gelsd(3)

NAME
gelsd - gelsd: least squares using SVD, divide and conquer

SYNOPSIS

  Functions
subroutine cgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)
CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
subroutine dgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)
DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
subroutine sgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)
SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
subroutine zgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)
ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Detailed Description

Function Documentation

  subroutine cgelsd (integer m, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) s, real rcond, integer rank, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

     CGELSD computes the minimum-norm solution to a real linear least
     squares problem:
         minimize 2-norm(| b - A*x |)
     using the singular value decomposition (SVD) of A. A is an M-by-N
     matrix which may be rank-deficient.
     Several right hand side vectors b and solution vectors x can be
     handled in a single call; they are stored as the columns of the
     M-by-NRHS right hand side matrix B and the N-by-NRHS solution
     matrix X.
     The problem is solved in three steps:
     (1) Reduce the coefficient matrix A to bidiagonal form with
         Householder transformations, reducing the original problem
         into a 'bidiagonal least squares problem' (BLS)
     (2) Solve the BLS using a divide and conquer approach.
     (3) Apply back all the Householder transformations to solve
         the original least squares problem.
     The effective rank of A is determined by treating as zero those
     singular values which are less than RCOND times the largest singular
     value.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X. NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A has been destroyed.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the M-by-NRHS right hand side matrix B.
              On exit, B is overwritten by the N-by-NRHS solution matrix X.
              If m >= n and RANK = n, the residual sum-of-squares for
              the solution in the i-th column is given by the sum of
              squares of the modulus of elements n+1:m in that column.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M,N).

    S

              S is REAL array, dimension (min(M,N))
              The singular values of A in decreasing order.
              The condition number of A in the 2-norm = S(1)/S(min(m,n)).

    RCOND

              RCOND is REAL
              RCOND is used to determine the effective rank of A.
              Singular values S(i) <= RCOND*S(1) are treated as zero.
              If RCOND < 0, machine precision is used instead.

    RANK

              RANK is INTEGER
              The effective rank of A, i.e., the number of singular values
              which are greater than RCOND*S(1).

    WORK

              WORK is COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK must be at least 1.
              The exact minimum amount of workspace needed depends on M,
              N and NRHS. As long as LWORK is at least
                  2 * N + N * NRHS
              if M is greater than or equal to N or
                  2 * M + M * NRHS
              if M is less than N, the code will execute correctly.
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the array WORK and the
              minimum sizes of the arrays RWORK and IWORK, and returns
              these values as the first entries of the WORK, RWORK and
              IWORK arrays, and no error message related to LWORK is issued
              by XERBLA.

    RWORK

              RWORK is REAL array, dimension (MAX(1,LRWORK))
              LRWORK >=
                 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
                 MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
              if M is greater than or equal to N or
                 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
                 MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
              if M is less than N, the code will execute correctly.
              SMLSIZ is returned by ILAENV and is equal to the maximum
              size of the subproblems at the bottom of the computation
              tree (usually about 25), and
                 NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
              On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
              where MINMN = MIN( M,N ).
              On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value.
              > 0:  the algorithm for computing the SVD failed to converge;
                    if INFO = i, i off-diagonal elements of an intermediate
                    bidiagonal form did not converge to zero.

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

Definition at line 217 of file cgelsd.f.

  subroutine dgelsd (integer m, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) s, double precision rcond, integer rank, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

     DGELSD computes the minimum-norm solution to a real linear least
     squares problem:
         minimize 2-norm(| b - A*x |)
     using the singular value decomposition (SVD) of A. A is an M-by-N
     matrix which may be rank-deficient.
     Several right hand side vectors b and solution vectors x can be
     handled in a single call; they are stored as the columns of the
     M-by-NRHS right hand side matrix B and the N-by-NRHS solution
     matrix X.
     The problem is solved in three steps:
     (1) Reduce the coefficient matrix A to bidiagonal form with
         Householder transformations, reducing the original problem
         into a 'bidiagonal least squares problem' (BLS)
     (2) Solve the BLS using a divide and conquer approach.
     (3) Apply back all the Householder transformations to solve
         the original least squares problem.
     The effective rank of A is determined by treating as zero those
     singular values which are less than RCOND times the largest singular
     value.

