--- name: "lasd6" 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/lasd6" --- # lasd6(3) lasd6: D&C step: merge subproblems ## NAME lasd6 - lasd6: D&C step: merge subproblems ## SYNOPSIS ### Functions subroutine **dlasd6** (icompq, nl, nr, sqre, d, vf, vl, alpha, beta, idxq, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, iwork, info)\ **DLASD6** computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc. subroutine **slasd6** (icompq, nl, nr, sqre, d, vf, vl, alpha, beta, idxq, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, iwork, info)\ **SLASD6** computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc. ## Detailed Description ## Function Documentation ### subroutine dlasd6 (integer icompq, integer nl, integer nr, integer sqre, double precision, dimension( \* ) d, double precision, dimension( \* ) vf, double precision, dimension( \* ) vl, double precision alpha, double precision beta, integer, dimension( \* ) idxq, integer, dimension( \* ) perm, integer givptr, integer, dimension( ldgcol, \* ) givcol, integer ldgcol, double precision, dimension( ldgnum, \* ) givnum, integer ldgnum, double precision, dimension( ldgnum, \* ) poles, double precision, dimension( \* ) difl, double precision, dimension( \* ) difr, double precision, dimension( \* ) z, integer k, double precision c, double precision s, double precision, dimension( \* ) work, integer, dimension( \* ) iwork, integer info) **DLASD6** computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc. **Purpose:** ``` !> !> DLASD6 computes the SVD of an updated upper bidiagonal matrix B !> obtained by merging two smaller ones by appending a row. This !> routine is used only for the problem which requires all singular !> values and optionally singular vector matrices in factored form. !> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE. !> A related subroutine, DLASD1, handles the case in which all singular !> values and singular vectors of the bidiagonal matrix are desired. !> !> DLASD6 computes the SVD as follows: !> !> ( D1(in) 0 0 0 ) !> B = U(in) * ( Z1**T a Z2**T b ) * VT(in) !> ( 0 0 D2(in) 0 ) !> !> = U(out) * ( D(out) 0) * VT(out) !> !> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M !> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros !> elsewhere; and the entry b is empty if SQRE = 0. !> !> The singular values of B can be computed using D1, D2, the first !> components of all the right singular vectors of the lower block, and !> the last components of all the right singular vectors of the upper !> block. These components are stored and updated in VF and VL, !> respectively, in DLASD6. Hence U and VT are not explicitly !> referenced. !> !> The singular values are stored in D. The algorithm consists of two !> stages: !> !> The first stage consists of deflating the size of the problem !> when there are multiple singular values or if there is a zero !> in the Z vector. For each such occurrence the dimension of the !> secular equation problem is reduced by one. This stage is !> performed by the routine DLASD7. !> !> The second stage consists of calculating the updated !> singular values. This is done by finding the roots of the !> secular equation via the routine DLASD4 (as called by DLASD8). !> This routine also updates VF and VL and computes the distances !> between the updated singular values and the old singular !> values. !> !> DLASD6 is called from DLASDA. !> ``` **Parameters** *ICOMPQ* ``` !> ICOMPQ is INTEGER !> Specifies whether singular vectors are to be computed in !> factored form: !> = 0: Compute singular values only. !> = 1: Compute singular vectors in factored form as well. !> ``` *NL* ``` !> NL is INTEGER !> The row dimension of the upper block. NL >= 1. !> ``` *NR* ``` !> NR is INTEGER !> The row dimension of the lower block. NR >= 1. !> ``` *SQRE* ``` !> SQRE is INTEGER !> = 0: the lower block is an NR-by-NR square matrix. !> = 1: the lower block is an NR-by-(NR+1) rectangular matrix. !> !> The bidiagonal matrix has row dimension N = NL + NR + 1, !> and column dimension M = N + SQRE. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension ( NL+NR+1 ). !> On entry D(1:NL,1:NL) contains the singular values of the !> upper block, and D(NL+2:N) contains the singular values !> of the lower block. On exit D(1:N) contains the singular !> values of the modified matrix. !> ``` *VF* ``` !> VF is DOUBLE PRECISION array, dimension ( M ) !> On entry, VF(1:NL+1) contains the first components of all !> right singular vectors of the upper block; and VF(NL+2:M) !> contains the first components of all right singular vectors !> of the lower block. On exit, VF contains the first components !> of all right singular vectors of the bidiagonal matrix. !> ``` *VL* ``` !> VL is DOUBLE PRECISION array, dimension ( M ) !> On entry, VL(1:NL+1) contains the last components of all !> right singular vectors of the upper block; and VL(NL+2:M) !> contains the last components of all right singular vectors of !> the lower block. On exit, VL contains the last components of !> all right singular vectors of the bidiagonal matrix. !> ``` *ALPHA* ``` !> ALPHA is DOUBLE PRECISION !> Contains the diagonal element associated with the added row. !> ``` *BETA* ``` !> BETA is DOUBLE PRECISION !> Contains the off-diagonal element associated with the added !> row. !