CLAUNHR_COL_GETRFNP 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 'modified' Gaussian elimination is


    at least one in absolute value (so that division-by-zero not


    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 blocked right-looking version of the algorithm,


 calling Level 3 BLAS to update the submatrix. To factorize a block,


 this routine calls the recursive routine CLAUNHR_COL_GETRFNP2.


 [1] 'Reconstructing Householder vectors from tall-skinny QR',


     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,


     E. Solomonik, J. Parallel Distrib. Comput.,


     vol. 85, pp. 3-31, 2015.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 November 2019, Igor Kozachenko,


                Computer Science Division,


                University of California, Berkeley

 DLAORHR_COL_GETRFNP 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 'modified' Gaussian elimination is


    at least one in absolute value (so that division-by-zero not


    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 blocked right-looking version of the algorithm,


 calling Level 3 BLAS to update the submatrix. To factorize a block,


 this routine calls the recursive routine DLAORHR_COL_GETRFNP2.


 [1] 'Reconstructing Householder vectors from tall-skinny QR',


     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,


     E. Solomonik, J. Parallel Distrib. Comput.,


     vol. 85, pp. 3-31, 2015.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 November 2019, Igor Kozachenko,


                Computer Science Division,


                University of California, Berkeley

 SLAORHR_COL_GETRFNP 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 'modified' Gaussian elimination is


    at least one in absolute value (so that division-by-zero not


    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 blocked right-looking version of the algorithm,


 calling Level 3 BLAS to update the submatrix. To factorize a block,


 this routine calls the recursive routine SLAORHR_COL_GETRFNP2.


 [1] 'Reconstructing Householder vectors from tall-skinny QR',


     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,


     E. Solomonik, J. Parallel Distrib. Comput.,


     vol. 85, pp. 3-31, 2015.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 November 2019, Igor Kozachenko,


                Computer Science Division,


                University of California, Berkeley

 ZLAUNHR_COL_GETRFNP 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 'modified' Gaussian elimination is


    at least one in absolute value (so that division-by-zero not


    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 blocked right-looking version of the algorithm,


 calling Level 3 BLAS to update the submatrix. To factorize a block,


 this routine calls the recursive routine ZLAUNHR_COL_GETRFNP2.


 [1] 'Reconstructing Householder vectors from tall-skinny QR',


     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,


     E. Solomonik, J. Parallel Distrib. Comput.,


     vol. 85, pp. 3-31, 2015.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 November 2019, Igor Kozachenko,


                Computer Science Division,


                University of California, Berkeley