--- name: "bdsvdx" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/bdsvdx" --- # bdsvdx(3) bdsvdx: bidiagonal SVD, bisection ## NAME bdsvdx - bdsvdx: bidiagonal SVD, bisection ## SYNOPSIS ### Functions subroutine **dbdsvdx** (uplo, jobz, range, n, d, e, vl, vu, il, iu, ns, s, z, ldz, work, iwork, info)\ **DBDSVDX** subroutine **sbdsvdx** (uplo, jobz, range, n, d, e, vl, vu, il, iu, ns, s, z, ldz, work, iwork, info)\ **SBDSVDX** ## Detailed Description ## Function Documentation ### 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) **DBDSVDX** **Purpose:** ``` !> !> 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.) !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': B is upper bidiagonal; !> = 'L': B is lower bidiagonal. !> ``` *JOBZ* ``` !> JOBZ is CHARACTER*1 !> = 'N': Compute singular values only; !> = 'V': Compute singular values and singular vectors. !> ``` *RANGE* ``` !> 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. !> ``` *N* ``` !> N is INTEGER !> The order of the bidiagonal matrix. N >= 0. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension (N) !> The n diagonal elements of the bidiagonal matrix B. !> ``` *E* ``` !> 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. !> ``` *VL* ``` !> 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'. !> ``` *VU* ``` !> 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'. !> ``` *IL* ``` !> 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'. !> ``` *IU* ``` !> 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'. !> ``` *NS* ``` !> 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. !> ``` *S* ``` !> S is DOUBLE PRECISION array, dimension (N) !> The first NS elements contain the selected singular values in !> ascending order. !> ``` *Z* ``` !> 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. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> JOBZ = 'V', LDZ >= max(2,N*2). !> ``` *WORK* ``` !> WORK is DOUBLE PRECISION array, dimension (14*N) !> ``` *IWORK* ``` !> 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. !> ``` *INFO* ``` !> 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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **224** of file **dbdsvdx.f**. ### 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) **SBDSVDX** **Purpose:** ``` !> !> 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.) !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': B is upper bidiagonal; !> = 'L': B is lower bidiagonal. !> ``` *JOBZ* ``` !> JOBZ is CHARACTER*1 !> = 'N': Compute singular values only; !> = 'V': Compute singular values and singular vectors. !> ``` *RANGE* ``` !> 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. !> ``` *N* ``` !> N is INTEGER !> The order of the bidiagonal matrix. N >= 0. !> ``` *D* ``` !> D is REAL array, dimension (N) !> The n diagonal elements of the bidiagonal matrix B. !> ``` *E* ``` !> 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. !> ``` *VL* ``` !> 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'. !> ``` *VU* ``` !> 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'. !> ``` *IL* ``` !> 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'. !> ``` *IU* ``` !> 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'. !> ``` *NS* ``` !> 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. !> ``` *S* ``` !> S is REAL array, dimension (N) !> The first NS elements contain the selected singular values in !> ascending order. !> ``` *Z* ``` !> 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. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> JOBZ = 'V', LDZ >= max(2,N*2). !> ``` *WORK* ``` !> WORK is REAL array, dimension (14*N) !> ``` *IWORK* ``` !> 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. !> ``` *INFO* ``` !> 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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **224** of file **sbdsvdx.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.