getsls(3)

NAME
getsls - getsls: least squares using tall-skinny QR/LQ

SYNOPSIS

  Functions

subroutine cgetsls (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
CGETSLS
subroutine dgetsls (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
DGETSLS
subroutine sgetsls (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
SGETSLS
subroutine zgetsls (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
ZGETSLS

Detailed Description

Function Documentation

  subroutine cgetsls (character trans, integer m, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) work, integer lwork, integer info)

CGETSLS

Purpose:

    !>
    !> CGETSLS solves overdetermined or underdetermined complex linear systems
    !> involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
    !> factorization of A.
    !>
    !> It is assumed that A has full rank, and only a rudimentary protection
    !> against rank-deficient matrices is provided. This subroutine only detects
    !> exact rank-deficiency, where a diagonal element of the triangular factor
    !> of A is exactly zero.
    !>
    !> It is conceivable for one (or more) of the diagonal elements of the triangular
    !> factor of A to be subnormally tiny numbers without this subroutine signalling
    !> an error. The solutions computed for such almost-rank-deficient matrices may
    !> be less accurate due to a loss of numerical precision.
    !>
    !>
    !> The following options are provided:
    !>
    !> 1. If TRANS = 'N' and m >= n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A*X ||.
    !>
    !> 2. If TRANS = 'N' and m < n:  find the minimum norm solution of
    !>    an underdetermined system A * X = B.
    !>
    !> 3. If TRANS = 'C' and m >= n:  find the minimum norm solution of
    !>    an undetermined system A**T * X = B.
    !>
    !> 4. If TRANS = 'C' and m < n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A**T * X ||.
    !>
    !> 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': the linear system involves A;
    !>          = 'C': the linear system involves A**H.
    !>

    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 is overwritten by details of its QR or LQ
    !>          factorization as returned by CGEQR or CGELQ.
    !>

    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 matrix B of right hand side vectors, stored
    !>          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
    !>          if TRANS = 'C'.
    !>          On exit, if INFO = 0, B is overwritten by the solution
    !>          vectors, stored columnwise:
    !>          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
    !>          squares solution vectors.
    !>          if TRANS = 'N' and m < n, rows 1 to N of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'C' and m >= n, rows 1 to M of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'C' and m < n, rows 1 to M of B contain the
    !>          least squares solution vectors.
    !>

    LDB

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

    WORK

    !>          (workspace) COMPLEX array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
    !>          or optimal, if query was assumed) LWORK.
    !>          See LWORK for details.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= 1.
    !>          If LWORK = -1 or -2, then a workspace query is assumed.
    !>          If LWORK = -1, the routine calculates optimal size of WORK for the
    !>          optimal performance and returns this value in WORK(1).
    !>          If LWORK = -2, the routine calculates minimal size of WORK and
    !>          returns this value in WORK(1).
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO =  i, the i-th diagonal element of the
    !>                triangular factor of A is exactly zero, so that A does not have
    !>                full rank; the least squares solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dgetsls (character trans, integer m, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) work, integer lwork, integer info)

DGETSLS

Purpose:

    !>
    !> DGETSLS solves overdetermined or underdetermined real linear systems
    !> involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
    !> factorization of A.
    !>
    !> It is assumed that A has full rank, and only a rudimentary protection
    !> against rank-deficient matrices is provided. This subroutine only detects
    !> exact rank-deficiency, where a diagonal element of the triangular factor
    !> of A is exactly zero.
    !>
    !> It is conceivable for one (or more) of the diagonal elements of the triangular
    !> factor of A to be subnormally tiny numbers without this subroutine signalling
    !> an error. The solutions computed for such almost-rank-deficient matrices may
    !> be less accurate due to a loss of numerical precision.
    !>
    !>
    !> The following options are provided:
    !>
    !> 1. If TRANS = 'N' and m >= n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A*X ||.
    !>
    !> 2. If TRANS = 'N' and m < n:  find the minimum norm solution of
    !>    an underdetermined system A * X = B.
    !>
    !> 3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
    !>    an undetermined system A**T * X = B.
    !>
    !> 4. If TRANS = 'T' and m < n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A**T * X ||.
    !>
    !> 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': the linear system involves A;
    !>          = 'T': the linear system involves A**T.
    !>

    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,
    !>          A is overwritten by details of its QR or LQ
    !>          factorization as returned by DGEQR or DGELQ.
    !>

    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 matrix B of right hand side vectors, stored
    !>          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
    !>          if TRANS = 'T'.
    !>          On exit, if INFO = 0, B is overwritten by the solution
    !>          vectors, stored columnwise:
    !>          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
    !>          squares solution vectors.
    !>          if TRANS = 'N' and m < n, rows 1 to N of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'T' and m >= n, rows 1 to M of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'T' and m < n, rows 1 to M of B contain the
    !>          least squares solution vectors.
    !>

    LDB

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

    WORK

    !>          (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
    !>          or optimal, if query was assumed) LWORK.
    !>          See LWORK for details.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= 1.
    !>          If LWORK = -1 or -2, then a workspace query is assumed.
    !>          If LWORK = -1, the routine calculates optimal size of WORK for the
    !>          optimal performance and returns this value in WORK(1).
    !>          If LWORK = -2, the routine calculates minimal size of WORK and
    !>          returns this value in WORK(1).
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO =  i, the i-th diagonal element of the
    !>                triangular factor of A is exactly zero, so that A does not have
    !>                full rank; the least squares solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine sgetsls (character trans, integer m, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) work, integer lwork, integer info)

