lals0#

Functions

void slals0(
    const INT           icompq,
    const INT           nl,
    const INT           nr,
    const INT           sqre,
    const INT           nrhs,
          f32* restrict B,
    const INT           ldb,
          f32* restrict BX,
    const INT           ldbx,
    const INT* restrict perm,
    const INT           givptr,
    const INT* restrict givcol,
    const INT           ldgcol,
    const f32* restrict givnum,
    const INT           ldgnum,
    const f32* restrict poles,
    const f32* restrict difl,
    const f32* restrict difr,
    const f32* restrict Z,
    const INT           k,
    const f32           c,
    const f32           s,
          f32* restrict work,
          INT*          info
);
void slals0(const INT icompq, const INT nl, const INT nr, const INT sqre, const INT nrhs, f32 *restrict B, const INT ldb, f32 *restrict BX, const INT ldbx, const INT *restrict perm, const INT givptr, const INT *restrict givcol, const INT ldgcol, const f32 *restrict givnum, const INT ldgnum, const f32 *restrict poles, const f32 *restrict difl, const f32 *restrict difr, const f32 *restrict Z, const INT k, const f32 c, const f32 s, f32 *restrict work, INT *info)#

SLALS0 applies back the multiplying factors of either the left or the right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide-and-conquer SVD approach.

For the left singular vector matrix, three types of orthogonal matrices are involved: 

(1L) Givens rotations: the number of such rotations is GIVPTR; the
     pairs of columns/rows they were applied to are stored in GIVCOL;
     and the C- and S-values of these rotations are stored in GIVNUM.

(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
     row, and for J=2:N, PERM(J)-th row of B is to be moved to the
     J-th row.

(3L) The left singular vector matrix of the remaining matrix.

For the right singular vector matrix, four types of orthogonal matrices are involved: 

(1R) The right singular vector matrix of the remaining matrix.

(2R) If SQRE = 1, one extra Givens rotation to generate the right
     null space.

(3R) The inverse transformation of (2L).

(4R) The inverse transformation of (1L).

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Parameters

in
icompq

Specifies whether singular vectors are to be computed in factored form: icompq=0: Left singular vector matrix. icompq=1: Right singular vector matrix.

in
nl

The row dimension of the upper block. nl>=1.

in
nr

The row dimension of the lower block. nr>=1.

in
sqre

sqre=0: the lower block is an nr-by-nr square matrix. sqre=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.

in
nrhs

The number of columns of B and BX. nrhs must be at least 1.

inout
B

Real array of dimension (ldb, nrhs). On input, B contains the right hand sides of the least squares problem in rows 0 through m-1. On output, B contains the solution X in rows 0 through n-1.

in
ldb

The leading dimension of B. ldb must be at least max(1,max(m,n)).

out
BX

Real array of dimension (ldbx, nrhs).

in
ldbx

The leading dimension of BX.

in
perm

Integer array of dimension (n). The permutations (from deflation and sorting) applied to the two blocks.

in
givptr

The number of Givens rotations which took place in this subproblem.

in
givcol

Integer array of dimension (ldgcol, 2). Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.

in
ldgcol

The leading dimension of givcol, must be at least n.

in
givnum

Real array of dimension (ldgnum, 2). Each number indicates the C or S value used in the corresponding Givens rotation.

in
ldgnum

The leading dimension of arrays difr, poles and givnum, must be at least k.

in
poles

Real array of dimension (ldgnum, 2). On entry, poles[0:k-1,0] is an array containing the old singular values which will be used to form the secular equation in SLASD4; poles[0:k-1,1] is an array containing the new singular values obtained from solving the secular equation.

in
difl

Real array of dimension (k). On entry, difl[i] is the distance between i-th updated (undeflated) singular value and the i-th (undeflated) old singular value.

in
difr

Real array of dimension (ldgnum, 2). On entry, difr[i,0] contains the distances between i-th updated (undeflated) singular value and the (i+1)-th (undeflated) old singular value. And difr[i,1] is the normalizing factor for the i-th right singular vector.

in
Z

Real array of dimension (k). Contain the components of the deflation-adjusted updating row vector.

in
k

Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1<=k<=n.

in
c

c contains garbage if sqre=0 and the C-value of a Givens rotation related to the right null space if sqre=1.

in
s

s contains garbage if sqre=0 and the S-value of a Givens rotation related to the right null space if sqre=1.

out
work

Real array of dimension (k).

out
info

info=0: successful exit. info<0: if info=-i, the i-th argument had an illegal value.

