.TH "bdsvdx" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME bdsvdx \- bdsvdx: bidiagonal SVD, bisection .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBdbdsvdx\fP (uplo, jobz, range, n, d, e, vl, vu, il, iu, ns, s, z, ldz, work, iwork, info)" .br .RI "\fBDBDSVDX\fP " .ti -1c .RI "subroutine \fBsbdsvdx\fP (uplo, jobz, range, n, d, e, vl, vu, il, iu, ns, s, z, ldz, work, iwork, info)" .br .RI "\fBSBDSVDX\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine dbdsvdx (character uplo, character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, integer ns, double precision, dimension( * ) s, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)" .PP \fBDBDSVDX\fP .PP \fBPurpose:\fP .RS 4 .PP .nf DBDSVDX computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT, where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively\&. Given an upper bidiagonal B with diagonal D = [ d_1 d_2 \&.\&.\&. d_N ] and superdiagonal E = [ e_1 e_2 \&.\&.\&. e_N-1 ], DBDSVDX computes the singular value decomposition of B through the eigenvalues and eigenvectors of the N*2-by-N*2 tridiagonal matrix | 0 d_1 | | d_1 0 e_1 | TGK = | e_1 0 d_2 | | d_2 \&. \&. | | \&. \&. \&. | If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / sqrt(2) = ( v_1 u_1 v_2 u_2 \&.\&.\&. v_n u_n ) / sqrt(2), and P = [ e_{n+1} e_{1} e_{n+2} e_{2} \&.\&.\&. ]\&. Given a TGK matrix, one can either a) compute -s,-v and change signs so that the singular values (and corresponding vectors) are already in descending order (as in DGESVD/DGESDD) or b) compute s,v and reorder the values (and corresponding vectors)\&. DBDSVDX implements a) by calling DSTEVX (bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm\&. (See P\&. Willems and B\&. Lang, A framework for the MR^3 algorithm: theory and implementation, SIAM J\&. Sci\&. Comput\&., 35:740-766, 2013\&.) .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 \fIJOBZ\fP .PP .nf JOBZ is CHARACTER*1 = 'N': Compute singular values only; = 'V': Compute singular values and singular vectors\&. .fi .PP .br \fIRANGE\fP .PP .nf RANGE is CHARACTER*1 = 'A': all singular values will be found\&. = 'V': all singular values in the half-open interval [VL,VU) will be found\&. = 'I': the IL-th through IU-th singular values will be found\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the bidiagonal matrix\&. N >= 0\&. .fi .PP .br \fID\fP .PP .nf D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the bidiagonal matrix B\&. .fi .PP .br \fIE\fP .PP .nf E is DOUBLE PRECISION array, dimension (max(1,N-1)) The (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1\&. .fi .PP .br \fIVL\fP .PP .nf VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for singular values\&. VU > VL\&. Not referenced if RANGE = 'A' or 'I'\&. .fi .PP .br \fIVU\fP .PP .nf VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for singular values\&. VU > VL\&. Not referenced if RANGE = 'A' or 'I'\&. .fi .PP .br \fIIL\fP .PP .nf IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned\&. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0\&. Not referenced if RANGE = 'A' or 'V'\&. .fi .PP .br \fIIU\fP .PP .nf IU is INTEGER If RANGE='I', the index of the largest singular value to be returned\&. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0\&. Not referenced if RANGE = 'A' or 'V'\&. .fi .PP .br \fINS\fP .PP .nf NS is INTEGER The total number of singular values found\&. 0 <= NS <= N\&. If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1\&. .fi .PP .br \fIS\fP .PP .nf S is DOUBLE PRECISION array, dimension (N) The first NS elements contain the selected singular values in ascending order\&. .fi .PP .br \fIZ\fP .PP .nf Z is DOUBLE PRECISION array, dimension (2*N,K) If JOBZ = 'V', then if INFO = 0 the first NS columns of Z contain the singular vectors of the matrix B corresponding to the selected singular values, with U in rows 1 to N and V in rows N+1 to N*2, i\&.e\&. Z = [ U ] [ V ] If JOBZ = 'N', then Z is not referenced\&. Note: The user must ensure that at least K = NS+1 columns are supplied in the array Z; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used\&. .fi .PP .br \fILDZ\fP .PP .nf LDZ is INTEGER The leading dimension of the array Z\&. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is DOUBLE PRECISION array, dimension (14*N) .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (12*N) If JOBZ = 'V', then if INFO = 0, the first NS elements of IWORK are zero\&. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DSTEVX\&. .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: if INFO = i, then i eigenvectors failed to converge in DSTEVX\&. The indices of the eigenvectors (as returned by DSTEVX) are stored in the array IWORK\&. if INFO = N*2 + 1, an internal error occurred\&. .