launhr_col_getrfnp2(3)

NAME
launhr_col_getrfnp2 - la{un,or}hr_col_getrfnp2: LU factor without pivoting, level 2

SYNOPSIS

  Functions

recursive subroutine claunhr_col_getrfnp2 (m, n, a, lda, d, info)
CLAUNHR_COL_GETRFNP2
recursive subroutine dlaorhr_col_getrfnp2 (m, n, a, lda, d, info)
DLAORHR_COL_GETRFNP2
recursive subroutine slaorhr_col_getrfnp2 (m, n, a, lda, d, info)
SLAORHR_COL_GETRFNP2
recursive subroutine zlaunhr_col_getrfnp2 (m, n, a, lda, d, info)
ZLAUNHR_COL_GETRFNP2

Detailed Description

Function Documentation

  recursive subroutine claunhr_col_getrfnp2 (integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) d, integer info)

CLAUNHR_COL_GETRFNP2

Purpose:

    !>
    !> CLAUNHR_COL_GETRFNP2 computes the modified LU factorization without
    !> pivoting of a complex general M-by-N matrix A. The factorization has
    !> the form:
    !>
    !>     A - S = L * U,
    !>
    !> where:
    !>    S is a m-by-n diagonal sign matrix with the diagonal D, so that
    !>    D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
    !>    as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
    !>    i-1 steps of Gaussian elimination. This means that the diagonal
    !>    element at each step of  Gaussian elimination is at
    !>    least one in absolute value (so that division-by-zero not
    !>    possible during the division by the diagonal element);
    !>
    !>    L is a M-by-N lower triangular matrix with unit diagonal elements
    !>    (lower trapezoidal if M > N);
    !>
    !>    and U is a M-by-N upper triangular matrix
    !>    (upper trapezoidal if M < N).
    !>
    !> This routine is an auxiliary routine used in the Householder
    !> reconstruction routine CUNHR_COL. In CUNHR_COL, this routine is
    !> applied to an M-by-N matrix A with orthonormal columns, where each
    !> element is bounded by one in absolute value. With the choice of
    !> the matrix S above, one can show that the diagonal element at each
    !> step of Gaussian elimination is the largest (in absolute value) in
    !> the column on or below the diagonal, so that no pivoting is required
    !> for numerical stability [1].
    !>
    !> For more details on the Householder reconstruction algorithm,
    !> including the modified LU factorization, see [1].
    !>
    !> This is the recursive version of the LU factorization algorithm.
    !> Denote A - S by B. The algorithm divides the matrix B into four
    !> submatrices:
    !>
    !>        [  B11 | B12  ]  where B11 is n1 by n1,
    !>    B = [ -----|----- ]        B21 is (m-n1) by n1,
    !>        [  B21 | B22  ]        B12 is n1 by n2,
    !>                               B22 is (m-n1) by n2,
    !>                               with n1 = min(m,n)/2, n2 = n-n1.
    !>
    !>
    !> The subroutine calls itself to factor B11, solves for B21,
    !> solves for B12, updates B22, then calls itself to factor B22.
    !>
    !> For more details on the recursive LU algorithm, see [2].
    !>
    !> CLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked
    !> routine CLAUNHR_COL_GETRFNP, which uses blocked code calling
    !> Level 3 BLAS to update the submatrix. However, CLAUNHR_COL_GETRFNP2
    !> is self-sufficient and can be used without CLAUNHR_COL_GETRFNP.
    !>
    !> [1] ,
    !>     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
    !>     E. Solomonik, J. Parallel Distrib. Comput.,
    !>     vol. 85, pp. 3-31, 2015.
    !>
    !> [2] , F. Gustavson, IBM J. of Res. and Dev.,
    !>     vol. 41, no. 6, pp. 737-755, 1997.
    !>

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.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix to be factored.
    !>          On exit, the factors L and U from the factorization
    !>          A-S=L*U; the unit diagonal elements of L are not stored.
    !>

    LDA

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

    D

    !>          D is COMPLEX array, dimension min(M,N)
    !>          The diagonal elements of the diagonal M-by-N sign matrix S,
    !>          D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be
    !>          only ( +1.0, 0.0 ) or (-1.0, 0.0 ).
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

    !>
    !> November 2019, Igor Kozachenko,
    !>                Computer Science Division,
    !>                University of California, Berkeley
    !>
    !>

  recursive subroutine dlaorhr_col_getrfnp2 (integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, integer info)

DLAORHR_COL_GETRFNP2

Purpose:

