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- W2044327062 abstract "Let U be an $n times n$ upper-triangular matrix with unit diagonal, D an $n times n$ diagonal positive-definite matrix and ${b {fz}}$ an n-vector. The downdating problem is to compute the new triangular factors of the matrix $U^T DU - {bf {zz}}^T $. A common way to start the downdating process is to compute a vector ${bf y}$ such that $U^T DU - {bf {zz}}^T = U^T (D - {bf {yy}}^T )U$. If ${bf y}^T D^{ - 1} {bf y} leqq 1$ there are many suitable ways to carry out the downdating. This paper is concerned with the case when, theoretically, $U^T DU - {bf {zz}}^T $ should be positive (semi) definite but, because of computer rounding errors, ${bf y}^T D^{ - 1} {bf y} > 1$. In this case the modified algorithm replaces D by $D + T$ where $T = {operatorname{diag}}{ t_1 , cdots ,t_n } $ solves the following problem: [ {text{minimize}}sum_{i = 1}^n {t_i } ][ {text{subject to:}}{bf y}^T ( D + T )^{ - 1} {b fy} = 1{text{ and }}t_i geqq 0{text{ for }}i = 1, cdots ,n. ] The computation of T is simple and needs only a few operations. The elements of T satisfy $sumlimits_{i = 1}^n {t_i leqq n} |lambda |$ where $lambda $ is the negative eigenvalue of $D - {bf {yy}}^T $. This ensures that the correction is not much larger than necessary. The case when U has fewer rows than columns is also discussed." @default.
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- W2044327062 date "1983-03-01" @default.
- W2044327062 modified "2023-09-26" @default.
- W2044327062 title "A Diagonal Modification for the Downdating Algorithm" @default.
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- W2044327062 doi "https://doi.org/10.1137/0904006" @default.
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