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NAME SBDSQR - compute the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B SYNOPSIS SUBROUTINE SBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO ) CHARACTER UPLO INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU REAL C( LDC, * ), D( * ), E( * ), U( LDU, * ), VT( LDVT, * ), WORK( * ) PURPOSE SBDSQR computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the transpose of P), where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and Q and P are orthogonal matrices. The routine computes S, and optionally computes U * Q, P' * VT, or Q' * C, for given real input matrices U, VT, and C. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3, for a detailed description of the algorithm. ARGUMENTS UPLO (input) CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal. N (input) INTEGER The order of the matrix B. N >= 0. NCVT (input) INTEGER The number of columns of the matrix VT. NCVT >= 0. NRU (input) INTEGER The number of rows of the matrix U. NRU >= 0. NCC (input) INTEGER The number of columns of the matrix C. NCC >= 0. D (input/output) REAL array, dimension (N) On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B in decreasing order. E (input/output) REAL array, dimension (N-1) On entry, the (n-1) off-diagonal elements of the bidiagonal matrix B. On normal exit, E is des- troyed. VT (input/output) REAL array, dimension (LDVT, NCVT) On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P' * VT. VT is not referenced if NCVT = 0. LDVT (input) INTEGER The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. U (input/output) REAL array, dimension (LDU, N) On entry, an NRU-by-N matrix U. On exit, U is overwritten by U * Q. U is not referenced if NRU = 0. LDU (input) INTEGER The leading dimension of the array U. LDU >= max(1,NRU). C (input/output) REAL array, dimension (LDC, NCC) On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q' * C. C is not referenced if NCC = 0. LDC (input) INTEGER The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. WORK (workspace) REAL array, dimension (MAX( 1, 4*N-4 )) WORK is not referenced if NCVT = NRU = NCC = 0. INFO (output) INTEGER = 0: successful exit < 0: If INFO = -i, the i-th argument had an illegal value > 0: the algorithm did not converge; D and E con- tain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero. PARAMETERS TOLMUL REAL, default = max(10,min(100,EPS**(-1/8))) TOLMUL controls the convergence criterion of the QR loop. If it is positive, TOLMUL*EPS is the desired relative precision in the computed singular values. If it is negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value. abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (for there to be some accuracy in the results). Default is to lose at either one eighth or 2 of the available decimal digits in each com- puted singular value (whichever is smaller). MAXITR INTEGER, default = 6 MAXITR controls the maximum number of passes of the algorithm through its inner loop. The algorithms stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2.