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- W2014100981 abstract "We present two new algorithms for floating-point computation of the generalized singular values of a real pair $(A,B)$ of full column rank matrices and for floating-point solution of the generalized eigenvalue problem $Hx=lambda Mx$ with symmetric, positive definite matrices H and M. The pair $(A,B)$ is replaced with an equivalent pair $(A',B')$, and the generalized singular values are computed as the singular values of the explicitly computed matrix $F=A' B'^{-1}$. The singular values of F are computed using the Jacobi method. The relative accuracy of the computed singular value approximations does not depend on column scalings of A and B; that is, the accuracy is nearly the same for all pairs $(AD_1,BD_2)$, with $D_1$, $D_2$ arbitrary diagonal, nonsingular matrices. Similarly, the pencil $H-lambda M$ is replaced with an equivalent pencil $H'-lambda M'$, and the eigenvalues of $H-lambda M$ are computed as the squares of the singular values of $G=L_H L_M^{-1}$, where $L_H$, $L_M$ are the Cholesky factors of $H'$, $M'$, respectively, and the matrix G is explicitly computed as the solution of a linear system of equations. For the computed approximation $lambda+deltalambda$ of any exact eigenvalue $lambda$, the relative error $|deltalambda|/lambda$ is of order $p(n) {mbox{boldmath$varepsilon$}} max{min_{Deltain{cal D}}kappa_2(Delta HDelta), min_{Deltain{cal D}}kappa_2(Delta MDelta)}$, where $p(n)$ is a modestly growing polynomial of the dimension of the problem, {mbox{boldmath$varepsilon$}} is the round-off unit of floating-point arithmetic, ${cal D}$ denotes the set of diagonal nonsingular matrices, and $kappa_2(cdot)$ is the spectral condition number. Furthermore, floating-point computation corresponds to an exact computation with $H+delta H$, $M+delta M$, where, for all i, j, $|delta H_{ij}|/sqrt{H_{ii}H_{jj}}$ and $|delta M_{ij}|/sqrt{M_{ii}M_{jj}}$ are of order of {mbox{boldmath$varepsilon$}} times a modest function of n." @default.
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- W2014100981 creator A5068311458 @default.
- W2014100981 date "1998-10-01" @default.
- W2014100981 modified "2023-09-24" @default.
- W2014100981 title "A Tangent Algorithm for Computing the Generalized Singular Value Decomposition" @default.
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- W2014100981 doi "https://doi.org/10.1137/s0036142995289883" @default.
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