of the Cauchy problem, linear stability theory for finite-difference approximations, and simplification of block-tridiagonal systems that arise in implicit time-split algorithms.
Abstract: The hyperbolic nature of the unsteady, inviscid, gas-dynamic equations implies the existence of a similarity transformation for diagonalizing an arbitrary linear combination of coefficient matrices. It is shown that the individual matrices are simultaneously symmetrized by the similarity transformation. The transformations and their norms can be applied to the well-posedness of the Cauchy problem, linear stability theory for finite-difference approximations, and simplification of block-tridiagonal systems that arise in implicit time-split algorithms.
TL;DR: In this paper, a wider zero-free region for the Riemann zeta function was given, and improved methods for using this and a recent determination that 3,500,000 zeros lie on the critical line to develop better bounds for functions of primes.
Abstract: : The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before They give improved methods for using this and a recent determination that 3,500,000 zeros lie on the critical line to develop better bounds for functions of primes
TL;DR: In this article, the authors define the process of bisecting a triangle with vertices at A1, A2 and A3 as follows: first locate the longest edge, A'Ai+i of AA 1A2A3 where A'+3 = Al, set D = (A' + Ai+l)/2, and then define two new tri- angles, AA'DAi+2 and ADAi+1Ai +2.
Abstract: Let AA A A be a triangle with vertices at A1, A2 and A3. The process of "bisecting AA A A is defined as follows. We first locate the longest edge, A'Ai+i of AA 1A2A3 where A'+3 = Al, set D = (A' + Ai+l)/2, and then define two new tri- angles, AA'DAi+2 and ADAi+1Ai+2.
TL;DR: In this paper, a Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation V * (a(x, u)Vu) = f.
Abstract: A Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation V * (a(x, u)Vu) = f. The asymptotic error estimates are of the same form as in the linear case. Newton's method can be used to solve the nonlinear algebraic equations.
TL;DR: In this paper, an extraordinarily compact solution of the problem in the title, given (a trifle hidden) in the work of the late Jacques Touchard, is presented. But its main weight is on properties of the several kinds of number sequences appearing.
Abstract: This paper, in the first place, calls attention to an extraordinarily compact solution of the problem in the title, given (a trifle hidden) in the work of the late Jacques Touchard. Its main weight, however, is on properties of the several kinds of number sequences appearing.
TL;DR: The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximation problem has a stable solution which is close to the exact solution.
Abstract: General nonlinear problems in the abstract form F(x) = 0 and corresponding families of approximating problems in the form Ffi(xn) = 0 are considered (in an appropriate Banach space setting) The relation between "isolation" and "stability" of solutions is briefly studied The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximating problem has a stable solution which is close to the exact solution Error estimates are obtained and Newton's method is shown to converge quadratic-ally These results are then used to justify a broad class of difference schemes (resembling linear multistep methods) for general nonlinear two-point boundary value problems
TL;DR: In this paper, an efficient fourth order method for the multi-dimensional wave equation is presented, which is used to solve first order hyperbolic systems for application to inviscid flow calculations.
Abstract: : The results of this report are part of an effort to develop high accuracy efficient methods for solving hyperbolic equations. Here an efficient fourth order method for the multi-dimensional wave equation is presented. Similar techniques are being tried to solve first order hyperbolic systems for application to inviscid flow calculations.
TL;DR: In this article, asymptotic expansions for the incomplete gamma functions and the incomplete beta functions are derived for each of which contains the complementary error function and an asymptonic series, which are uniformly valid with respect to certain domains of the parameters.
Abstract: New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function. In each case the expansion contains the complementary error function and an asymptotic series. The expansions are uniformly valid with respect to certain domains of the parameters.
TL;DR: Methods are given for computing the LDV factorization of a matrix B and modifying the factorization when columns of B are added or deleted and it is shown how these techniques lead to two numerically stable methods for updating the Cholesky factorizationof a matrix following the addition or subtraction,respectively, of a Matrix of rank one.
Abstract: Methods are given for computing the LDV factorization of a matrix B and modifying the factorization when columns of B are added or deleted. The methods may be viewed as a means for updating the orthogonal (LQ) factorization of B without the use of square roots. It is also shown how these techniques lead to two numerically stable methods for updating the Cholesky factorization of a matrix following the addition or subtraction,respectively, of a matrix of rank one. The first method turns out to be one given recently by Fletcher and Powell; the second method has not appeared before.
