Cleve B. Moler

Matrix computations with Fortran and paging

The efficiency of conventional Fortran programs for matrix computations can often be improved by reversing the order of nested loops. Such modifications produce modest savings in many common situations and very significant savings for large problems run under an operating system which uses paging.

Use of Splines and Numerical Integration in Geometrical Acoustics

Two contributions are made to ray theoretic techniques for propagation through continuously stratified media. The first makes use of spline polynomials to approximate speed‐of‐sound profiles smoothly. The second eliminates the singularity occuring in the usual range integral. A versatile automatic numerical procedure for solving a wide variety of acoustic propagation problems results.

Elementary interactions in quasi-linear hyperbolic systems

where t>0, o o < x < o o , U(t, x)=(u(t, x), v(t, x)), and the function F(U)= (f(u, v), g(u, v)) is smooth. GLIMM [2] has proved existence of a weak solution provided that the variation of the initial data Uo(x) is small. GLIMM & LAX [3] then improved this result by requiring that the oscillation of Uo(x) be small. SMOLLER [7] has proved the existence of a weak solution provided that Uo(x ) is constant except for a single arbitrary jump (the Riemann problem). Finally, NISmDA [6] has proved existence for general data Uo but for the special case f= l / v , g=u. Here we will prove existence under the assumption that Uo(x) is constant except for two jumps; that is, Uo (x) is of the form

Accurate solution of linear algebraic systems: a survey

Many problems encountered in computing involve the solution of the simultaneous linear equations (1) A x = b where A is a n-by-n matrix, [EQUATION] and b and x are vectors. Most people interested in computing are familiar with the basic concepts involved in solving such a system, but there are several useful refinements and extensions that are not so well known. It is the purpose of this article to introduce the non-expert to these ideas. Several of the newer ideas, as well as clarifications of older ones, are due to J. H. Wilkinson and are summarized in his book. Many people, including this writer, are indebted to G. E. Forsythe for their knowledge of this field. Forsythes recent survey contains an extensive bibliography together with material on related topics. A forthcoming text contains most of the details we will omit here, as well as computer programs which implement the techniques.

Semi-symbolic methods in partial differential equations

We discuss some partly numeric, partly symbolic methods which could conceivably be used for the solution of a reasonably large class of elliptic boundary value problems. The methods have already proved to be quite effective for some simpler forms of these problems. But as more general problems are approached, we exceed the capabilities of all the symbolic manipulation systems with which we are familiar.