Provided by: liblapack-doc_3.11.0-2_all bug

NAME

       variantsGEcomputational - Variants Computational routines

SYNOPSIS

   Functions
       subroutine cgetrf (M, N, A, LDA, IPIV, INFO)
           CGETRF VARIANT: Crout Level 3 BLAS version of the algorithm.
       subroutine dgetrf (M, N, A, LDA, IPIV, INFO)
           DGETRF VARIANT: Crout Level 3 BLAS version of the algorithm.
       subroutine sgetrf (M, N, A, LDA, IPIV, INFO)
           SGETRF VARIANT: Crout Level 3 BLAS version of the algorithm.
       subroutine zgetrf (M, N, A, LDA, IPIV, INFO)
           ZGETRF VARIANT: Crout Level 3 BLAS version of the algorithm.
       subroutine cgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           CGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm.
       subroutine dgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           DGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm.
       subroutine sgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           SGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm.
       subroutine zgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
           ZGEQRF VARIANT: left-looking Level 3 BLAS of the algorithm.

Detailed Description

       This is the group of Variants Computational routines

Function Documentation

   subroutine cgeqrf (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex,
       dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)
       CGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm. Purpose:

        CGEQRF computes a QR factorization of a real M-by-N matrix A:
        A = Q * R.

        This is the left-looking Level 3 BLAS version of the algorithm.

       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 A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(M,N)-by-N upper trapezoidal matrix R (R is
                     upper triangular if m >= n); the elements below the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of min(m,n) elementary reflectors (see Further
                     Details).

           LDA

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

           TAU

                     TAU is COMPLEX array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           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 if MIN(M,N) = 0,
                     otherwise the dimension can be divided into three parts.

                     1) The part for the triangular factor T. If the very last T is not bigger
                        than any of the rest, then this part is NB x ceiling(K/NB), otherwise,
                        NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T

                     2) The part for the very last T when T is bigger than any of the rest T.
                        The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB,
                        where K = min(M,N), NX is calculated by
                              NX = MAX( 0, ILAENV( 3, 'CGEQRF', ' ', M, N, -1, -1 ) )

                     3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

                     So LWORK = part1 + part2 + part3

                     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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Further Details

         The matrix Q is represented as a product of elementary reflectors

            Q = H(1) H(2) . . . H(k), where k = min(m,n).

         Each H(i) has the form

            H(i) = I - tau * v * v'

         where tau is a real scalar, and v is a real vector with
         v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
         and tau in TAU(i).

   subroutine cgetrf (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, integer,
       dimension( * ) IPIV, integer INFO)
       CGETRF VARIANT: Crout Level 3 BLAS version of the algorithm. CGETRF VARIANT: iterative
       version of Sivan Toledo's recursive LU algorithm

       CGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

       Purpose:

        CGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the Crout Level 3 BLAS version of the algorithm.

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Purpose:

        CGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the left-looking Level 3 BLAS version of the algorithm.

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Purpose:

        CGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This code implements an iterative version of Sivan Toledo's recursive
        LU algorithm[1].  For square matrices, this iterative versions should
        be within a factor of two of the optimum number of memory transfers.

        The pattern is as follows, with the large blocks of U being updated
        in one call to DTRSM, and the dotted lines denoting sections that
        have had all pending permutations applied:

         1 2 3 4 5 6 7 8
        +-+-+---+-------+------
        | |1|   |       |
        |.+-+ 2 |       |
        | | |   |       |
        |.|.+-+-+   4   |
        | | | |1|       |
        | | |.+-+       |
        | | | | |       |
        |.|.|.|.+-+-+---+  8
        | | | | | |1|   |
        | | | | |.+-+ 2 |
        | | | | | | |   |
        | | | | |.|.+-+-+
        | | | | | | | |1|
        | | | | | | |.+-+
        | | | | | | | | |
        |.|.|.|.|.|.|.|.+-----
        | | | | | | | | |

        The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
        the binary expansion of the current column.  Each Schur update is
        applied as soon as the necessary portion of U is available.

        [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
        Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
        1065-1081. http://dx.doi.org/10.1137/S0895479896297744

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

   subroutine dgeqrf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer
       LWORK, integer INFO)
       DGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm. Purpose:

        DGEQRF computes a QR factorization of a real M-by-N matrix A:
        A = Q * R.

        This is the left-looking Level 3 BLAS version of the algorithm.

       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 A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(M,N)-by-N upper trapezoidal matrix R (R is
                     upper triangular if m >= n); the elements below the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of min(m,n) elementary reflectors (see Further
                     Details).

