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  (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

  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  (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

  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  (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

  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  (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

Author

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