LAPACK

NAME¶

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

SYNOPSIS¶

Functions¶

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.

Detailed Description¶

Function Documentation¶

subroutine clalsd (character uplo, integer smlsiz, integer n, integer nrhs, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldb, * ) b, integer ldb, real rcond, integer rank, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)¶

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 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 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 is REAL array, dimension (N-1)


         Contains the super-diagonal entries of the bidiagonal matrix.


         On exit, E has been destroyed.



          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

subroutine dlalsd (character uplo, integer smlsiz, integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldb, * ) b, integer ldb, double precision rcond, integer rank, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)¶

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 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 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 is DOUBLE PRECISION array, dimension (N-1)


         Contains the super-diagonal entries of the bidiagonal matrix.


         On exit, E has been destroyed.



          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

subroutine slalsd (character uplo, integer smlsiz, integer n, integer nrhs, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldb, * ) b, integer ldb, real rcond, integer rank, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)¶

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 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 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 is REAL array, dimension (N-1)


         Contains the super-diagonal entries of the bidiagonal matrix.


         On exit, E has been destroyed.



          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

subroutine zlalsd (character uplo, integer smlsiz, integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, complex16, dimension( ldb, ) b, integer ldb, double precision rcond, integer rank, complex16, dimension( ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)¶

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 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 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 is DOUBLE PRECISION array, dimension (N-1)


         Contains the super-diagonal entries of the bidiagonal matrix.


         On exit, E has been destroyed.



          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

Author¶

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