Functions

void dlals0(
    const INT           icompq,
    const INT           nl,
    const INT           nr,
    const INT           sqre,
    const INT           nrhs,
          f64* restrict B,
    const INT           ldb,
          f64* restrict BX,
    const INT           ldbx,
    const INT* restrict perm,
    const INT           givptr,
    const INT* restrict givcol,
    const INT           ldgcol,
    const f64* restrict givnum,
    const INT           ldgnum,
    const f64* restrict poles,
    const f64* restrict difl,
    const f64* restrict difr,
    const f64* restrict Z,
    const INT           k,
    const f64           c,
    const f64           s,
          f64* restrict work,
          INT*          info
);
void dlals0(const INT icompq, const INT nl, const INT nr, const INT sqre, const INT nrhs, f64 *restrict B, const INT ldb, f64 *restrict BX, const INT ldbx, const INT *restrict perm, const INT givptr, const INT *restrict givcol, const INT ldgcol, const f64 *restrict givnum, const INT ldgnum, const f64 *restrict poles, const f64 *restrict difl, const f64 *restrict difr, const f64 *restrict Z, const INT k, const f64 c, const f64 s, f64 *restrict work, INT *info)#

DLALS0 applies back the multiplying factors of either the left or the right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide-and-conquer SVD approach.

For the left singular vector matrix, three types of orthogonal matrices are involved: 

(1L) Givens rotations: the number of such rotations is GIVPTR; the
     pairs of columns/rows they were applied to are stored in GIVCOL;
     and the C- and S-values of these rotations are stored in GIVNUM.

(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
     row, and for J=2:N, PERM(J)-th row of B is to be moved to the
     J-th row.

(3L) The left singular vector matrix of the remaining matrix.

For the right singular vector matrix, four types of orthogonal matrices are involved: 

(1R) The right singular vector matrix of the remaining matrix.

(2R) If SQRE = 1, one extra Givens rotation to generate the right
     null space.

(3R) The inverse transformation of (2L).

(4R) The inverse transformation of (1L).

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Parameters

in
icompq

Specifies whether singular vectors are to be computed in factored form: icompq=0: Left singular vector matrix. icompq=1: Right singular vector matrix.

in
nl

The row dimension of the upper block. nl>=1.

in
nr

The row dimension of the lower block. nr>=1.

in
sqre

sqre=0: the lower block is an nr-by-nr square matrix. sqre=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.

in
nrhs

The number of columns of B and BX. nrhs must be at least 1.

inout
B

Real array of dimension (ldb, nrhs). On input, B contains the right hand sides of the least squares problem in rows 0 through m-1. On output, B contains the solution X in rows 0 through n-1.

in
ldb

The leading dimension of B. ldb must be at least max(1,max(m,n)).

out
BX

Real array of dimension (ldbx, nrhs).

in
ldbx

The leading dimension of BX.

in
perm

Integer array of dimension (n). The permutations (from deflation and sorting) applied to the two blocks.

in
givptr

The number of Givens rotations which took place in this subproblem.

in
givcol

Integer array of dimension (ldgcol, 2). Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.

in
ldgcol

The leading dimension of givcol, must be at least n.

in
givnum

Real array of dimension (ldgnum, 2). Each number indicates the C or S value used in the corresponding Givens rotation.

in
ldgnum

The leading dimension of arrays difr, poles and givnum, must be at least k.

in
poles

Real array of dimension (ldgnum, 2). On entry, poles[0:k-1,0] is an array containing the old singular values which will be used to form the secular equation in DLASD4; poles[0:k-1,1] is an array containing the new singular values obtained from solving the secular equation.

in
difl

Real array of dimension (k). On entry, difl[i] is the distance between i-th updated (undeflated) singular value and the i-th (undeflated) old singular value.

in
difr

Real array of dimension (ldgnum, 2). On entry, difr[i,0] contains the distances between i-th updated (undeflated) singular value and the (i+1)-th (undeflated) old singular value. And difr[i,1] is the normalizing factor for the i-th right singular vector.

in
Z

Real array of dimension (k). Contain the components of the deflation-adjusted updating row vector.

in
k

Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1<=k<=n.

in
c

c contains garbage if sqre=0 and the C-value of a Givens rotation related to the right null space if sqre=1.

in
s

s contains garbage if sqre=0 and the S-value of a Givens rotation related to the right null space if sqre=1.

out
work

Real array of dimension (k).

out
info

info=0: successful exit. info<0: if info=-i, the i-th argument had an illegal value.

