APPENDIX An Axiomatic Theory of Numerical Computation with an Application to the Quotient-Difference Algorithm Editor's Foreword H. Rutishauser occupied himself with the topic of this appendix already in 1968 (report [21] in the bibliography to the appendix) and dis­ cussed part of this material in a course held during the spring semester of 1969. Later, however, the content and text were revised by Rutishauser and significantly extended. He also intended to present the very interest­ ing third chapter on "Finite Arithmetic" at a meeting in Oberwolfach which took place in November, 1970, shortly after his death. It is not known, however, in which fonn he wanted to publish the whole work, which does not require for the reader to have any extensive previous knowledge, but which exceeds the usual length of a journal article. What is certain is only that the work remained unfinished. At least seven chapters were contemplated, but the manuscript breaks off in the fifth, entitled "Forcing Coincidence". Fortunately, the text nevertheless is fairly well rounded. It contains in the first two chapters an introduction to the qd-algorithm in a partly novel exposition. The third chapter, as already mentioned, gives an axiomatic approach to numerical computa­ tion. The principal goal is not an examination of completeness and independence of the axiomatic system, but rather the possibility of prov­ ing for an algorithm that it never fails in spite of the presence of rounding errors. The discussions of the qd-algorithm and its stationary fonn in the fourth and fifth chapters then also point into the same direction. The text of the appendix agrees over long stretches word for word with the handwritten manuscript of H. Rutishauser. In a few places, how­ ever, theorems and proofs were fonnulated a bit more accurately and with more details. A larger reorganization was necessary only in the third chapter, where the statements now contained in Theorems A13, A14 and 462 Editor's Foreword A16 have been regrouped. The editors, in addition, prepared the bibliog­ raphy and inserted the references thereto (which were left open in the manuscript). Vancouver, B.C., February, 1976 M. Gutknecht CHAPTER Al Introduction §A1.1. The eigenvalues ofa qd-row An important area of application of the qd-algorithm(l) is the com­ putation of the eigenvalues of a tridiagonal matrix. By a trivial (diagonal) similarity transformation, such a matrix almost always can be brought into the form ql -ql 0 -el q2 + el -q2 -e2 q3 + e2 -q3 A= (1) -qn-l 0 -en-l qn + en-l in which only 2n-1 independent quantities occur. These are collected in a qd-row 1 The quotient-difference algorithm (briefly qd-algorithm) in principle is a computational method for the determination of the poles of a meromorphic function, but has many other applications. It is due to H. Rutishauser [8-12]. (The report [14], among other things, contains [8-11] in partly revised form. [21] represents a precursor of the unfinished work printed here.) (Editors'remark) 464 Chapter AI. Introduction (2) By the eigenvalues of Z one means the eigenvalues of the matrix A asso­ ciated with (2) according to (1). This tenninology suggests itself very naturally, since the computation of the eigenvalues is accomplished exclusively with the help of data structures of the fonn (2). The present work deals with the computation of the eigenvalues of a qd-row (2), particular attention being paid to the sequential reliability of the numerical process (that is, the process contaminated by rounding errors). It is possible to prove in an important special case that the com­ putational process, even when perturbed by rounding errors, must run its course without any mishaps, and must furnish approximately the correct eigenvalues. §A1.2. The progressive form of the qd-algorithm The detennination of the eigenvalues AI, ... , An of a qd-row (2) is effected in principle by means of an iterative process which consists of infinitely many steps of the following kind: A progressive qd-step is defined by the following computational algorithmct): comment it is assumed that en = 0; qi := qI + eI; for k := 2 step 1 until n do begin (3) ek-I := (ek-tlqk-I) x qk; qk := (qk - ek-I) + ek; end for k; Provided that these operations are executable (which presupposes qi, qz, ... , q~-I "* 0), (3) produces a new qd-row Z' = {qi,ei,qz, ... , q~}, which is expressed symbolically by 1 Computational algorithms are given as pieces of ALGOL programs in which declara­ tions and input and output operations are omitted, and indices, etc. are written in a fonn not permissible in ALGOL. Lower indices are true indices, while upper indices, primes, asterisks, etc. distinguish quantities which during the computation are stored in the same register. (Editors' remark) §Al.2. The progressive form of the qd-algorithm 465 o Z ------'> Z', (4) the number 0 indicating that one is dealing with a qd-step without shift. It is to be noted that the type of parenthesizing prescribed in (3) is of cru­ cial importance for the numerical execution (see Ch. A4). The matrix associated with the row Z' is qlI -qlI o -ei q2 + ei -q2I q3 + e2 A'= -qn-lI o On the basis of the rhombus rules q" + e"-1 = qk + eko q"e" = qk+l eko which follow from (3), it transpires that A' is diagonally similare) to the matrix ql + el -q2 o -el q2 + e2 B= (5) o 1 2 The diagonal matrix D with B = D- A'D has the diagonal elements d l = I, '-1 '-1 d. = IT e;/e; = IT q;+l/q; (k = 2, ... ,n). (Editors' remark) i=1 i=l 466 Chapter AI. Introduction which in tum is similar to A in (1), because A = XV, B = YX with qI ° 1 -1 -el q2 1 -1 ° -e2 q3 1 -1 X= , Y= -1 ° -en-I qn ° 1 We thus have: Theorem AI. If the operations (3) are executable, then Z and Z' have the same eigenvalues. The computational process (4) is now continued iteratively: 000 Z ~ Z'; Z' ~ Z"; Z" ~ Z'" ; ... , which produces an infinite sequence of qd-rows, all having the same eigenvalues, for which under certain conditions lIm. Z (j) - {AI, 0, ~, 0, ~, 0, ... , 0, An}, (6) j-')OO that is, the qk-values tend to the eigenvalues Ak as the iteration progresses (see [14], Ch. I)e). 3 A detailed convergence proof is given in [4], §7.6. [21] contains a simple convergence proof for the special case of positive qd-rows (cf. §A1.4). (Editors'remark) §A1.3. The generating function of a qd-row 467 §A1.3. The generating function of a qd-row One associates with the qd-row (2) as generating function the finite continued fraction J (z) = _1_ J..2.....:!... q2 e2 (7) z- 1- z- 1- z- z- 1 (see [14, 20]). (7) represents a rational function with a denominator of degree n; its poles are also the eigenvalues of (2). The qd-algorithm therefore also permits the calculation of the poles of a rational function; if they are simple, one can in this way even compute the residues(l). Between the generating function J (z) of Z and J'(z) of Z' there holds the relatione) J'(z) = zJ(z) - 1 , (8) ql which was used in [22] to prove the convergence of the qd-algorithm. §AI.4. Positive qd-rows The computational algorithm (3), and hence also its iterative con­ tinuation, is numerically endangered because of the possibility of one of the denominators qk-l vanishing or almost vanishing. There exists, how­ ever, a special case in which this danger (even in numerical computation) does not arise, namely the case when all elements of the row Z are posi­ tive: Definition. A qd-row Z = {ql,el,q2, ... ,qn} is called positive (in symbols: Z > 0) if 1 The author had the intention to describe this in a 7th chapter of this work. He dealt with this problem already briefly in [14], Ch. II, §1O, and in [22]. (Editors'remark) 2 r here does not denote the derivative of f (Editors' remark) 468 Chapter AI. Introduction qk > 0 (k = 1, ... , n), (9) ek > 0 (k = 1, ... , n - 1). One then has by [6], Ch. 9, Theorems 1 and 5: Theorem A2. The eigenvalues of a positive qd-row are all real, positive, and simple. Proof In the case of a positive qd-row the associated matrix (1) is diagonally similar to ql ~ 0 ~ q2 + el "'q2e2 H= (10) "'qn-l en-l 0 "'qn-l en-l qn + en-l (with all square roots positive), and this matrix admits a Cholesky decom­ position H = RTR with ~ ~ 0 ~. ~ R= (11) "'en-l 0 ~ §Al.4. Positive qd-rows 469 where all qko ek are positive; q.e.d.(l) For positive rows one now obtains the following important fact: Theorem A3. If the qd-row Z is positive, then the qd-rows Z(j) generated from it by the progressive qd-algorithm, that is, by o 0 0 Z ------> Z', Z' ------> Z", Z" ------> Z'" , . , are likewise positive, and one has unconditionally: . (j) - hm Z - P"l, 0, ~, 0, ... , An}, (12) j~oo where Al > ~ > ... > An > °are the eigenvalues of Z. Proof (a) By (3) and (9), qi = ql + el > e I > 0, ei = q2(edql) < q2, as well as ei > 0, qz = (q2 - el) + e2 > e2 > 0, ez = q3(e2/qZ) < q3, as well as ez > 0, Therefore, Z' > 0, and likewise Z" > 0, etc. b) For the convergence of lim li = tk and lim eY) = 0, (13) j~oo j~oo see, for example, [21].