SGETSLS

Purpose:

    !>
    !> SGETSLS solves overdetermined or underdetermined real linear systems
    !> involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
    !> factorization of A.
    !>
    !> It is assumed that A has full rank, and only a rudimentary protection
    !> against rank-deficient matrices is provided. This subroutine only detects
    !> exact rank-deficiency, where a diagonal element of the triangular factor
    !> of A is exactly zero.
    !>
    !> It is conceivable for one (or more) of the diagonal elements of the triangular
    !> factor of A to be subnormally tiny numbers without this subroutine signalling
    !> an error. The solutions computed for such almost-rank-deficient matrices may
    !> be less accurate due to a loss of numerical precision.
    !>
    !>
    !> The following options are provided:
    !>
    !> 1. If TRANS = 'N' and m >= n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A*X ||.
    !>
    !> 2. If TRANS = 'N' and m < n:  find the minimum norm solution of
    !>    an underdetermined system A * X = B.
    !>
    !> 3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
    !>    an undetermined system A**T * X = B.
    !>
    !> 4. If TRANS = 'T' and m < n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A**T * X ||.
    !>
    !> 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': the linear system involves A;
    !>          = 'T': the linear system involves A**T.
    !>

    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,
    !>          A is overwritten by details of its QR or LQ
    !>          factorization as returned by SGEQR or SGELQ.
    !>

    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 matrix B of right hand side vectors, stored
    !>          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
    !>          if TRANS = 'T'.
    !>          On exit, if INFO = 0, B is overwritten by the solution
    !>          vectors, stored columnwise:
    !>          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
    !>          squares solution vectors.
    !>          if TRANS = 'N' and m < n, rows 1 to N of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'T' and m >= n, rows 1 to M of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'T' and m < n, rows 1 to M of B contain the
    !>          least squares solution vectors.
    !>

    LDB

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

    WORK

    !>          (workspace) REAL array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
    !>          or optimal, if query was assumed) LWORK.
    !>          See LWORK for details.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= 1.
    !>          If LWORK = -1 or -2, then a workspace query is assumed.
    !>          If LWORK = -1, the routine calculates optimal size of WORK for the
    !>          optimal performance and returns this value in WORK(1).
    !>          If LWORK = -2, the routine calculates minimal size of WORK and
    !>          returns this value in WORK(1).
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO =  i, the i-th diagonal element of the
    !>                triangular factor of A is exactly zero, so that A does not have
    !>                full rank; the least squares solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine zgetsls (character trans, integer m, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integer lwork, integer info)

ZGETSLS

Purpose:

    !>
    !> ZGETSLS solves overdetermined or underdetermined complex linear systems
    !> involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
    !> factorization of A.
    !>
    !> It is assumed that A has full rank, and only a rudimentary protection
    !> against rank-deficient matrices is provided. This subroutine only detects
    !> exact rank-deficiency, where a diagonal element of the triangular factor
    !> of A is exactly zero.
    !>
    !> It is conceivable for one (or more) of the diagonal elements of the triangular
    !> factor of A to be subnormally tiny numbers without this subroutine signalling
    !> an error. The solutions computed for such almost-rank-deficient matrices may
    !> be less accurate due to a loss of numerical precision.
    !>
    !>
    !> The following options are provided:
    !>
    !> 1. If TRANS = 'N' and m >= n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A*X ||.
    !>
    !> 2. If TRANS = 'N' and m < n:  find the minimum norm solution of
    !>    an underdetermined system A * X = B.
    !>
    !> 3. If TRANS = 'C' and m >= n:  find the minimum norm solution of
    !>    an undetermined system A**T * X = B.
    !>
    !> 4. If TRANS = 'C' and m < n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A**T * X ||.
    !>
    !> 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': the linear system involves A;
    !>          = 'C': the linear system involves A**H.
    !>

    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 is overwritten by details of its QR or LQ
    !>          factorization as returned by ZGEQR or ZGELQ.
    !>

    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 matrix B of right hand side vectors, stored
    !>          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
    !>          if TRANS = 'C'.
    !>          On exit, if INFO = 0, B is overwritten by the solution
    !>          vectors, stored columnwise:
    !>          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
    !>          squares solution vectors.
    !>          if TRANS = 'N' and m < n, rows 1 to N of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'C' and m >= n, rows 1 to M of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'C' and m < n, rows 1 to M of B contain the
    !>          least squares solution vectors.
    !>

    LDB

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

    WORK

    !>          (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
    !>          or optimal, if query was assumed) LWORK.
    !>          See LWORK for details.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= 1.
    !>          If LWORK = -1 or -2, then a workspace query is assumed.
    !>          If LWORK = -1, the routine calculates optimal size of WORK for the
    !>          optimal performance and returns this value in WORK(1).
    !>          If LWORK = -2, the routine calculates minimal size of WORK and
    !>          returns this value in WORK(1).
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO =  i, the i-th diagonal element of the
    !>                triangular factor of A is exactly zero, so that A does not have
    !>                full rank; the least squares solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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