# Numerical work of Hans F. Weinberger

Saturday, October 4, 2008 - 2:00pm - 2:50pm

EE/CS 3-180

John Osborn (University of Maryland)

In this talk we will survey several papers (listed

below) by

Hans Weinberger dealing with numerical and approximation

issues. We have

divided them into three categories: (i) approximation of

eigenvalues; (ii)

approximation theory issues; and (iii) error bounds for

iterative methods for

matrix inversion.

The seven papers listed are only a small part of Hans’ work—but

they

were very influential. We, of course, cannot discuss any of

these papers

in detail, but will instead concentrate on those results that

are especially

insightful and elegant.

Approximation of Eigenvalues

[1] Upper and lower bounds for eigenvalues by finite difference

methods.

Communications on Pure and Appl. Math. 9 (1956), pp. 613-623.

[2] Lower bounds for higher eigenvalues by finite difference

methods. Pacific

J. Math. 8 (1958), pp. 339-368.

Approximation Theory Issues

[3] Optimal approximations and error bounds (joint with M.

Golumb). In

Proc. Symposium on Numerical Approximation, Univ. of Wisconsin

Press, 1959, pp. 117-190.

[4] Optimal approximation for functions prescribed at equally

spaced points.

Nat. Bureau of Standards J. of Research 65B, 2 (1961), pp.

99-104.

[5] On optimal numerical solution of partial differential

equations. SIAM J.

Numer. Anal. 9 (1972), pp. 182-198.

[6] Optimal numerical approximation of a linear operator.

Linear Alg. and

its Appl. 52/53 (1983), pp. 717-737.

Error Bounds for Iterative Methods for Matrix Inversion

[7] A posteriori error bounds in iterative matrix inversion. In

Numerical

Treatment of Partial Differential Equations, Academic Press

1965,

pp. 153-163. Proceedings of Symposium on Numerical Solution of

Partial

Differential Equations, held at the Univ. of Maryland in 1965

(Edited by

J. Bramble).

below) by

Hans Weinberger dealing with numerical and approximation

issues. We have

divided them into three categories: (i) approximation of

eigenvalues; (ii)

approximation theory issues; and (iii) error bounds for

iterative methods for

matrix inversion.

The seven papers listed are only a small part of Hans’ work—but

they

were very influential. We, of course, cannot discuss any of

these papers

in detail, but will instead concentrate on those results that

are especially

insightful and elegant.

**References**

Approximation of Eigenvalues

[1] Upper and lower bounds for eigenvalues by finite difference

methods.

Communications on Pure and Appl. Math. 9 (1956), pp. 613-623.

[2] Lower bounds for higher eigenvalues by finite difference

methods. Pacific

J. Math. 8 (1958), pp. 339-368.

[3] Optimal approximations and error bounds (joint with M.

Golumb). In

Proc. Symposium on Numerical Approximation, Univ. of Wisconsin

Press, 1959, pp. 117-190.

[4] Optimal approximation for functions prescribed at equally

spaced points.

Nat. Bureau of Standards J. of Research 65B, 2 (1961), pp.

99-104.

[5] On optimal numerical solution of partial differential

equations. SIAM J.

Numer. Anal. 9 (1972), pp. 182-198.

[6] Optimal numerical approximation of a linear operator.

Linear Alg. and

its Appl. 52/53 (1983), pp. 717-737.

Error Bounds for Iterative Methods for Matrix Inversion

[7] A posteriori error bounds in iterative matrix inversion. In

Numerical

Treatment of Partial Differential Equations, Academic Press

1965,

pp. 153-163. Proceedings of Symposium on Numerical Solution of

Partial

Differential Equations, held at the Univ. of Maryland in 1965

(Edited by

J. Bramble).