Parameters
    M

              M is INTEGER
              The number of rows of A. M >= 0.

    N

              N is INTEGER
              The number of columns of A. N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X. NRHS >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A has been destroyed.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the M-by-NRHS right hand side matrix B.
              On exit, B is overwritten by the N-by-NRHS solution
              matrix X.  If m >= n and RANK = n, the residual
              sum-of-squares for the solution in the i-th column is given
              by the sum of squares of elements n+1:m in that column.

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,max(M,N)).

    S

              S is DOUBLE PRECISION array, dimension (min(M,N))
              The singular values of A in decreasing order.
              The condition number of A in the 2-norm = S(1)/S(min(m,n)).

    RCOND

              RCOND is DOUBLE PRECISION
              RCOND is used to determine the effective rank of A.
              Singular values S(i) <= RCOND*S(1) are treated as zero.
              If RCOND < 0, machine precision is used instead.

    RANK

              RANK is INTEGER
              The effective rank of A, i.e., the number of singular values
              which are greater than RCOND*S(1).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK must be at least 1.
              The exact minimum amount of workspace needed depends on M,
              N and NRHS. As long as LWORK is at least
                  12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
              if M is greater than or equal to N or
                  12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
              if M is less than N, the code will execute correctly.
              SMLSIZ is returned by ILAENV and is equal to the maximum
              size of the subproblems at the bottom of the computation
              tree (usually about 25), and
                 NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
              where MINMN = MIN( M,N ).
              On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  the algorithm for computing the SVD failed to converge;
                    if INFO = i, i off-diagonal elements of an intermediate
                    bidiagonal form did not converge to zero.

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

Definition at line 201 of file dgelsd.f.

  subroutine sgelsd (integer m, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) s, real rcond, integer rank, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

     SGELSD computes the minimum-norm solution to a real linear least
     squares problem:
         minimize 2-norm(| b - A*x |)
     using the singular value decomposition (SVD) of A. A is an M-by-N
     matrix which may be rank-deficient.
     Several right hand side vectors b and solution vectors x can be
     handled in a single call; they are stored as the columns of the
     M-by-NRHS right hand side matrix B and the N-by-NRHS solution
     matrix X.
     The problem is solved in three steps:
     (1) Reduce the coefficient matrix A to bidiagonal form with
         Householder transformations, reducing the original problem
         into a 'bidiagonal least squares problem' (BLS)
     (2) Solve the BLS using a divide and conquer approach.
     (3) Apply back all the Householder transformations to solve
         the original least squares problem.
     The effective rank of A is determined by treating as zero those
     singular values which are less than RCOND times the largest singular
     value.

Parameters
    M

              M is INTEGER
              The number of rows of A. M >= 0.

    N

              N is INTEGER
              The number of columns of A. N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X. NRHS >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A has been destroyed.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the M-by-NRHS right hand side matrix B.
              On exit, B is overwritten by the N-by-NRHS solution
              matrix X.  If m >= n and RANK = n, the residual
              sum-of-squares for the solution in the i-th column is given
              by the sum of squares of elements n+1:m in that column.

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,max(M,N)).

    S

              S is REAL array, dimension (min(M,N))
              The singular values of A in decreasing order.
              The condition number of A in the 2-norm = S(1)/S(min(m,n)).

    RCOND

              RCOND is REAL
              RCOND is used to determine the effective rank of A.
              Singular values S(i) <= RCOND*S(1) are treated as zero.
              If RCOND < 0, machine precision is used instead.