    !>
    !> DLAORHR_COL_GETRFNP2 computes the modified LU factorization without
    !> pivoting of a real general M-by-N matrix A. The factorization has
    !> the form:
    !>
    !>     A - S = L * U,
    !>
    !> where:
    !>    S is a m-by-n diagonal sign matrix with the diagonal D, so that
    !>    D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
    !>    as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
    !>    i-1 steps of Gaussian elimination. This means that the diagonal
    !>    element at each step of  Gaussian elimination is at
    !>    least one in absolute value (so that division-by-zero not
    !>    possible during the division by the diagonal element);
    !>
    !>    L is a M-by-N lower triangular matrix with unit diagonal elements
    !>    (lower trapezoidal if M > N);
    !>
    !>    and U is a M-by-N upper triangular matrix
    !>    (upper trapezoidal if M < N).
    !>
    !> This routine is an auxiliary routine used in the Householder
    !> reconstruction routine DORHR_COL. In DORHR_COL, this routine is
    !> applied to an M-by-N matrix A with orthonormal columns, where each
    !> element is bounded by one in absolute value. With the choice of
    !> the matrix S above, one can show that the diagonal element at each
    !> step of Gaussian elimination is the largest (in absolute value) in
    !> the column on or below the diagonal, so that no pivoting is required
    !> for numerical stability [1].
    !>
    !> For more details on the Householder reconstruction algorithm,
    !> including the modified LU factorization, see [1].
    !>
    !> This is the recursive version of the LU factorization algorithm.
    !> Denote A - S by B. The algorithm divides the matrix B into four
    !> submatrices:
    !>
    !>        [  B11 | B12  ]  where B11 is n1 by n1,
    !>    B = [ -----|----- ]        B21 is (m-n1) by n1,
    !>        [  B21 | B22  ]        B12 is n1 by n2,
    !>                               B22 is (m-n1) by n2,
    !>                               with n1 = min(m,n)/2, n2 = n-n1.
    !>
    !>
    !> The subroutine calls itself to factor B11, solves for B21,
    !> solves for B12, updates B22, then calls itself to factor B22.
    !>
    !> For more details on the recursive LU algorithm, see [2].
    !>
    !> DLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
    !> routine DLAORHR_COL_GETRFNP, which uses blocked code calling
    !> Level 3 BLAS to update the submatrix. However, DLAORHR_COL_GETRFNP2
    !> is self-sufficient and can be used without DLAORHR_COL_GETRFNP.
    !>
    !> [1] ,
    !>     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
    !>     E. Solomonik, J. Parallel Distrib. Comput.,
    !>     vol. 85, pp. 3-31, 2015.
    !>
    !> [2] , F. Gustavson, IBM J. of Res. and Dev.,
    !>     vol. 41, no. 6, pp. 737-755, 1997.
    !>

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.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix to be factored.
    !>          On exit, the factors L and U from the factorization
    !>          A-S=L*U; the unit diagonal elements of L are not stored.
    !>

    LDA

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

    D

    !>          D is DOUBLE PRECISION array, dimension min(M,N)
    !>          The diagonal elements of the diagonal M-by-N sign matrix S,
    !>          D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
    !>          be only plus or minus one.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

    !>
    !> November 2019, Igor Kozachenko,
    !>                Computer Science Division,
    !>                University of California, Berkeley
    !>
    !>

  recursive subroutine slaorhr_col_getrfnp2 (integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, integer info)

SLAORHR_COL_GETRFNP2

Purpose:

    !>
    !> SLAORHR_COL_GETRFNP2 computes the modified LU factorization without
    !> pivoting of a real general M-by-N matrix A. The factorization has
    !> the form:
    !>
    !>     A - S = L * U,
    !>
    !> where:
    !>    S is a m-by-n diagonal sign matrix with the diagonal D, so that
    !>    D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
    !>    as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
    !>    i-1 steps of Gaussian elimination. This means that the diagonal
    !>    element at each step of  Gaussian elimination is at
    !>    least one in absolute value (so that division-by-zero not
    !>    possible during the division by the diagonal element);
    !>
    !>    L is a M-by-N lower triangular matrix with unit diagonal elements
    !>    (lower trapezoidal if M > N);
    !>
    !>    and U is a M-by-N upper triangular matrix
    !>    (upper trapezoidal if M < N).
    !>
    !> This routine is an auxiliary routine used in the Householder
    !> reconstruction routine SORHR_COL. In SORHR_COL, this routine is
    !> applied to an M-by-N matrix A with orthonormal columns, where each
    !> element is bounded by one in absolute value. With the choice of
    !> the matrix S above, one can show that the diagonal element at each
    !> step of Gaussian elimination is the largest (in absolute value) in
    !> the column on or below the diagonal, so that no pivoting is required
    !> for numerical stability [1].
    !>
    !> For more details on the Householder reconstruction algorithm,
    !> including the modified LU factorization, see [1].
    !>
    !> This is the recursive version of the LU factorization algorithm.
    !> Denote A - S by B. The algorithm divides the matrix B into four
    !> submatrices:
    !>
    !>        [  B11 | B12  ]  where B11 is n1 by n1,
    !>    B = [ -----|----- ]        B21 is (m-n1) by n1,
    !>        [  B21 | B22  ]        B12 is n1 by n2,
    !>                               B22 is (m-n1) by n2,
    !>                               with n1 = min(m,n)/2, n2 = n-n1.
    !>
    !>
    !> The subroutine calls itself to factor B11, solves for B21,
    !> solves for B12, updates B22, then calls itself to factor B22.
    !>
    !> For more details on the recursive LU algorithm, see [2].
    !>
    !> SLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
    !> routine SLAORHR_COL_GETRFNP, which uses blocked code calling
    !> Level 3 BLAS to update the submatrix. However, SLAORHR_COL_GETRFNP2
    !> is self-sufficient and can be used without SLAORHR_COL_GETRFNP.
    !>
    !> [1] ,
    !>     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
    !>     E. Solomonik, J. Parallel Distrib. Comput.,
    !>     vol. 85, pp. 3-31, 2015.
    !>
    !> [2] , F. Gustavson, IBM J. of Res. and Dev.,
    !>     vol. 41, no. 6, pp. 737-755, 1997.
    !>

