Error analysis of some techniques for updating orthogonal decompositions

Author:
C. C. Paige

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

MSC:
Primary 65F30

MathSciNet review:
559196

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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. Golub, W. Murray, and M. A. Saunders,*Methods for modifying matrix factorizations*, Math. Comp.**28**(1974), 505–535. MR**0343558**, 10.1090/S0025-5718-1974-0343558-6**[2]**Charles L. Lawson and Richard J. Hanson,*Solving least squares problems*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Prentice-Hall Series in Automatic Computation. MR**0366019****[3]**C. C. Paige,*Computer solution and perturbation analysis of generalized linear least squares problems*, Math. Comp.**33**(1979), no. 145, 171–183. MR**514817**, 10.1090/S0025-5718-1979-0514817-3**[4]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422**

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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