> ``` *IDXQ* ``` !> IDXQ is INTEGER array, dimension ( N ) !> This contains the permutation which will reintegrate the !> subproblem just solved back into sorted order, i.e. !> D( IDXQ( I = 1, N ) ) will be in ascending order. !> ``` *PERM* ``` !> PERM is INTEGER array, dimension ( N ) !> The permutations (from deflation and sorting) to be applied !> to each block. Not referenced if ICOMPQ = 0. !> ``` *GIVPTR* ``` !> GIVPTR is INTEGER !> The number of Givens rotations which took place in this !> subproblem. Not referenced if ICOMPQ = 0. !> ``` *GIVCOL* ``` !> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) !> Each pair of numbers indicates a pair of columns to take place !> in a Givens rotation. Not referenced if ICOMPQ = 0. !> ``` *LDGCOL* ``` !> LDGCOL is INTEGER !> leading dimension of GIVCOL, must be at least N. !> ``` *GIVNUM* ``` !> GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) !> Each number indicates the C or S value to be used in the !> corresponding Givens rotation. Not referenced if ICOMPQ = 0. !> ``` *LDGNUM* ``` !> LDGNUM is INTEGER !> The leading dimension of GIVNUM and POLES, must be at least N. !> ``` *POLES* ``` !> POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) !> On exit, POLES(1,*) is an array containing the new singular !> values obtained from solving the secular equation, and !> POLES(2,*) is an array containing the poles in the secular !> equation. Not referenced if ICOMPQ = 0. !> ``` *DIFL* ``` !> DIFL is DOUBLE PRECISION array, dimension ( N ) !> On exit, DIFL(I) is the distance between I-th updated !> (undeflated) singular value and the I-th (undeflated) old !> singular value. !> ``` *DIFR* ``` !> DIFR is DOUBLE PRECISION array, !> dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and !> dimension ( K ) if ICOMPQ = 0. !> On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not !> defined and will not be referenced. !> !> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the !> normalizing factors for the right singular vector matrix. !> !> See DLASD8 for details on DIFL and DIFR. !> ``` *Z* ``` !> Z is DOUBLE PRECISION array, dimension ( M ) !> The first elements of this array contain the components !> of the deflation-adjusted updating row vector. !> ``` *K* ``` !> K is INTEGER !> Contains the dimension of the non-deflated matrix, !> This is the order of the related secular equation. 1 <= K <=N. !> ``` *C* ``` !> C is DOUBLE PRECISION !> C contains garbage if SQRE =0 and the C-value of a Givens !> rotation related to the right null space if SQRE = 1. !> ``` *S* ``` !> S is DOUBLE PRECISION !> S contains garbage if SQRE =0 and the S-value of a Givens !> rotation related to the right null space if SQRE = 1. !> ``` *WORK* ``` !> WORK is DOUBLE PRECISION array, dimension ( 4 * M ) !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension ( 3 * N ) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value. !> > 0: if INFO = 1, a singular value did not converge !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Contributors:** Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA Definition at line **309** of file **dlasd6.f**. ### subroutine slasd6 (integer icompq, integer nl, integer nr, integer sqre, real, dimension( \* ) d, real, dimension( \* ) vf, real, dimension( \* ) vl, real alpha, real beta, integer, dimension( \* ) idxq, integer, dimension( \* ) perm, integer givptr, integer, dimension( ldgcol, \* ) givcol, integer ldgcol, real, dimension( ldgnum, \* ) givnum, integer ldgnum, real, dimension( ldgnum, \* ) poles, real, dimension( \* ) difl, real, dimension( \* ) difr, real, dimension( \* ) z, integer k, real c, real s, real, dimension( \* ) work, integer, dimension( \* ) iwork, integer info) **SLASD6** computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc. **Purpose:** ``` !> !> SLASD6 computes the SVD of an updated upper bidiagonal matrix B !> obtained by merging two smaller ones by appending a row. This !> routine is used only for the problem which requires all singular !> values and optionally singular vector matrices in factored form. !> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE. !> A related subroutine, SLASD1, handles the case in which all singular !> values and singular vectors of the bidiagonal matrix are desired. !> !> SLASD6 computes the SVD as follows: !> !> ( D1(in) 0 0 0 ) !> B = U(in) * ( Z1**T a Z2**T b ) * VT(in) !> ( 0 0 D2(in) 0 ) !> !> = U(out) * ( D(out) 0) * VT(out) !> !> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M !> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros !> elsewhere; and the entry b is empty if SQRE = 0. !> !> The singular values of B can be computed using D1, D2, the first !> components of all the right singular vectors of the lower block, and !> the last components of all the right singular vectors of the upper !> block. These components are stored and updated in VF and VL, !> respectively, in SLASD6. Hence U and VT are not explicitly !> referenced. !> !> The singular values are stored in D. The algorithm consists of two !> stages: !> !> The first stage consists of deflating the size of the problem !> when there are multiple singular values or if there is a zero !> in the Z vector. For each such occurrence the dimension of the !> secular equation problem is reduced by one. This stage is !