.TH "SRC/cbdsqr.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/cbdsqr.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBcbdsqr\fP (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info)" .br .RI "\fBCBDSQR\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine cbdsqr (character uplo, integer n, integer ncvt, integer nru, integer ncc, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldvt, * ) vt, integer ldvt, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldc, * ) c, integer ldc, real, dimension( * ) rwork, integer info)" .PP \fBCBDSQR\fP .PP \fBPurpose:\fP .RS 4 .PP .nf CBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm\&. The SVD of B has the form B = Q * S * P**H where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors\&. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P**H*VT instead of P**H, for given complex input matrices U and VT\&. When U and VT are the unitary matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by CGEBRD, then A = (U*Q) * S * (P**H*VT) is the SVD of A\&. Optionally, the subroutine may also compute Q**H*C for a given complex input matrix C\&. See 'Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy,' by J\&. Demmel and W\&. Kahan, LAPACK Working Note #3 (or SIAM J\&. Sci\&. Statist\&. Comput\&. vol\&. 11, no\&. 5, pp\&. 873-912, Sept 1990) and 'Accurate singular values and differential qd algorithms,' by B\&. Parlett and V\&. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix B\&. N >= 0\&. .fi .PP .br \fINCVT\fP .PP .nf NCVT is INTEGER The number of columns of the matrix VT\&. NCVT >= 0\&. .fi .PP .br \fINRU\fP .PP .nf NRU is INTEGER The number of rows of the matrix U\&. NRU >= 0\&. .fi .PP .br \fINCC\fP .PP .nf NCC is INTEGER The number of columns of the matrix C\&. NCC >= 0\&. .fi .PP .br \fID\fP .PP .nf D is 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\&. .fi .PP .br \fIE\fP .PP .nf E is REAL array, dimension (N-1) On entry, the N-1 offdiagonal elements of the bidiagonal matrix B\&. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input\&. .fi .PP .br \fIVT\fP .PP .nf VT is COMPLEX array, dimension (LDVT, NCVT) On entry, an N-by-NCVT matrix VT\&. On exit, VT is overwritten by P**H * VT\&. Not referenced if NCVT = 0\&. .fi .PP .br \fILDVT\fP .PP .nf LDVT is INTEGER The leading dimension of the array VT\&. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0\&. .fi .PP .br \fIU\fP .PP .nf U is COMPLEX array, dimension (LDU, N) On entry, an NRU-by-N matrix U\&. On exit, U is overwritten by U * Q\&. Not referenced if NRU = 0\&. .fi .PP .br \fILDU\fP .PP .nf LDU is INTEGER The leading dimension of the array U\&. LDU >= max(1,NRU)\&. .fi .PP .br \fIC\fP .PP .nf C is COMPLEX array, dimension (LDC, NCC) On entry, an N-by-NCC matrix C\&. On exit, C is overwritten by Q**H * C\&. Not referenced if NCC = 0\&. .fi .PP .br \fILDC\fP .PP .nf LDC is INTEGER The leading dimension of the array C\&. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0\&. .fi .PP .br \fIRWORK\fP .PP .nf RWORK is REAL array, dimension (4*N) .fi .PP .br \fIINFO\fP .PP .nf INFO is 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 contain 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\&. .fi .PP .RE .PP \fBInternal Parameters:\fP .RS 4 .PP .nf 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 computed 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\&. .fi .PP .RE .PP \fBNote:\fP .RS 4 .PP .nf Bug report from Cezary Dendek\&. On November 3rd 2023, the INTEGER variable MAXIT = MAXITR*N**2 is removed since it can overflow pretty easily (for N larger or equal than 18,919)\&. We instead use MAXITDIVN = MAXITR*N\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .PP Definition at line \fB231\fP of file \fBcbdsqr\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.