Functions

void clals0(
    const INT           icompq,
    const INT           nl,
    const INT           nr,
    const INT           sqre,
    const INT           nrhs,
          c64* restrict B,
    const INT           ldb,
          c64* restrict BX,
    const INT           ldbx,
    const INT* restrict perm,
    const INT           givptr,
    const INT* restrict givcol,
    const INT           ldgcol,
    const f32* restrict givnum,
    const INT           ldgnum,
    const f32* restrict poles,
    const f32* restrict difl,
    const f32* restrict difr,
    const f32* restrict Z,
    const INT           k,
    const f32           c,
    const f32           s,
          f32* restrict rwork,
          INT*          info
);
void clals0(const INT icompq, const INT nl, const INT nr, const INT sqre, const INT nrhs, c64 *restrict B, const INT ldb, c64 *restrict BX, const INT ldbx, const INT *restrict perm, const INT givptr, const INT *restrict givcol, const INT ldgcol, const f32 *restrict givnum, const INT ldgnum, const f32 *restrict poles, const f32 *restrict difl, const f32 *restrict difr, const f32 *restrict Z, const INT k, const f32 c, const f32 s, f32 *restrict rwork, INT *info)#

CLALS0 applies back the multiplying factors of either the left or the right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide-and-conquer SVD approach.

For the left singular vector matrix, three types of orthogonal matrices are involved: 

(1L) Givens rotations: the number of such rotations is GIVPTR; the
     pairs of columns/rows they were applied to are stored in GIVCOL;
     and the C- and S-values of these rotations are stored in GIVNUM.

(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
     row, and for J=2:N, PERM(J)-th row of B is to be moved to the
     J-th row.

(3L) The left singular vector matrix of the remaining matrix.

For the right singular vector matrix, four types of orthogonal matrices are involved: 

(1R) The right singular vector matrix of the remaining matrix.

(2R) If SQRE = 1, one extra Givens rotation to generate the right
     null space.

(3R) The inverse transformation of (2L).

(4R) The inverse transformation of (1L).

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Parameters

in
icompq

Specifies whether singular vectors are to be computed in factored form: icompq=0: Left singular vector matrix. icompq=1: Right singular vector matrix.

in
nl

The row dimension of the upper block. nl>=1.

in
nr

The row dimension of the lower block. nr>=1.

in
sqre

sqre=0: the lower block is an nr-by-nr square matrix. sqre=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.

in
nrhs

The number of columns of B and BX. nrhs must be at least 1.

inout
B

Complex array of dimension (ldb, nrhs). On input, B contains the right hand sides of the least squares problem in rows 0 through m-1. On output, B contains the solution X in rows 0 through n-1.

in
ldb

The leading dimension of B. ldb must be at least max(1,max(m,n)).

out
BX

Complex array of dimension (ldbx, nrhs).

in
ldbx

The leading dimension of BX.

in
perm

Integer array of dimension (n). The permutations (from deflation and sorting) applied to the two blocks.

in
givptr

The number of Givens rotations which took place in this subproblem.

in
givcol

Integer array of dimension (ldgcol, 2). Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.

in
ldgcol

The leading dimension of givcol, must be at least n.

in
givnum

Real array of dimension (ldgnum, 2). Each number indicates the C or S value used in the corresponding Givens rotation.

in
ldgnum

The leading dimension of arrays difr, poles and givnum, must be at least k.

in
poles

Real array of dimension (ldgnum, 2). On entry, poles[0:k-1,0] is an array containing the old singular values which will be used to form the secular equation in CLASD4; poles[0:k-1,1] is an array containing the new singular values obtained from solving the secular equation.

in
difl

Real array of dimension (k). On entry, difl[i] is the distance between i-th updated (undeflated) singular value and the i-th (undeflated) old singular value.

in
difr

Real array of dimension (ldgnum, 2). On entry, difr[i,0] contains the distances between i-th updated (undeflated) singular value and the (i+1)-th (undeflated) old singular value. And difr[i,1] is the normalizing factor for the i-th right singular vector.

in
Z

Real array of dimension (k). Contain the components of the deflation-adjusted updating row vector.

in
k

Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1<=k<=n.

in
c

c contains garbage if sqre=0 and the C-value of a Givens rotation related to the right null space if sqre=1.

in
s

s contains garbage if sqre=0 and the S-value of a Givens rotation related to the right null space if sqre=1.

out
rwork

Real array of dimension (k*(1+nrhs) + 2*nrhs).

out
info

info=0: successful exit. info<0: if info=-i, the i-th argument had an illegal value.

