Showing papers in "SIAM Journal on Numerical Analysis in 1967"
421 citations
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TL;DR: In this paper, an invited paper presented at the Symposium on Numerical Solution of Differential Equations, SIAM 1966 National Meeting at the University of Iowa, sponsored by the Air Force Office of Scientific Research.
Abstract: * Invited paper presented May 13, 1966, at the Symposium on Numerical Solution of Differential Equations, SIAM 1966 National Meeting at the University of Iowa, sponsored by the Air Force Office of Scientific Research. f Oxford University Computing Laboratory. : Eidgenhssische Technische Hochschule, Zurich, and IBM Zurich Research Laboratory, 8803 Rfischlikon-ZH, Switzerland. Eidgenhssische Technische Hochschule, Zurich. Present address: University of Michigan, Ann Arbor, Michigan.
218 citations
TL;DR: General convergence theorems for Newton method applied to convex operators for partially ordered topological linear spaces have been applied to the convex operator for convex topological spaces.
Abstract: General convergence theorems for Newton method applied to convex operators for partially ordered topological linear spaces
151 citations
TL;DR: When the kernel has several continuous derivatives, the method reduces to replacing the integral with a numerical integral and then to solving a finite linear system; see as discussed by the authors for a generalized form of numerical inltegration for functioins of oine variable.
Abstract: When the kernel has several continuous derivatives, the method reduces to replacing the integral with a numerical integral and then to solving a finite linear system; see [1], [2], [3], [5], [11], [13]. In the followiing section, a generalized form of numerical inltegration is given for functioins of oine variable. It is applied to (1) in ?3, aild convergence of the resulting method is showii in ?4. Section 5 conitains computational notes and a numerical example.
140 citations
TL;DR: Monotone iterations for nonlinear elliptic differential equations in boundary value problems applied to Gauss-Seidel methods as discussed by the authors were used to solve boundary-value problems in boundary-valued problems.
Abstract: Monotone iterations for nonlinear elliptic differential equations in boundary-value problems applied to Gauss-Seidel methods
120 citations
TL;DR: In this article, a procedure for obtaining spline function approximations for solutions of the initial value problem in ordinary differential equations is presented, which is related to the well-known trapezoidal rule and Milne-Simpson method.
Abstract: A procedure for obtaining spline function approximations for solutions of the initial value problem in ordinary differential equations is presented. The proposed method with quadratic and cubic splines is shown to be related to the well-known trapezoidal rule and Milne-Simpson method, respectively. The method is shown to be divergent, however, when higher degree spline functions are used.
113 citations
TL;DR: In this paper, the authors considered a Cauchy-like problem for the heat equation and showed that it is possible to solve the problem in polynomial time, in the sense that
Abstract: Part I is devoted to the consideration of a Cauchy-like problem for the heat equation. Let $u(x,t)$ satisfy the heat equation in $D = \{ (x,t):0 < x < s(t),0 < t \leqq T\} $ and let $u(x,0) = \varp...
98 citations
81 citations
TL;DR: In this article, a region of convergence is derived for the Newton-Raphson and secant iteration methods, and a modification of one of these theorems is given and applied to study the effect of roundoff errors in the Gauss-Seidel iteration methods.
Abstract: LaSalle [5], [6], [7] and others have developed a generalization of the “second method” of Liapunov which utilizes certain invariance properties of solutions of ordinary differential equations. Invariance properties of solutions of ordinary difference equations are utilized here to develop stability theorems similar to those in LaSalle [5]. As illustrations of the application of these theorems, a region of convergence is derived for the Newton-Raphson and secant iteration methods. A modification of one of these theorems is given and applied to study the effect of roundoff errors in the Newton-Raphson and Gauss-Seidel iteration methods.
TL;DR: A class of iterative methods for solving nonlinear least squares problems is presented and a convergence theorem with error estimations is proved and the theorem is specialized for the case of the Gauss-Newton method.
Abstract: : A class of iterative methods for solving nonlinear least squares problems is presented and a convergence theorem with error estimations is proved. The theorem is then specialized for the case of the Gauss-Newton method, and an algorithm for checking automatically the conditions for convergence is included. Some numerical examples are discussed. (Author)
TL;DR: Acceleration techniques for discretization algorithms used in the approximate solution of nonlinear operator equations are considered and applied to the approximation solution of mildly nonlinear elliptic equations by finite differences.
