ISSN:
1572-9125
Keywords:
Least square
;
linear equations
;
orthogonalization
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract A general analysis of the condition of the linear least squares problem is given. The influence of rounding errors is studied in detail for a modified version of the Gram-Schmidt orthogonalization to obtain a factorizationA=QR of a givenm×n matrixA, whereR is upper triangular andQ T Q=I. Letx be the vector which minimizes ‖b−Ax‖2 andr=b−Ax. It is shown that if inner-products are accumulated in double precision then the errors in the computedx andr are less than the errors resulting from some simultaneous initial perturbation δA, δb such that $$\parallel \delta A\parallel _E /\parallel A\parallel _E \approx \parallel \delta b\parallel _2 /\parallel b\parallel _2 \approx 2 \cdot n^{3/2} machine units.$$ No reorthogonalization is needed and the result is independent of the pivoting strategy used.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01934122
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