Functions

void zlals0(
    const INT            icompq,
    const INT            nl,
    const INT            nr,
    const INT            sqre,
    const INT            nrhs,
          c128* restrict B,
    const INT            ldb,
          c128* restrict BX,
    const INT            ldbx,
    const INT*  restrict perm,
    const INT            givptr,
    const INT*  restrict givcol,
    const INT            ldgcol,
    const f64*  restrict givnum,
    const INT            ldgnum,
    const f64*  restrict poles,
    const f64*  restrict difl,
    const f64*  restrict difr,
    const f64*  restrict Z,
    const INT            k,
    const f64            c,
    const f64            s,
          f64*  restrict rwork,
          INT*           info
);
void zlals0(const INT icompq, const INT nl, const INT nr, const INT sqre, const INT nrhs, c128 *restrict B, const INT ldb, c128 *restrict BX, const INT ldbx, const INT *restrict perm, const INT givptr, const INT *restrict givcol, const INT ldgcol, const f64 *restrict givnum, const INT ldgnum, const f64 *restrict poles, const f64 *restrict difl, const f64 *restrict difr, const f64 *restrict Z, const INT k, const f64 c, const f64 s, f64 *restrict rwork, INT *info)#

ZLALS0 applies back the multiplying factors of either the left or the right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide-and-conquer SVD approach.

For the left singular vector matrix, three types of orthogonal matrices are involved: 

(1L) Givens rotations: the number of such rotations is GIVPTR; the
     pairs of columns/rows they were applied to are stored in GIVCOL;
     and the C- and S-values of these rotations are stored in GIVNUM.

(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
     row, and for J=2:N, PERM(J)-th row of B is to be moved to the
     J-th row.

(3L) The left singular vector matrix of the remaining matrix.

For the right singular vector matrix, four types of orthogonal matrices are involved: 

(1R) The right singular vector matrix of the remaining matrix.

(2R) If SQRE = 1, one extra Givens rotation to generate the right
     null space.

(3R) The inverse transformation of (2L).

(4R) The inverse transformation of (1L).

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Parameters

in
icompq

Specifies whether singular vectors are to be computed in factored form: icompq=0: Left singular vector matrix. icompq=1: Right singular vector matrix.

in
nl

The row dimension of the upper block. nl>=1.

in
nr

The row dimension of the lower block. nr>=1.

in
sqre

sqre=0: the lower block is an nr-by-nr square matrix. sqre=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.

in
nrhs

The number of columns of B and BX. nrhs must be at least 1.

inout
B

Complex array of dimension (ldb, nrhs). On input, B contains the right hand sides of the least squares problem in rows 0 through m-1. On output, B contains the solution X in rows 0 through n-1.

in
ldb

The leading dimension of B. ldb must be at least max(1,max(m,n)).

out
BX

Complex array of dimension (ldbx, nrhs).

in
ldbx

The leading dimension of BX.

in
perm

Integer array of dimension (n). The permutations (from deflation and sorting) applied to the two blocks.

in
givptr

The number of Givens rotations which took place in this subproblem.

in
givcol

Integer array of dimension (ldgcol, 2). Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.

in
ldgcol

The leading dimension of givcol, must be at least n.

in
givnum

Real array of dimension (ldgnum, 2). Each number indicates the C or S value used in the corresponding Givens rotation.

in
ldgnum

The leading dimension of arrays difr, poles and givnum, must be at least k.

in
poles

Real array of dimension (ldgnum, 2). On entry, poles[0:k-1,0] is an array containing the old singular values which will be used to form the secular equation in ZLASD4; poles[0:k-1,1] is an array containing the new singular values obtained from solving the secular equation.

in
difl

Real array of dimension (k). On entry, difl[i] is the distance between i-th updated (undeflated) singular value and the i-th (undeflated) old singular value.

in
difr

Real array of dimension (ldgnum, 2). On entry, difr[i,0] contains the distances between i-th updated (undeflated) singular value and the (i+1)-th (undeflated) old singular value. And difr[i,1] is the normalizing factor for the i-th right singular vector.

in
Z

Real array of dimension (k). Contain the components of the deflation-adjusted updating row vector.

in
k

Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1<=k<=n.

in
c

c contains garbage if sqre=0 and the C-value of a Givens rotation related to the right null space if sqre=1.

in
s

s contains garbage if sqre=0 and the S-value of a Givens rotation related to the right null space if sqre=1.

out
rwork

Real array of dimension (k*(1+nrhs) + 2*nrhs).

out
info

info=0: successful exit. info<0: if info=-i, the i-th argument had an illegal value.