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1959

Volume 7, Issue 4, pp. 345-498


Best Approximation of Mixed Type

Herbert E. Salzer

J. Soc. Indust. and Appl. Math. 7, pp. 345-360 (16 pages)

Online Publication Date: July 10, 2006

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Show Abstract
Consider $Ax = b$, an overdetermined system of $n$ linear equations in $m$ unknowns, $n > m$, where $x \equiv ( x_1 ,x_2 , \cdots ,x_m ),b \equiv ( b_1 ,b_2 , \cdots ,b_n ),A \equiv \| a_{ij} \|$ is of rank $m$, and $r_i \equiv \Sigma _{j = 1}^m a_{ij} x_j - b_i ,i = 1,2, \cdots ,n$. Many practical problems might tolerate $\max | r_i | = R,M < R < L$, where $M = \max | r_i |$ for $x_M $ minimizing $\max | r_i |$, and$L = \max | r_i |$ for $x_L$ minimizing $\Sigma _{i = 1}^n r_i^2 $. The residually bounded least squares solution $( r.b.l.s.s. )x_R $ minimizes $\Sigma _{i = 1}^n r_i^2 $ subject to $| r_i |\leqq R$ (a special problem in quadratic programming). The $r.b.l.s.s.$ may be obtained in a finite number of steps by a method of successive reductions from an $n \times m$ to an $( n - 1) \times ( m - 1)$ system, and the employment of a finite algorithm for $m = 1$. The proof utilizes the property that $S_R = \min \Sigma _{i = 1}^n r_i^2 $ is a strictly decreasing function of $R$ for $M < R < L$. This monotonicity property solves the inverse problem, of finding $x_D $ minimizing $D = \max | r_i |$ for preassigned $S = \min \Sigma _{i = 1}^n r_i^2 $, the answer being $D$ is equal to that $R$ for which $S_R = S$, and $x_D = x_R $. The $r.b.l.s.s.$ for $n$ large or infinite (e.g. determining the coefficients $a_i $ in $R_m ( x ) \equiv x^m + \Sigma _{i = 0}^{m - 1} a_i x^i $ minimizing $\int_{ - 1}^1 {R_m ( x )^2 dx} $ subject to $| R_m ( x )|\leqq R, - 1\leqq x\leqq 1 )$ might be approximated by linear inverse interpolation between $x_M $ and $x_L $ (subject to certain limitations).

On the Numerical Integration of the Orr-Sommerfeld Equation

S. D. Conte and J. W. Miles

J. Soc. Indust. and Appl. Math. 7, pp. 361-366 (6 pages) | Cited 4 times

Online Publication Date: July 10, 2006

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Abstract Unavailable

Estimates of the Roots of Certain Polynomials

Czerna Flanagan and J. E. Maxfield

J. Soc. Indust. and Appl. Math. 7, pp. 367-373 (7 pages)

Online Publication Date: July 10, 2006

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Abstract Unavailable

The Distribution of Quadratic Forms in Normal Variates: A Small Sample Theory with Applications to Spectral Analysis

Ulf Grenander, H. O. Pollak, and D. Slepian

J. Soc. Indust. and Appl. Math. 7, pp. 374-401 (28 pages) | Cited 6 times

Online Publication Date: July 10, 2006

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Abstract Unavailable

Error Bounds for a Family of Three-Point Integration Procedures

T. E. Hull and A. C. R. Newbery

J. Soc. Indust. and Appl. Math. 7, pp. 402-412 (11 pages) | Cited 3 times

Online Publication Date: July 10, 2006

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Abstract Unavailable

Optimal Search Strategies

E. N. Gilbert

J. Soc. Indust. and Appl. Math. 7, pp. 413-424 (12 pages) | Cited 6 times

Online Publication Date: July 10, 2006

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Abstract Unavailable

Method of Functional Extrapolation for the Numerical Integration of Ordinary Differential Equations

J. R. M. Radok

J. Soc. Indust. and Appl. Math. 7, pp. 425-430 (6 pages)

Online Publication Date: July 10, 2006

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Abstract Unavailable

A Linear Algebraic Formulation of the Theory of Sampled-Data Control

T. F. Bridgland, Jr.

J. Soc. Indust. and Appl. Math. 7, pp. 431-446 (16 pages) | Cited 2 times

Online Publication Date: July 10, 2006

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Abstract Unavailable

Automatic Computation of Nerve Excitation—Detailed Corrections and Additions

R. Fitzhugh and H. A. Antosiewicz

J. Soc. Indust. and Appl. Math. 7, pp. 447-458 (12 pages) | Cited 20 times

Online Publication Date: July 10, 2006

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Abstract Unavailable

Iterative Solutions of the Dirichlet Problem for $\Delta u = u^2 $

C. M. Ablow and C. L. Perry

J. Soc. Indust. and Appl. Math. 7, pp. 459-467 (9 pages) | Cited 3 times

Online Publication Date: July 10, 2006

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Abstract Unavailable

On the Numerical Solution of Simultaneous Linear Differential Equations

K. T. Chang

J. Soc. Indust. and Appl. Math. 7, pp. 468-472 (5 pages)

Online Publication Date: July 10, 2006

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Abstract Unavailable

An Integro-Differential Equation for a Markov Process

R. H. Boyer

J. Soc. Indust. and Appl. Math. 7, pp. 473-486 (14 pages) | Cited 1 time

Online Publication Date: July 10, 2006

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Abstract Unavailable

On a Theory of Boolean Functions

Sheldon B. Akers, Jr.

J. Soc. Indust. and Appl. Math. 7, pp. 487-498 (12 pages) | Cited 20 times

Online Publication Date: July 10, 2006

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Abstract Unavailable
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