lalsd(3) LAPACK lalsd(3)

lalsd - lalsd: uses SVD for least squares, step in gelsd


subroutine clalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, rwork, iwork, info)
CLALSD uses the singular value decomposition of A to solve the least squares problem. subroutine dlalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, iwork, info)
DLALSD uses the singular value decomposition of A to solve the least squares problem. subroutine slalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, iwork, info)
SLALSD uses the singular value decomposition of A to solve the least squares problem. subroutine zlalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, rwork, iwork, info)
ZLALSD uses the singular value decomposition of A to solve the least squares problem.

CLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

 CLALSD uses the singular value decomposition of A to solve the least
 squares problem of finding X to minimize the Euclidean norm of each
 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
 are N-by-NRHS. The solution X overwrites B.
 The singular values of A smaller than RCOND times the largest
 singular value are treated as zero in solving the least squares
 problem; in this case a minimum norm solution is returned.
 The actual singular values are returned in D in ascending order.

Parameters

UPLO 

          UPLO is CHARACTER*1
         = 'U': D and E define an upper bidiagonal matrix.
         = 'L': D and E define a  lower bidiagonal matrix.

SMLSIZ

          SMLSIZ is INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

N

          N is INTEGER
         The dimension of the  bidiagonal matrix.  N >= 0.

NRHS

          NRHS is INTEGER
         The number of columns of B. NRHS must be at least 1.

D

          D is REAL array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit, if INFO = 0, D contains its singular values.

E

          E is REAL array, dimension (N-1)
         Contains the super-diagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

B

          B is COMPLEX array, dimension (LDB,NRHS)
         On input, B contains the right hand sides of the least
         squares problem. On output, B contains the solution X.

LDB

          LDB is INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,N).

RCOND

          RCOND is REAL
         The singular values of A less than or equal to RCOND times
         the largest singular value are treated as zero in solving
         the least squares problem. If RCOND is negative,
         machine precision is used instead.
         For example, if diag(S)*X=B were the least squares problem,
         where diag(S) is a diagonal matrix of singular values, the
         solution would be X(i) = B(i) / S(i) if S(i) is greater than
         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
         RCOND*max(S).

RANK

          RANK is INTEGER
         The number of singular values of A greater than RCOND times
         the largest singular value.

WORK

          WORK is COMPLEX array, dimension (N * NRHS).

RWORK

          RWORK is REAL array, dimension at least
         (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
         MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),
         where
         NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

IWORK

          IWORK is INTEGER array, dimension (3*N*NLVL + 11*N).

INFO

          INFO is INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  The algorithm failed to compute a singular value while
               working on the submatrix lying in rows and columns
               INFO/(N+1) through MOD(INFO,N+1).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

DLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

 DLALSD uses the singular value decomposition of A to solve the least
 squares problem of finding X to minimize the Euclidean norm of each
 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
 are N-by-NRHS. The solution X overwrites B.
 The singular values of A smaller than RCOND times the largest
 singular value are treated as zero in solving the least squares
 problem; in this case a minimum norm solution is returned.
 The actual singular values are returned in D in ascending order.

Parameters

UPLO 

          UPLO is CHARACTER*1
         = 'U': D and E define an upper bidiagonal matrix.
         = 'L': D and E define a  lower bidiagonal matrix.

SMLSIZ

          SMLSIZ is INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

N

          N is INTEGER
         The dimension of the  bidiagonal matrix.  N >= 0.

NRHS

          NRHS is INTEGER
         The number of columns of B. NRHS must be at least 1.

D

          D is DOUBLE PRECISION array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit, if INFO = 0, D contains its singular values.

E

          E is DOUBLE PRECISION array, dimension (N-1)
         Contains the super-diagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
         On input, B contains the right hand sides of the least
         squares problem. On output, B contains the solution X.

LDB

          LDB is INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,N).

RCOND

          RCOND is DOUBLE PRECISION
         The singular values of A less than or equal to RCOND times
         the largest singular value are treated as zero in solving
         the least squares problem. If RCOND is negative,
         machine precision is used instead.
         For example, if diag(S)*X=B were the least squares problem,
         where diag(S) is a diagonal matrix of singular values, the
         solution would be X(i) = B(i) / S(i) if S(i) is greater than
         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
         RCOND*max(S).

RANK

          RANK is INTEGER
         The number of singular values of A greater than RCOND times
         the largest singular value.

