# Problem Book in High-School Mathematics by Prilepko A.I. (ed.) By Prilepko A.I. (ed.)

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CHOLESKY FACTORIZATION (ROW-WISE VERSION). , they will compute the same factor R, even taking rounding errors into account. We remark that from it follows that the elements in R are bounded in terms of the diagonal elements in C. This is essential for the numerical stability of the Cholesky factorization. 1.

A relationship between the Cholesky factorization of ATA and the QR decomposition of A is given next. 20 CHAPTER 1. 2. Let A e R mxn have rank n. Then if the R-factor in the QR decomposition of A has positive diagonal elements it equals the Cholesky factor ofATA. 4 the Cholesky factor of ATA is unique. 2) it follows that which concludes the proof. 4) Hence we can express Q\ uniquely in terms of A and R. However the matrix Q<2 will not, in general, be uniquely determined. 5) which shows that the columns of Q\ and \$2 form orthonormal bases for Tl(A) and its complement.

17) is as useful as the SVD. 5. 7. 24 CHAPTER 1. 4. Sensitivity of Least Squares Solutions In this section we give results on the sensitivity of pseudoinverses and least squares solutions to perturbations in A and b. Many of the results below were first given by Wedin [824, 1973]. Stewart in [731, 1977] gives a unified treatment with interesting historical comments on the perturbation theory for pseudoinverses and least squares solutions. A more recent and excellent source of information is Stewart and Sun [754, 1990, Chap.