> performed by the routine SLASD7. !> !> The second stage consists of calculating the updated !> singular values. This is done by finding the roots of the !> secular equation via the routine SLASD4 (as called by SLASD8). !> This routine also updates VF and VL and computes the distances !> between the updated singular values and the old singular !> values. !> !> SLASD6 is called from SLASDA. !> ``` **Parameters** *ICOMPQ* ``` !> ICOMPQ is INTEGER !> Specifies whether singular vectors are to be computed in !> factored form: !> = 0: Compute singular values only. !> = 1: Compute singular vectors in factored form as well. !> ``` *NL* ``` !> NL is INTEGER !> The row dimension of the upper block. NL >= 1. !> ``` *NR* ``` !> NR is INTEGER !> The row dimension of the lower block. NR >= 1. !> ``` *SQRE* ``` !> SQRE is INTEGER !> = 0: the lower block is an NR-by-NR square matrix. !> = 1: the lower block is an NR-by-(NR+1) rectangular matrix. !> !> The bidiagonal matrix has row dimension N = NL + NR + 1, !> and column dimension M = N + SQRE. !> ``` *D* ``` !> D is REAL array, dimension (NL+NR+1). !> On entry D(1:NL,1:NL) contains the singular values of the !> upper block, and D(NL+2:N) contains the singular values !> of the lower block. On exit D(1:N) contains the singular !> values of the modified matrix. !> ``` *VF* ``` !> VF is REAL array, dimension (M) !> On entry, VF(1:NL+1) contains the first components of all !> right singular vectors of the upper block; and VF(NL+2:M) !> contains the first components of all right singular vectors !> of the lower block. On exit, VF contains the first components !> of all right singular vectors of the bidiagonal matrix. !> ``` *VL* ``` !> VL is REAL array, dimension (M) !> On entry, VL(1:NL+1) contains the last components of all !> right singular vectors of the upper block; and VL(NL+2:M) !> contains the last components of all right singular vectors of !> the lower block. On exit, VL contains the last components of !> all right singular vectors of the bidiagonal matrix. !> ``` *ALPHA* ``` !> ALPHA is REAL !> Contains the diagonal element associated with the added row. !> ``` *BETA* ``` !> BETA is REAL !> Contains the off-diagonal element associated with the added !> row. !> ``` *IDXQ* ``` !> IDXQ is INTEGER array, dimension (N) !> This contains the permutation which will reintegrate the !> subproblem just solved back into sorted order, i.e. !> D( IDXQ( I = 1, N ) ) will be in ascending order. !> ``` *PERM* ``` !> PERM is INTEGER array, dimension ( N ) !> The permutations (from deflation and sorting) to be applied !> to each block. Not referenced if ICOMPQ = 0. !> ``` *GIVPTR* ``` !> GIVPTR is INTEGER !> The number of Givens rotations which took place in this !> subproblem. Not referenced if ICOMPQ = 0. !> ``` *GIVCOL* ``` !> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) !> Each pair of numbers indicates a pair of columns to take place !> in a Givens rotation. Not referenced if ICOMPQ = 0. !> ``` *LDGCOL* ``` !> LDGCOL is INTEGER !> leading dimension of GIVCOL, must be at least N. !> ``` *GIVNUM* ``` !> GIVNUM is REAL array, dimension ( LDGNUM, 2 ) !> Each number indicates the C or S value to be used in the !> corresponding Givens rotation. Not referenced if ICOMPQ = 0. !> ``` *LDGNUM* ``` !> LDGNUM is INTEGER !> The leading dimension of GIVNUM and POLES, must be at least N. !> ``` *POLES* ``` !> POLES is REAL array, dimension ( LDGNUM, 2 ) !> On exit, POLES(1,*) is an array containing the new singular !> values obtained from solving the secular equation, and !> POLES(2,*) is an array containing the poles in the secular !> equation. Not referenced if ICOMPQ = 0. !> ``` *DIFL* ``` !> DIFL is REAL array, dimension ( N ) !> On exit, DIFL(I) is the distance between I-th updated !> (undeflated) singular value and the I-th (undeflated) old !> singular value. !> ``` *DIFR* ``` !> DIFR is REAL array, !> dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and !> dimension ( K ) if ICOMPQ = 0. !> On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not !> defined and will not be referenced. !> !> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the !> normalizing factors for the right singular vector matrix. !> !> See SLASD8 for details on DIFL and DIFR. !> ``` *Z* ``` !> Z is REAL array, dimension ( M ) !> The first elements of this array contain the components !> of the deflation-adjusted updating row vector. !> ``` *K* ``` !> K is INTEGER !> Contains the dimension of the non-deflated matrix, !> This is the order of the related secular equation. 1 <= K <=N. !> ``` *C* ``` !> C is REAL !> C contains garbage if SQRE =0 and the C-value of a Givens !> rotation related to the right null space if SQRE = 1. !> ``` *S* ``` !> S is REAL !> S contains garbage if SQRE =0 and the S-value of a Givens !> rotation related to the right null space if SQRE = 1. !> ``` *WORK* ``` !> WORK is REAL array, dimension ( 4 * M ) !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension ( 3 * N ) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value. !> > 0: if INFO = 1, a singular value did not converge !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Contributors:** Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA Definition at line **309** of file **slasd6.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.