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

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subroutineclalsd(characteruplo,integersmlsiz,integern,integernrhs,real,dimension(*)d,real,dimension(*)e,complex,dimension(ldb,*)b,integerldb,realrcond,integerrank,complex,dimension(*)work,real,dimension(*)rwork,integer,dimension(*)iwork,integerinfo)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. ParametersUPLO 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 subroutinedlalsd(characteruplo,integersmlsiz,integern,integernrhs,doubleprecision,dimension(*)d,doubleprecision,dimension(*)e,doubleprecision,dimension(ldb,*)b,integerldb,doubleprecisionrcond,integerrank,doubleprecision,dimension(*)work,integer,dimension(*)iwork,integerinfo)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. ParametersUPLO 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 subroutineslalsd(characteruplo,integersmlsiz,integern,integernrhs,real,dimension(*)d,real,dimension(*)e,real,dimension(ldb,*)b,integerldb,realrcond,integerrank,real,dimension(*)work,integer,dimension(*)iwork,integerinfo)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. ParametersUPLO 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 subroutinezlalsd(characteruplo,integersmlsiz,integern,integernrhs,doubleprecision,dimension(*)d,doubleprecision,dimension(*)e,complex*16,dimension(ldb,*)b,integerldb,doubleprecisionrcond,integerrank,complex*16,dimension(*)work,doubleprecision,dimension(*)rwork,integer,dimension(*)iwork,integerinfo)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. ParametersUPLO 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

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

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.