    RANK

              RANK is INTEGER
              The effective rank of A, i.e., the number of singular values
              which are greater than RCOND*S(1).

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK must be at least 1.
              The exact minimum amount of workspace needed depends on M,
              N and NRHS. As long as LWORK is at least
                  12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
              if M is greater than or equal to N or
                  12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
              if M is less than N, the code will execute correctly.
              SMLSIZ is returned by ILAENV and is equal to the maximum
              size of the subproblems at the bottom of the computation
              tree (usually about 25), and
                 NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the array WORK and the
              minimum size of the array IWORK, and returns these values as
              the first entries of the WORK and IWORK arrays, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
              where MINMN = MIN( M,N ).
              On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  the algorithm for computing the SVD failed to converge;
                    if INFO = i, i off-diagonal elements of an intermediate
                    bidiagonal form did not converge to zero.

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

Definition at line 202 of file sgelsd.f.

  subroutine zgelsd (integer m, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) s, double precision rcond, integer rank, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

     ZGELSD computes the minimum-norm solution to a real linear least
     squares problem:
         minimize 2-norm(| b - A*x |)
     using the singular value decomposition (SVD) of A. A is an M-by-N
     matrix which may be rank-deficient.
     Several right hand side vectors b and solution vectors x can be
     handled in a single call; they are stored as the columns of the
     M-by-NRHS right hand side matrix B and the N-by-NRHS solution
     matrix X.
     The problem is solved in three steps:
     (1) Reduce the coefficient matrix A to bidiagonal form with
         Householder transformations, reducing the original problem
         into a 'bidiagonal least squares problem' (BLS)
     (2) Solve the BLS using a divide and conquer approach.
     (3) Apply back all the Householder transformations to solve
         the original least squares problem.
     The effective rank of A is determined by treating as zero those
     singular values which are less than RCOND times the largest singular
     value.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X. NRHS >= 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A has been destroyed.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

    B

              B is COMPLEX*16 array, dimension (LDB,NRHS)
              On entry, the M-by-NRHS right hand side matrix B.
              On exit, B is overwritten by the N-by-NRHS solution matrix X.
              If m >= n and RANK = n, the residual sum-of-squares for
              the solution in the i-th column is given by the sum of
              squares of the modulus of elements n+1:m in that column.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M,N).

    S

              S is DOUBLE PRECISION array, dimension (min(M,N))
              The singular values of A in decreasing order.
              The condition number of A in the 2-norm = S(1)/S(min(m,n)).

    RCOND

              RCOND is DOUBLE PRECISION
              RCOND is used to determine the effective rank of A.
              Singular values S(i) <= RCOND*S(1) are treated as zero.
              If RCOND < 0, machine precision is used instead.

    RANK

              RANK is INTEGER
              The effective rank of A, i.e., the number of singular values
              which are greater than RCOND*S(1).

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK must be at least 1.
              The exact minimum amount of workspace needed depends on M,
              N and NRHS. As long as LWORK is at least
                  2*N + N*NRHS
              if M is greater than or equal to N or
                  2*M + M*NRHS
              if M is less than N, the code will execute correctly.
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the array WORK and the
              minimum sizes of the arrays RWORK and IWORK, and returns
              these values as the first entries of the WORK, RWORK and
              IWORK arrays, and no error message related to LWORK is issued
              by XERBLA.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
              LRWORK >=
                 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
                 MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
              if M is greater than or equal to N or
                 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
                 MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
              if M is less than N, the code will execute correctly.
              SMLSIZ is returned by ILAENV and is equal to the maximum
              size of the subproblems at the bottom of the computation
              tree (usually about 25), and
                 NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
              On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
              where MINMN = MIN( M,N ).
              On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value.
              > 0:  the algorithm for computing the SVD failed to converge;
                    if INFO = i, i off-diagonal elements of an intermediate
                    bidiagonal form did not converge to zero.

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

Definition at line 217 of file zgelsd.f.

Author

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