Error analysis of some techniques for updating orthogonal decompositions

Author:
C. C. Paige

Journal:
Math. Comp. **34** (1980), 465-471

MSC:
Primary 65F30

DOI:
https://doi.org/10.1090/S0025-5718-1980-0559196-9

MathSciNet review:
559196

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider accurate and efficient methods for updating the result of the transformation , *Q* orthogonal, of a given matrix *B* when *Q* is available. Adding or deleting a row, or adding a column of *B* leads to a continuation of the original transformation, and as such is numerically stable. In particular, we discuss a well-known method for updating when a column of *B* is deleted, and show that it is as numerically stable as the problem allows. The results extend to two-sided transformations of the form . The methods and analyses are independent of the form or rank of *B* and *C*, and so are widely applicable.

**[1]**P. E. GILL, G. H. COLUB, W. MURRAY & M. A. SAUNDERS, ``Methods for modifying matrix factorizations,''*Math. Comp.*, v. 28, 1974, pp. 505-535. MR**0343558 (49:8299)****[2]**C. L. LAWSON & R. J. HANSON,*Solving Least Squares Problems*, Prentice-Hall, Englewood Cliffs, N. J., 1974. MR**0366019 (51:2270)****[3]**C. C. PAIGE, ``Computer solution and perturbation analysis of generalized linear least squares problems,''*Math. Comp.*, v. 33, 1979, pp. 171-183. MR**514817 (80b:65013)****[4]**J. H. WILKINSON,*The Algebraic Eigenvalue Problem*, Clarendon Press, Oxford, 1965. MR**32**#1894. MR**0184422 (32:1894)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1980-0559196-9

Keywords:
Error analysis,
updating techniques,
orthogonal decompositions,
orthogonal matrices

Article copyright:
© Copyright 1980
American Mathematical Society