.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\&.