gelss(3)

NAME
gelss - gelss: least squares using SVD, QR iteration

SYNOPSIS

  Functions

subroutine cgelss (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, info)
CGELSS solves overdetermined or underdetermined systems for GE matrices
subroutine dgelss (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, info)
DGELSS solves overdetermined or underdetermined systems for GE matrices
subroutine sgelss (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, info)
SGELSS solves overdetermined or underdetermined systems for GE matrices
subroutine zgelss (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, info)
ZGELSS solves overdetermined or underdetermined systems for GE matrices

Detailed Description

Function Documentation

  subroutine cgelss (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 info)

CGELSS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> CGELSS computes the minimum norm solution to a complex 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 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, the first min(m,n) rows of A are overwritten with
    !>          its right singular vectors, stored rowwise.
    !>

    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 >= 1, and also:
    !>          LWORK >=  2*min(M,N) + max(M,N,NRHS)
    !>          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.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (5*min(M,N))
    !>

    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.

  subroutine dgelss (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 info)

DGELSS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> DGELSS 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 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 DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit, the first min(m,n) rows of A are overwritten with
    !>          its right singular vectors, stored rowwise.
    !>

    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 >= 1, and also:
    !>          LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
    !>          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.
    !>

    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.

  subroutine sgelss (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 info)

SGELSS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> SGELSS 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 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 REAL array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit, the first min(m,n) rows of A are overwritten with
    !>          its right singular vectors, stored rowwise.
    !>

    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 >= 1, and also:
    !>          LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
    !>          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.
    !>

    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.

  subroutine zgelss (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 info)

ZGELSS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> ZGELSS computes the minimum norm solution to a complex 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 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, the first min(m,n) rows of A are overwritten with
    !>          its right singular vectors, stored rowwise.
    !>

    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 >= 1, and also:
    !>          LWORK >=  2*min(M,N) + max(M,N,NRHS)
    !>          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.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (5*min(M,N))
    !>

    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.

Author

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