Results 1 
2 of
2
Accurate Singular Values of Bidiagonal Matrices
 SIAM J. SCI. STAT. COMPUT
, 1990
"... Computing the singular values of a bidiagonal matrix is the fin al phase of the standard algow rithm for the singular value decomposition of a general matrix. We present a new algorithm hich computes all the singular values of a bidiagonal matrix to high relative accuracy independent of their magni ..."
Abstract

Cited by 100 (17 self)
 Add to MetaCart
Computing the singular values of a bidiagonal matrix is the fin al phase of the standard algow rithm for the singular value decomposition of a general matrix. We present a new algorithm hich computes all the singular values of a bidiagonal matrix to high relative accuracy independent of their magnitudes. In contrast, the standard algorithm for bidiagonal matrices may compute small singular values with no relative accuracy at all. Numerical experiments show that the new algorithm is comparable in speed to the standard algorithm , and frequently faster.
Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices
, 1980
"... When computing eigenvalues of sym metric matrices and singular values of general matrices in finite precision arithmetic we in general only expect to compute them with an error bound proportional to the product of machine precision and the norm of the matrix. In particular, we do not expect to comp ..."
Abstract

Cited by 80 (14 self)
 Add to MetaCart
When computing eigenvalues of sym metric matrices and singular values of general matrices in finite precision arithmetic we in general only expect to compute them with an error bound proportional to the product of machine precision and the norm of the matrix. In particular, we do not expect to compute tiny eigenvalues and singular values to high relative accuracy. There are some important classes of matrices where we can do much better, including bidiagonal matrices, scaled diagonally dominant matrices, and scaled diagonally dominant definite pencils. These classes include many graded matrices, and all sym metric positive definite matrices which can be consistently ordered (and thus all symmetric positive definite tridiagonal matrices). In particular, the singular values and eigenvalues are determined to high relative precision independent of their magnitudes, and there are algorithms to compute them this accurately. The eigenvectors are also determined more accurately than for general matrices, and may be computed more accurately as well. This work extends results of Kahan and Demmel for bidiagonal and tridiagonal matrices.