TL;DR: In this paper, a method for finding vectors of the smallest Euclidean norm in a given lattice is presented, where the norm is defined by means of a convex, compact, and symmetric subset of the given space.
Abstract: A method for calculating vectors of smallest norm in a given lattice is outlined. The norm is defined by means of a convex, compact, and symmetric subset of the given space. The main tool is the systematic use of the dual lattice. The method generalizes an algorithm presented by Coveyou and MacPherson, and improved by Knuth, for the determination of vectors of smallest Euclidean norm. 1. Formulation of the Problem. Let G be a lattice in the n-dimensional Euclidean space Rn, generated by n linearly independent vectors e (1) G = {x = zie z1 integers}. The norm in Rn is defined by a convex, compact set B which has positive measure and is symmetric about the origin: (2) lIxIl = min{X E R I x E XB}. Examples of these norms for x = (xl, . . ., xn) are (i) The Euclidean norm llxl = (X2 + ? ? x2)1/2. (ii) The Maximum norm lxii = max{ixi I I i = 1, ... , n}. Here Boo = { . . ., Xn)I Ixi? 1 for all i}. (iii) The norm lixil = lx11 + -'_ + lXn 1Here B1 = {(xl, . . . , Xn) IXI I + ?* + Xn I 1 The problem is to find a nonzero vector of shortest length (norm) in G. The main tool of the presented method is the use of the dual lattice, (3) G* ={x*= E z4e*Iz*integers), k=l where the e* are defined by eie* = 6ik; here 6ik is equal to 1 if i = k and equal to 0 if i # k, and e e* denotes the scalar product V7n= e.1e* . The polar of B, namely (4) B* = {b* E Rn I Ibb*l < 1, V bEB}, induces a length or norm in G* by (5) 1ix*i*1 min{X* E R Ix* G X*B*}. It should be noted that the Euclidean norm corresponds to itself, whereas the Maximum norm lxii = maxi IxiI corresponds to iix*ii* = Ixv + i ? + Ix*l and vice versa. Received May 16, 1974; revised August 6, 1974. AMS (MOS) subject classifications (1970). Primary 10E0S, 10E20, 10E25; Secondary 65C10.
TL;DR: In this article, a polynomial interpolant which interpolates a function F E CN(T), and its derivatives of order N and less, on the boundary aT of a triangle T was derived.
Abstract: Boolean sum interpolation theory is used to derive a polynomial interpolant which interpolates a function F E CN(T), and its derivatives of order N and less, on the boundary aT of a triangle T. A triangle with one curved side is also considered.
TL;DR: In this paper, a generalization of Gershgorin's theorem is developed for the eigenvalue problem Ax = XBx and is applied to obtain perturbation bounds for multiple eigenvalues.
Abstract: A generalization of Gershgorin's theorem is developed for the eigenvalue problem Ax = XBx and is applied to obtain perturbation bounds for multiple eigenvalues. The results are interpreted in terms of the chordal metric on the Riemann sphere, which is especially convenient for treating infinite eigenvalues.
TL;DR: In this paper, the authors give a continued fraction algorithm for the system with tox + tlx2 + t2x3 + * for any positive integer s. The algorithm produces a nontrivial solution of the system E ti+j j 0O < i < s-1, j=O for every positive integer n 1.
Abstract: Let to, t1, t2, be a sequence of elements of a field F. We give a continued fraction algorithm for tox + tlx2 + t2x3 + * . . If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence. More generally the algorithm produces a nontrivial solution of the system E ti+j j 0O < i < s-1, j=O for every positive integer s. 1. Let tO, tl, t2, be a sequence of elements of a field F. Set
TL;DR: In this article, a generalization of Gershgorin's theorem is developed for the eigenvalue problem Ax = XBx and is applied to obtain perturbation bounds for multiple eigenvalues.
Abstract: A generalization of Gershgorin's theorem is developed for the eigenvalue problem Ax = XBx and is applied to obtain perturbation bounds for multiple eigenvalues. The results are interpreted in terms of the chordal metric on the Riemann sphere, which is especially convenient for treating infinite eigenvalues.