WORK

          WORK is DOUBLE PRECISION array, dimension at least
         (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
         where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

IWORK

          IWORK is INTEGER array, dimension at least
         (3*N*NLVL + 11*N)

INFO

          INFO is INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  The algorithm failed to compute a singular value while
               working on the submatrix lying in rows and columns
               INFO/(N+1) through MOD(INFO,N+1).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

SLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

 SLALSD uses the singular value decomposition of A to solve the least
 squares problem of finding X to minimize the Euclidean norm of each
 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
 are N-by-NRHS. The solution X overwrites B.
 The singular values of A smaller than RCOND times the largest
 singular value are treated as zero in solving the least squares
 problem; in this case a minimum norm solution is returned.
 The actual singular values are returned in D in ascending order.

Parameters

UPLO 

          UPLO is CHARACTER*1
         = 'U': D and E define an upper bidiagonal matrix.
         = 'L': D and E define a  lower bidiagonal matrix.

SMLSIZ

          SMLSIZ is INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

N

          N is INTEGER
         The dimension of the  bidiagonal matrix.  N >= 0.

NRHS

          NRHS is INTEGER
         The number of columns of B. NRHS must be at least 1.

D

          D is REAL array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit, if INFO = 0, D contains its singular values.

E

          E is REAL array, dimension (N-1)
         Contains the super-diagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

B

          B is REAL array, dimension (LDB,NRHS)
         On input, B contains the right hand sides of the least
         squares problem. On output, B contains the solution X.

LDB

          LDB is INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,N).

RCOND

          RCOND is REAL
         The singular values of A less than or equal to RCOND times
         the largest singular value are treated as zero in solving
         the least squares problem. If RCOND is negative,
         machine precision is used instead.
         For example, if diag(S)*X=B were the least squares problem,
         where diag(S) is a diagonal matrix of singular values, the
         solution would be X(i) = B(i) / S(i) if S(i) is greater than
         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
         RCOND*max(S).

RANK

          RANK is INTEGER
         The number of singular values of A greater than RCOND times
         the largest singular value.

WORK

          WORK is REAL array, dimension at least
         (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
         where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

IWORK

          IWORK is INTEGER array, dimension at least
         (3*N*NLVL + 11*N)

INFO

          INFO is INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  The algorithm failed to compute a singular value while
               working on the submatrix lying in rows and columns
               INFO/(N+1) through MOD(INFO,N+1).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

ZLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

 ZLALSD uses the singular value decomposition of A to solve the least
 squares problem of finding X to minimize the Euclidean norm of each
 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
 are N-by-NRHS. The solution X overwrites B.
 The singular values of A smaller than RCOND times the largest
 singular value are treated as zero in solving the least squares
 problem; in this case a minimum norm solution is returned.
 The actual singular values are returned in D in ascending order.

Parameters

UPLO 

          UPLO is CHARACTER*1
         = 'U': D and E define an upper bidiagonal matrix.
         = 'L': D and E define a  lower bidiagonal matrix.

SMLSIZ

          SMLSIZ is INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

N

          N is INTEGER
         The dimension of the  bidiagonal matrix.  N >= 0.

NRHS

          NRHS is INTEGER
         The number of columns of B. NRHS must be at least 1.

D

          D is DOUBLE PRECISION array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit, if INFO = 0, D contains its singular values.

E

          E is DOUBLE PRECISION array, dimension (N-1)
         Contains the super-diagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

B

          B is COMPLEX*16 array, dimension (LDB,NRHS)
         On input, B contains the right hand sides of the least
         squares problem. On output, B contains the solution X.

LDB

          LDB is INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,N).

RCOND

          RCOND is DOUBLE PRECISION
         The singular values of A less than or equal to RCOND times
         the largest singular value are treated as zero in solving
         the least squares problem. If RCOND is negative,
         machine precision is used instead.
         For example, if diag(S)*X=B were the least squares problem,
         where diag(S) is a diagonal matrix of singular values, the
         solution would be X(i) = B(i) / S(i) if S(i) is greater than
         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
         RCOND*max(S).

RANK

          RANK is INTEGER
         The number of singular values of A greater than RCOND times
         the largest singular value.

WORK

          WORK is COMPLEX*16 array, dimension (N * NRHS)

RWORK

          RWORK is DOUBLE PRECISION array, dimension at least
         (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
         MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),
         where
         NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

IWORK

          IWORK is INTEGER array, dimension at least
         (3*N*NLVL + 11*N).

INFO

          INFO is INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  The algorithm failed to compute a singular value while
               working on the submatrix lying in rows and columns
               INFO/(N+1) through MOD(INFO,N+1).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Generated automatically by Doxygen for LAPACK from the source code.

Fri Aug 9 2024 02:33:22 Version 3.12.0