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Solving algebraic problems with high accuracy. (English) Zbl 0597.65018
A new approach to scientific computation, Proc. Symp., Yorktown Heights, N.Y. 1982, Notes Rep. Comput. Sci. Appl. Math. 7, 51-120 (1983).
[For the entire collection see Zbl 0538.00023.]
The paper discusses algorithms for solving algebraic problems with high accuracy based on a computer arithmetic by U. Kulisch and W. Miranker [Computer arithmetic in theory and practice (1981; Zbl 0487.65026)]. These algorithms are used in program packages written by the author on PASCAL-SC language for a Z80 minicomputer. Most of the presented algorithms use an iteration and, as the author writes, allow to compute solutions with so called ”last significant bit accuracy” in less than ”ten iterations”.
The author stresses that ”the new algorithms perform the verification automatically without any efforts on the part of a user, without any knowledge about the condition of the problem and, most importantly, without either a deep mathematical background or an extensive investigation”.
The approaches used in the paper are of interest for computer scientists, but it’s difficult to agree with the author that ”the user without any knowledge about the problem” can make non-trivial reasoning about the result because ”the purpose of computing is insight, not numbers”.
Reviewer: A.Beljaev

65Fxx Numerical linear algebra
65F05 Direct numerical methods for linear systems and matrix inversion
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F30 Other matrix algorithms (MSC2010)
65H05 Numerical computation of solutions to single equations