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 \fB224\fP of file \fBdbdsvdx\&.f\fP\&. .SS "subroutine sbdsvdx (character uplo, character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, integer ns, real, dimension( * ) s, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)" .PP \fBSBDSVDX\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SBDSVDX computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT, where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively\&. Given an upper bidiagonal B with diagonal D = [ d_1 d_2 \&.\&.\&. d_N ] and superdiagonal E = [ e_1 e_2 \&.\&.\&. e_N-1 ], SBDSVDX computes the singular value decomposition of B through the eigenvalues and eigenvectors of the N*2-by-N*2 tridiagonal matrix | 0 d_1 | | d_1 0 e_1 | TGK = | e_1 0 d_2 | | d_2 \&. \&. | | \&. \&. \&. | If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / sqrt(2) = ( v_1 u_1 v_2 u_2 \&.\&.\&. v_n u_n ) / sqrt(2), and P = [ e_{n+1} e_{1} e_{n+2} e_{2} \&.\&.\&. ]\&. Given a TGK matrix, one can either a) compute -s,-v and change signs so that the singular values (and corresponding vectors) are already in descending order (as in SGESVD/SGESDD) or b) compute s,v and reorder the values (and corresponding vectors)\&. SBDSVDX implements a) by calling SSTEVX (bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm\&. (See P\&. Willems and B\&. Lang, A framework for the MR^3 algorithm: theory and implementation, SIAM J\&. Sci\&. Comput\&., 35:740-766, 2013\&.) .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 \fIJOBZ\fP .PP .nf JOBZ is CHARACTER*1 = 'N': Compute singular values only; = 'V': Compute singular values and singular vectors\&. .fi .PP .br \fIRANGE\fP .PP .nf RANGE is CHARACTER*1 = 'A': all singular values will be found\&. = 'V': all singular values in the half-open interval [VL,VU) will be found\&. = 'I': the IL-th through IU-th singular values will be found\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the bidiagonal matrix\&. N >= 0\&. .fi .PP .br \fID\fP .PP .nf D is REAL array, dimension (N) The n diagonal elements of the bidiagonal matrix B\&. .fi .PP .br \fIE\fP .PP .nf E is REAL array, dimension (max(1,N-1)) The (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1\&. .fi .PP .br \fIVL\fP .PP .nf VL is REAL If RANGE='V', the lower bound of the interval to be searched for singular values\&. VU > VL\&. Not referenced if RANGE = 'A' or 'I'\&. .fi .PP .br \fIVU\fP .PP .nf VU is REAL If RANGE='V', the upper bound of the interval to be searched for singular values\&. VU > VL\&. Not referenced if RANGE = 'A' or 'I'\&. .fi .PP .br \fIIL\fP .PP .nf IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned\&. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0\&. Not referenced if RANGE = 'A' or 'V'\&. .fi .PP .br \fIIU\fP .PP .nf IU is INTEGER If RANGE='I', the index of the largest singular value to be returned\&. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0\&. Not referenced if RANGE = 'A' or 'V'\&. .fi .PP .br \fINS\fP .PP .nf NS is INTEGER The total number of singular values found\&. 0 <= NS <= N\&. If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1\&. .fi .PP .br \fIS\fP .PP .nf S is REAL array, dimension (N) The first NS elements contain the selected singular values in ascending order\&. .fi .PP .br \fIZ\fP .PP .nf Z is REAL array, dimension (2*N,K) If JOBZ = 'V', then if INFO = 0 the first NS columns of Z contain the singular vectors of the matrix B corresponding to the selected singular values, with U in rows 1 to N and V in rows N+1 to N*2, i\&.e\&. Z = [ U ] [ V ] If JOBZ = 'N', then Z is not referenced\&. Note: The user must ensure that at least K = NS+1 columns are supplied in the array Z; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used\&. .fi .PP .br \fILDZ\fP .PP .nf LDZ is INTEGER The leading dimension of the array Z\&. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is REAL array, dimension (14*N) .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (12*N) If JOBZ = 'V', then if INFO = 0, the first NS elements of IWORK are zero\&. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DSTEVX\&. .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: if INFO = i, then i eigenvectors failed to converge in SSTEVX\&. The indices of the eigenvectors (as returned by SSTEVX) are stored in the array IWORK\&. if INFO = N*2 + 1, an internal error occurred\&. .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 \fB224\fP of file \fBsbdsvdx\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.