gels(3)

NAME
gels - gels: least squares using QR/LQ

SYNOPSIS

  Functions
subroutine cgels (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
CGELS solves overdetermined or underdetermined systems for GE matrices
subroutine dgels (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
DGELS solves overdetermined or underdetermined systems for GE matrices
subroutine sgels (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
SGELS solves overdetermined or underdetermined systems for GE matrices
subroutine zgels (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
ZGELS solves overdetermined or underdetermined systems for GE matrices

Detailed Description

Function Documentation

  subroutine cgels (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)

CGELS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> CGELS solves overdetermined or underdetermined complex linear systems
    !> involving an M-by-N matrix A, or its conjugate-transpose, using a QR
    !> or 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 underdetermined system A**H * 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**H * 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.
    !>            if M >= N, A is overwritten by details of its QR
    !>                       factorization as returned by CGEQRF;
    !>            if M <  N, A is overwritten by details of its LQ
    !>                       factorization as returned by CGELQF.
    !>

    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; the residual sum of squares for the
    !>          solution in each column is given by the sum of squares of the
    !>          modulus of elements N+1 to M in that column;
    !>          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; the residual sum of squares
    !>          for the solution in each column is given by the sum of
    !>          squares of the modulus of elements M+1 to N in that column.
    !>

    LDB

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

    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 >= max( 1, MN + max( MN, NRHS ) ).
    !>          For optimal performance,
    !>          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
    !>          where MN = min(M,N) and NB is the optimum block size.
    !>
    !>          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:  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.

Definition at line 190 of file cgels.f.

  subroutine dgels (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)

DGELS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> DGELS solves overdetermined or underdetermined real linear systems
    !> involving an M-by-N matrix A, or its transpose, using a QR or 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 underdetermined 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,
    !>            if M >= N, A is overwritten by details of its QR
    !>                       factorization as returned by DGEQRF;
    !>            if M <  N, A is overwritten by details of its LQ
    !>                       factorization as returned by DGELQF.
    !>

    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; the residual sum of squares for the
    !>          solution in each column is given by the sum of squares of
    !>          elements N+1 to M in that column;
    !>          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; the residual sum of squares
    !>          for the solution in each column is given by the sum of
    !>          squares of elements M+1 to N in that column.
    !>

    LDB

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

    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 >= max( 1, MN + max( MN, NRHS ) ).
    !>          For optimal performance,
    !>          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
    !>          where MN = min(M,N) and NB is the optimum block size.
    !>
    !>          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:  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.

Definition at line 191 of file dgels.f.

  subroutine sgels (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)

SGELS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> SGELS solves overdetermined or underdetermined real linear systems
    !> involving an M-by-N matrix A, or its transpose, using a QR or 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 underdetermined 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,
    !>            if M >= N, A is overwritten by details of its QR
    !>                       factorization as returned by SGEQRF;
    !>            if M <  N, A is overwritten by details of its LQ
    !>                       factorization as returned by SGELQF.
    !>

    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; the residual sum of squares for the
    !>          solution in each column is given by the sum of squares of
    !>          elements N+1 to M in that column;
    !>          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; the residual sum of squares
    !>          for the solution in each column is given by the sum of
    !>          squares of elements M+1 to N in that column.
    !>

    LDB

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

    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 >= max( 1, MN + max( MN, NRHS ) ).
    !>          For optimal performance,
    !>          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
    !>          where MN = min(M,N) and NB is the optimum block size.
    !>
    !>          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:  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.

Definition at line 191 of file sgels.f.

  subroutine zgels (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)

ZGELS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> ZGELS solves overdetermined or underdetermined complex linear systems
    !> involving an M-by-N matrix A, or its conjugate-transpose, using a QR
    !> or 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 underdetermined system A**H * 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**H * 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.
    !>            if M >= N, A is overwritten by details of its QR
    !>                       factorization as returned by ZGEQRF;
    !>            if M <  N, A is overwritten by details of its LQ
    !>                       factorization as returned by ZGELQF.
    !>

    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; the residual sum of squares for the
    !>          solution in each column is given by the sum of squares of the
    !>          modulus of elements N+1 to M in that column;
    !>          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; the residual sum of squares
    !>          for the solution in each column is given by the sum of
    !>          squares of the modulus of elements M+1 to N in that column.
    !>

    LDB

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

    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 >= max( 1, MN + max( MN, NRHS ) ).
    !>          For optimal performance,
    !>          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
    !>          where MN = min(M,N) and NB is the optimum block size.
    !>
    !>          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:  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.

Definition at line 190 of file zgels.f.

Author

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