Naslov
Accurate computation of the product-induced singular value decomposition with applications

Autori
Drmač, Zlatko

Izvornik
SIAM journal on numerical analysis (0036-1429) 35 (1998), 5; 1969-1994

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
contragredient transformation; eigenvalue problem; product induced

Sažetak
We present a new algorithm for floating--point computation of the singular value decomposition (SVD) of the product $B^{ au}C$, where $B$ and $C$ are full row rank matrices. The algorithm replaces the pair $(B,C)$ with an equivalent pair $(B',C')$ and then it uses the Jacobi SVD lgorithm to compute the SVD of the explicitly computed matrix $B'^{ au}C'$. In this way, each nonzero singular value $sigma$ is approximated with some $sigma+deltasigma$, where the relative error $|deltasigma|/sigma$ is, up to a factor of the dimensions, of order $ off {min_{Deltain{cal D}}kappa_2(Delta B)+min_{Deltain{cal D}} kappa_2(Delta C)} $, where ${cal D}$ denotes the set of diagonal nonsingular matrices, $kappa_2(cdot)$ denotes the spectral condition number and $ off$ is the roundoff unit of floating--point arithmetic. The new algorithm is applied to the eigenvalue roblem $HM x = lambda x$ with symmetric positive efinite $H$ and $M$. It is shown that each eigenvalue $lambda$ is computed with high relative accuracy and that the relative error $|deltalambda|/lambda$ of the computed pproximation $lambda+deltalambda$ is, up to factor of the dimension, of order $ off{min_{Deltain{cal D}}kappa_2(Delta HDelta) + min_{Deltain{cal D}}kappa_2 (Delta MDelta)}$. The new algorithm can also be used for accurate SVD computation of a single matrix $G$ that admits an accurate factorization $G=B^{ au}C$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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