           LDA

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

           TAU

                     TAU is DOUBLE PRECISION array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           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 if MIN(M,N) = 0,
                     otherwise the dimension can be divided into three parts.

                     1) The part for the triangular factor T. If the very last T is not bigger
                        than any of the rest, then this part is NB x ceiling(K/NB), otherwise,
                        NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T

                     2) The part for the very last T when T is bigger than any of the rest T.
                        The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB,
                        where K = min(M,N), NX is calculated by
                              NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) )

                     3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

                     So LWORK = part1 + part2 + part3

                     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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Further Details

         The matrix Q is represented as a product of elementary reflectors

            Q = H(1) H(2) . . . H(k), where k = min(m,n).

         Each H(i) has the form

            H(i) = I - tau * v * v'

         where tau is a real scalar, and v is a real vector with
         v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
         and tau in TAU(i).

   subroutine dgetrf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, integer INFO)
       DGETRF VARIANT: Crout Level 3 BLAS version of the algorithm. DGETRF VARIANT: iterative
       version of Sivan Toledo's recursive LU algorithm

       DGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

       Purpose:

        DGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the Crout Level 3 BLAS version of the algorithm.

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Purpose:

        DGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the left-looking Level 3 BLAS version of the algorithm.

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Purpose:

        DGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This code implements an iterative version of Sivan Toledo's recursive
        LU algorithm[1].  For square matrices, this iterative versions should
        be within a factor of two of the optimum number of memory transfers.

        The pattern is as follows, with the large blocks of U being updated
        in one call to DTRSM, and the dotted lines denoting sections that
        have had all pending permutations applied:

         1 2 3 4 5 6 7 8
        +-+-+---+-------+------
        | |1|   |       |
        |.+-+ 2 |       |
        | | |   |       |
        |.|.+-+-+   4   |
        | | | |1|       |
        | | |.+-+       |
        | | | | |       |
        |.|.|.|.+-+-+---+  8
        | | | | | |1|   |
        | | | | |.+-+ 2 |
        | | | | | | |   |
        | | | | |.|.+-+-+
        | | | | | | | |1|
        | | | | | | |.+-+
        | | | | | | | | |
        |.|.|.|.|.|.|.|.+-----
        | | | | | | | | |

        The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
        the binary expansion of the current column.  Each Schur update is
        applied as soon as the necessary portion of U is available.

        [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
        Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
        1065-1081. http://dx.doi.org/10.1137/S0895479896297744

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

   subroutine sgeqrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real,
       dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
       SGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm. Purpose:

        SGEQRF computes a QR factorization of a real M-by-N matrix A:
        A = Q * R.

        This is the left-looking Level 3 BLAS version of the algorithm.

       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 A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(M,N)-by-N upper trapezoidal matrix R (R is
                     upper triangular if m >= n); the elements below the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of min(m,n) elementary reflectors (see Further
                     Details).

           LDA

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

           TAU

                     TAU is REAL array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           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 if MIN(M,N) = 0,
                     otherwise the dimension can be divided into three parts.

                     1) The part for the triangular factor T. If the very last T is not bigger
                        than any of the rest, then this part is NB x ceiling(K/NB), otherwise,
                        NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T

                     2) The part for the very last T when T is bigger than any of the rest T.
                        The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB,
                        where K = min(M,N), NX is calculated by
                              NX = MAX( 0, ILAENV( 3, 'SGEQRF', ' ', M, N, -1, -1 ) )

                     3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

                     So LWORK = part1 + part2 + part3

                     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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Further Details

         The matrix Q is represented as a product of elementary reflectors

            Q = H(1) H(2) . . . H(k), where k = min(m,n).

         Each H(i) has the form

            H(i) = I - tau * v * v'

         where tau is a real scalar, and v is a real vector with
         v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
         and tau in TAU(i).

   subroutine sgetrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer,
       dimension( * ) IPIV, integer INFO)
       SGETRF VARIANT: Crout Level 3 BLAS version of the algorithm. SGETRF VARIANT: iterative
       version of Sivan Toledo's recursive LU algorithm

       SGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

       Purpose:

        SGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the Crout Level 3 BLAS version of the algorithm.

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Purpose:

        SGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the left-looking Level 3 BLAS version of the algorithm.

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Purpose:

        SGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This code implements an iterative version of Sivan Toledo's recursive
        LU algorithm[1].  For square matrices, this iterative versions should
        be within a factor of two of the optimum number of memory transfers.

