swMATH ID: 11489
Software Authors: Brezinski, C.; Redivo-Zaglia, M.
Description: Treatment of near-breakdown in the CGS algorithm. Lanczos’ method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (x k ) such thatr k =b−Ax k =P k (A)r 0 wherer 0=b−Ax 0.P k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingr k =P k 2 (A)r0. In the recurrence relation forP k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can seriously affect the algorithm, a situation known as near-breakdown. In this paper it is shown how to avoid near-breakdown in the CGS algorithm in order to obtain a more stable method.
Homepage: http://www.netlib.org/numeralgo/index.html
Keywords: orthogonal polynomials; numerical examples; near-breakdown situation; conjugate gradient squared algorithm; CGS algorithm; Lanczos’ method; recurrence relations; method of recursive zoom algorithm
Related Software: na1; CGS; BiCGstab; GpBiCg; ARPACK; JDQR; RODAS; DSUBSP; LSQR; MA32; MA42; MA47; DRIC; ScaLAPACK; LAPACK; testmatrix; QMRPACK; JDQZ; eigs; SRRIT
Cited in: 24 Documents

Standard Articles

1 Publication describing the Software, including 1 Publication in zbMATH Year
Treatment of near-breakdown in the CGS algorithm. Zbl 0810.65028
Brezinski, C.; Redivo-Zaglia, M.

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