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.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix to be factored.
    !>          On exit, the factors L and U from the factorization
    !>          A-S=L*U; the unit diagonal elements of L are not stored.
    !>

    LDA

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

    D

    !>          D is REAL array, dimension min(M,N)
    !>          The diagonal elements of the diagonal M-by-N sign matrix S,
    !>          D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
    !>          be only plus or minus one.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

    !>
    !> November 2019, Igor Kozachenko,
    !>                Computer Science Division,
    !>                University of California, Berkeley
    !>
    !>

  recursive subroutine zlaunhr_col_getrfnp2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) d, integer info)

ZLAUNHR_COL_GETRFNP2

Purpose:

    !>
    !> ZLAUNHR_COL_GETRFNP2 computes the modified LU factorization without
    !> pivoting of a complex general M-by-N matrix A. The factorization has
    !> the form:
    !>
    !>     A - S = L * U,
    !>
    !> where:
    !>    S is a m-by-n diagonal sign matrix with the diagonal D, so that
    !>    D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
    !>    as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
    !>    i-1 steps of Gaussian elimination. This means that the diagonal
    !>    element at each step of  Gaussian elimination is at
    !>    least one in absolute value (so that division-by-zero not
    !>    possible during the division by the diagonal element);
    !>
    !>    L is a M-by-N lower triangular matrix with unit diagonal elements
    !>    (lower trapezoidal if M > N);
    !>
    !>    and U is a M-by-N upper triangular matrix
    !>    (upper trapezoidal if M < N).
    !>
    !> This routine is an auxiliary routine used in the Householder
    !> reconstruction routine ZUNHR_COL. In ZUNHR_COL, this routine is
    !> applied to an M-by-N matrix A with orthonormal columns, where each
    !> element is bounded by one in absolute value. With the choice of
    !> the matrix S above, one can show that the diagonal element at each
    !> step of Gaussian elimination is the largest (in absolute value) in
    !> the column on or below the diagonal, so that no pivoting is required
    !> for numerical stability [1].
    !>
    !> For more details on the Householder reconstruction algorithm,
    !> including the modified LU factorization, see [1].
    !>
    !> This is the recursive version of the LU factorization algorithm.
    !> Denote A - S by B. The algorithm divides the matrix B into four
    !> submatrices:
    !>
    !>        [  B11 | B12  ]  where B11 is n1 by n1,
    !>    B = [ -----|----- ]        B21 is (m-n1) by n1,
    !>        [  B21 | B22  ]        B12 is n1 by n2,
    !>                               B22 is (m-n1) by n2,
    !>                               with n1 = min(m,n)/2, n2 = n-n1.
    !>
    !>
    !> The subroutine calls itself to factor B11, solves for B21,
    !> solves for B12, updates B22, then calls itself to factor B22.
    !>
    !> For more details on the recursive LU algorithm, see [2].
    !>
    !> ZLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked
    !> routine ZLAUNHR_COL_GETRFNP, which uses blocked code calling
    !> Level 3 BLAS to update the submatrix. However, ZLAUNHR_COL_GETRFNP2
    !> is self-sufficient and can be used without ZLAUNHR_COL_GETRFNP.
    !>
    !> [1] ,
    !>     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
    !>     E. Solomonik, J. Parallel Distrib. Comput.,
    !>     vol. 85, pp. 3-31, 2015.
    !>
    !> [2] , F. Gustavson, IBM J. of Res. and Dev.,
    !>     vol. 41, no. 6, pp. 737-755, 1997.
    !>

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.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix to be factored.
    !>          On exit, the factors L and U from the factorization
    !>          A-S=L*U; the unit diagonal elements of L are not stored.
    !>

    LDA

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

    D

    !>          D is COMPLEX*16 array, dimension min(M,N)
    !>          The diagonal elements of the diagonal M-by-N sign matrix S,
    !>          D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be
    !>          only ( +1.0, 0.0 ) or (-1.0, 0.0 ).
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

    !>
    !> November 2019, Igor Kozachenko,
    !>                Computer Science Division,
    !>                University of California, Berkeley
    !>
    !>

Author

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