Abstract: : Acceleration techniques for discretization algorithms used in the approximate solution of nonlinear operator equations are considered. Practical problems arising in the solution of large systems of nonlinear algebraic equations are discussed. These techniques are applied to the approximate solution of mildly nonlinear elliptic equations by finite differences, and several numerical examples are given. (Author)
TL;DR: In this article, a certain class of integral equations with possibly discontinuous kernels is treated, where only real equations are treated, and the methods of analysis depend heavily on the order structure of the real line.
Abstract: In effect, (1.2) provides an interpolation formula for gn (x) in terms of the gn (tnk ) . There have been many studies of this approximation method, mostly with g (x), h (x) and K (x, y) continuous for 0 < x, y < 1. Recent references include Buickner [1], [2], [3], Kantorovitch and Krylov [4], Mysovskih [5], [6], [7], Brakhage [8], Anselone and Moore [9], and Anselone [10]. For further references, see [2], [3], [4], [10]. In this paper we deal with a certain class of integral equations with possibly discontinuous kernels. Only real equations are treated, sinlce the methods of analysis depend heavily on the order structure of the real line. Extensions to complex integral equations will be made elsewhere. As for the quadrature formula, assume
TL;DR: A method for computing the generalized inverse of a matrix is described, which uses the well-known Gauss-Jordan elimination scheme in conjunction with the conventional Gram-Schmidt orthogonalization process.
Abstract: A method for computing the generalized inverse of a matrix is described. It is especially suited for large sparse matrices. It uses the well-known Gauss-Jordan elimination scheme in conjunction with the conventional Gram-Schmidt orthogonalization process.
TL;DR: In this paper, the common zeros of a set of polynomials in n variables can be used as the points in an integration formula for an n-dimensional region.
Abstract: This paper gives a necessary and sufficient condition that the common zeros of a set of polynomials in n variables can be used as the points in an integration formula for an n-dimensional region. This result applies for all $n \geqq 1$.
TL;DR: In this article, it was shown that each circle in the Laguerre model has at least one zero of an Nth degree polynomial, and if the circle is not a perfect square, then the constant can be replaced by a slightly smaller constant.
Abstract: Given an Nth degree polynomial $P(z)$ one may use Laguerre’s method to generate a sequence of complex numbers $x_0 ,x_1 ,x_2 , \ldots $ which usually converges to a, zero of $P(z)$. This note shows that each circle $\left| {z - x_n } \right| \leqq \sqrt N \left| {x_{n + 1} - x_n } \right|$ contains at least one zero of $P(z)$. If N is not a perfect square, then the constant $\sqrt N $ can be replaced by a slightly smaller constant $R_N $.
TL;DR: The concept of a field of values was introduced by Bauer [1] in 1962 and generalized by Bauer as mentioned in this paper, who showed that G[A] also contains the convex hull 5C(A(A)) of A(A).
Abstract: 1. For a square matrix A the so-called field of values G[A] is defined as the following set of complex numbers:' (1.1) G[A] := {xHAx|xHx = 1}. It is well known that this set contains the spectrum A(A) = {Xj(A) } of A. Moreover, Toeplitz [11] proved in 1918 that G[A] also contains the convex hull 5C( A(A)) of A(A). Hausdorff [5] generalized this result by proving that G[A] is convex. The concept of a field of values was generalized by Bauer [1] in 1962. Starting with an arbitrary norm || v 1l in Cn (or R') and its dual norm (defined on the dual space of Cn (or Rn) of row vectors yH) IIyH D =max Re y X
TL;DR: In this paper, the speed of convergence of the continued fraction of positive real elements is estimated for complex real elements and for complex elements contained in regions for which $K({1 / {b_n }})$ converges.
Abstract: The speed of convergence of the continued fraction $K({1 / {b_n }})$ is estimated for positive real elements $b_n $ and for complex elements contained in regions for which $K({1 / {b_n }})$ converges. The estimates are determined by calculating the ratio of successive diameters of a nested sequence of sets $\{Z_n \}$ , where the nth approximant of $K({1 / {b_n }})$ is contained in the set $Z_n $.