# Handbook for Automatic Computation: Volume II: Linear by Dr. J. H. Wilkinson F.R.S., Dr. C. Reinsch (auth.), Prof.

By Dr. J. H. Wilkinson F.R.S., Dr. C. Reinsch (auth.), Prof. Dr. F. L. Bauer, A. S. Householder, F. W. J. Olver, Prof. Dr. H. Rutishauser, K. Samelson, E. Stiefel (eds.)

The improvement of the the world over standardized language ALGOL has made it attainable to arrange methods which might be used with no amendment each time a working laptop or computer with an ALGOL translator is on the market. quantity Ia during this sequence gave information of the constrained model of ALGOL that is to be hired through the guide, and quantity Ib defined its implementation on a working laptop or computer. all the next volumes should be dedicated to a presentation of the elemental algorithms in a few particular parts of numerical research. this is often the 1st such quantity and it was once feIt that the subject Linear Algebra used to be a usual selection, because the correct algorithms are might be the main everyday in numerical research and feature the benefit of forming a weil outlined dass. The algorithms defined right here fall into major different types, linked to the answer of linear platforms and the algebraic eigenvalue challenge respectively and every set is preceded by means of an introductory bankruptcy giving a comparative assessment.

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This is easily seen by putting X kH = ... = X n = 0 in the first k equations of (1) or, respectively, Yl = ... = Yk =0 in the last n -k equations of (4). Linear Algebra 1/3 46 From this follows that All and A 22 are both positive definite (for all k) if A is so. Consequently, the first diagonal entry of A 22 is positive showing that for positive definite matriees the "next" step is always possible. By induction it also follows that (6) henee for positive definite matriees entries above the diagonal need not be eomputed.

If we denote the first computed inverse by X(ll then we have the iterative refinement procedure defined by X(s+l) = X(·) + XIs) (1 - = X(·) +XIs) B(s), = X(s)+Z(s). AX('») , In order to be effective it is essential that the right-hand side of (3) should not be expressed in the form 2X(s)_X(s)AX(s). The "residual matrix" 1 -AX(') must be computed using double-precision or accumulation of inner-products, each component being rounded only on completion. The conditions for convergence are the same as those for the iterative refinement of the solution ofAx=b, but if the condition is satisfied the convergence is effectively quadratic.

H. Rutishauser for pointing out that fact. Inversion of Positive Definite Matrices by the Gauss- Jordan Method 47 4. ALGOL Programs procedure gfdefl (n) trans: (a) exit: (fail); value n; integer n; array a; label fail; comment Gauss-Jordan algorithm for the in situ inversion of a positive definite matrix a[1 :n, 1 :n]. Only the lower triangle is transformed; begin integer i, f, k; real p, q; array h[1 :nJ; for k : = n step -1 until 1 do begin p:= a[1, 1J; ifp~O then goto fail; for i:= 2 step 1 until n do begin q:= a[i, 1J; h[iJ:= (ifi>k then q else -q)/P; for f := 2 step 1 until i do a[i -1, f -1J:= a[i, +qxhCiJ end i; a[n, nJ:= 1/P; for i:= 2 step 1 until n do a[n, i -1J:= h[iJ end k end gidefl; n procedure gidef2 (n) trans: (a) exit: (fail); value n; integer n; arraya; label fail; comment Gauss-Jordan algorithm for the in situ inversion of a positive definite matrix.