        The pattern is as follows, with the large blocks of U being updated
        in one call to STRSM, and the dotted lines denoting sections that
        have had all pending permutations applied:

         1 2 3 4 5 6 7 8
        +-+-+---+-------+------
        | |1|   |       |
        |.+-+ 2 |       |
        | | |   |       |
        |.|.+-+-+   4   |
        | | | |1|       |
        | | |.+-+       |
        | | | | |       |
        |.|.|.|.+-+-+---+  8
        | | | | | |1|   |
        | | | | |.+-+ 2 |
        | | | | | | |   |
        | | | | |.|.+-+-+
        | | | | | | | |1|
        | | | | | | |.+-+
        | | | | | | | | |
        |.|.|.|.|.|.|.|.+-----
        | | | | | | | | |

        The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
        the binary expansion of the current column.  Each Schur update is
        applied as soon as the necessary portion of U is available.

        [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
        Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
        1065-1081. http://dx.doi.org/10.1137/S0895479896297744

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

   subroutine zgeqrf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
       complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer
       INFO)
       ZGEQRF VARIANT: left-looking Level 3 BLAS of the algorithm. Purpose:

        ZGEQRF computes a QR factorization of a real M-by-N matrix A:
        A = Q * R.

        This is the left-looking Level 3 BLAS version of the algorithm.

       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 A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(M,N)-by-N upper trapezoidal matrix R (R is
                     upper triangular if m >= n); the elements below the diagonal,
                     with the array TAU, represent the orthogonal matrix Q as a
                     product of min(m,n) elementary reflectors (see Further
                     Details).

           LDA

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

           TAU

                     TAU is COMPLEX*16 array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           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 if MIN(M,N) = 0,
                     otherwise the dimension can be divided into three parts.

                     1) The part for the triangular factor T. If the very last T is not bigger
                        than any of the rest, then this part is NB x ceiling(K/NB), otherwise,
                        NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T

                     2) The part for the very last T when T is bigger than any of the rest T.
                        The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB,
                        where K = min(M,N), NX is calculated by
                              NX = MAX( 0, ILAENV( 3, 'ZGEQRF', ' ', M, N, -1, -1 ) )

                     3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

                     So LWORK = part1 + part2 + part3

                     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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Further Details

         The matrix Q is represented as a product of elementary reflectors

            Q = H(1) H(2) . . . H(k), where k = min(m,n).

         Each H(i) has the form

            H(i) = I - tau * v * v'

         where tau is a real scalar, and v is a real vector with
         v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
         and tau in TAU(i).

   subroutine zgetrf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, integer INFO)
       ZGETRF VARIANT: Crout Level 3 BLAS version of the algorithm. ZGETRF VARIANT: iterative
       version of Sivan Toledo's recursive LU algorithm

       ZGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

       Purpose:

        ZGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the Crout Level 3 BLAS version of the algorithm.

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Purpose:

        ZGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This is the left-looking Level 3 BLAS version of the algorithm.

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

       Purpose:

        ZGETRF computes an LU factorization of a general M-by-N matrix A
        using partial pivoting with row interchanges.

        The factorization has the form
           A = P * L * U
        where P is a permutation matrix, L is lower triangular with unit
        diagonal elements (lower trapezoidal if m > n), and U is upper
        triangular (upper trapezoidal if m < n).

        This code implements an iterative version of Sivan Toledo's recursive
        LU algorithm[1].  For square matrices, this iterative versions should
        be within a factor of two of the optimum number of memory transfers.

        The pattern is as follows, with the large blocks of U being updated
        in one call to DTRSM, and the dotted lines denoting sections that
        have had all pending permutations applied:

         1 2 3 4 5 6 7 8
        +-+-+---+-------+------
        | |1|   |       |
        |.+-+ 2 |       |
        | | |   |       |
        |.|.+-+-+   4   |
        | | | |1|       |
        | | |.+-+       |
        | | | | |       |
        |.|.|.|.+-+-+---+  8
        | | | | | |1|   |
        | | | | |.+-+ 2 |
        | | | | | | |   |
        | | | | |.|.+-+-+
        | | | | | | | |1|
        | | | | | | |.+-+
        | | | | | | | | |
        |.|.|.|.|.|.|.|.+-----
        | | | | | | | | |

        The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
        the binary expansion of the current column.  Each Schur update is
        applied as soon as the necessary portion of U is available.

        [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
        Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
        1065-1081. http://dx.doi.org/10.1137/S0895479896297744

       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 = P*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).

           IPIV

                     IPIV is INTEGER array, dimension (min(M,N))
                     The pivot indices; for 1 <= i <= min(M,N), row i of the
                     matrix was interchanged with row IPIV(i).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                           has been completed, but the factor U is exactly
                           singular, and division by zero will occur if it is used
                           to solve a system of equations.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date
           December 2016

Author

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