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SIAM J. on Numerical Analysis

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1966

Volume 3, Issue 4, pp. 545-658


Stability and Convergence of Finite Difference Schemes with Singular Coefficients

Dennis Eisen

SIAM J. Numer. Anal. 3, pp. 545-552 (8 pages) | Cited 6 times

Online Publication Date: July 14, 2006

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Simultaneous Approximation of a Function and Its Derivatives

A. Meir and A. Sharma

SIAM J. Numer. Anal. 3, pp. 553-563 (11 pages)

Online Publication Date: July 14, 2006

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A Method for the Computation of the Greatest Root of a Nonnegative Matrix

Alfred Brauer

SIAM J. Numer. Anal. 3, pp. 564-569 (6 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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Multistage Alternating Direction Methods

Jim Douglas, Jr., A. O. Garder, and Carl Pearcy

SIAM J. Numer. Anal. 3, pp. 570-581 (12 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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A Method for Computing the Generalized Inverse of a Matrix

Ben Noble

SIAM J. Numer. Anal. 3, pp. 582-584 (3 pages) | Cited 11 times

Online Publication Date: July 14, 2006

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On Error Bounds for Generalized Inverses

Adi Ben-Israel

SIAM J. Numer. Anal. 3, pp. 585-592 (8 pages) | Cited 13 times

Online Publication Date: July 14, 2006

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An Order Five Runge-Kutta Process with Extended Region of Stability

J. Douglas Lawson

SIAM J. Numer. Anal. 3, pp. 593-597 (5 pages) | Cited 8 times

Online Publication Date: July 14, 2006

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Solutions of the Fifth-Order Runge-Kutta Equations

C. R. Cassity

SIAM J. Numer. Anal. 3, pp. 598-606 (9 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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(a) Under the restriction $R_2 = 0$, the sixteen equations in twenty-one unknowns which specify a fifth-order Runge-Kutta formula are so arranged as to give a solution which is completely rational in terms of five parameters. (b) If $R_2 \ne 0$, the solutions fall into two classes. The simpler class permits a solution involving a single quadratic irrationality. The other class is reduced to the simultaneous solution of two determinantal equations each involving two unknowns and five parameters.

Monotone Iterations and Two-Sided Convergence

L. F. Shampine

SIAM J. Numer. Anal. 3, pp. 607-615 (9 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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Norms of Powers of Matrices in a Special Class

Mary Louise Buchanan

SIAM J. Numer. Anal. 3, pp. 616-623 (8 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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Numerical Inversion of the Laplace Transform by Use of Jacobi Polynomials

Max K. Miller and W. T. Guy, Jr.

SIAM J. Numer. Anal. 3, pp. 624-635 (12 pages) | Cited 60 times

Online Publication Date: July 14, 2006

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Functional values of a function $f$ are determined from the values $F(s)$ of its Laplace transform at discrete points of $s$. Evaluation of $F(s)$ at points given by $s = (\beta + 1 + k)\delta ,\, k = 0,1, \cdots $, determine coefficients in an infinite series expansion of $f(t)$ in terms of Jacobi polynomials. The values of $\beta $ and $\delta $ determine the position along the real $s$-axis at which $F(s)$ is evaluated. An approximation to $f(t)$ is given by using a finite number of terms of the infinite series expansion of $f(t)$. Numerical examples are given and results are compared with some known numerical methods for approximating $f(t)$.

Numerical Inversion of a Laplace Transform

R. A. Spinelli

SIAM J. Numer. Anal. 3, pp. 636-649 (14 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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Interpolative Solution of Systems of Nonlinear Equations

Stephen M. Robinson

SIAM J. Numer. Anal. 3, pp. 650-658 (9 pages) | Cited 8 times

Online Publication Date: July 14, 2006

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This paper presents an iterative method for numerical solution of a system of $n$ nonlinear functional equations in $n$ unknowns, by repeated linear interpolation. It is a variation of Newton’s method, in which the partial derivatives are replaced by the multidimensional analogues of difference quotients. The only computations required are evaluation of the given functions and simple matrix operations. The method is a generalization of an idea of C. F. Gauss [6], and can also be regarded as generalizing the secant method, or regula falsi, as used for solving functions of one variable [4]. It produces superlinear convergence (of order $\frac{1}{2}(1 + \sqrt 5 )$, or approximately 1.62) to a simple zero, provided the given functions are twice continuously differentiable in a neighborhood of the zero, and the initial approximations are close enough. A convergence analysis for this method is given, and some numerical results are presented.
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