launhr_col_getrfnp#

Functions

void claunhr_col_getrfnp(
    const INT           m,
    const INT           n,
          c64* restrict A,
    const INT           lda,
          c64* restrict D,
          INT*          info
);
void claunhr_col_getrfnp(const INT m, const INT n, c64 *restrict A, const INT lda, c64 *restrict D, INT *info)#

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.

Parameters

in
m

The number of rows of the matrix A. m >= 0.

in
n

The number of columns of the matrix A. n >= 0.

inout
A

Single 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.

in
lda

The leading dimension of the array A. lda >= max(1, m).

out
D

Single complex array, dimension min(m, n). The diagonal elements of the diagonal M-by-N sign matrix S, D(i) = S(i,i), where 0 <= i < min(m, n). The elements can be only ( +1.0, 0.0 ) or (-1.0, 0.0 ).

out
info

= 0: successful exit < 0: if info = -i, the i-th argument had an illegal value

Functions

void zlaunhr_col_getrfnp(
    const INT            m,
    const INT            n,
          c128* restrict A,
    const INT            lda,
          c128* restrict D,
          INT*           info
);
void zlaunhr_col_getrfnp(const INT m, const INT n, c128 *restrict A, const INT lda, c128 *restrict D, INT *info)#

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.

Parameters

in
m

The number of rows of the matrix A. m >= 0.

in
n

The number of columns of the matrix A. n >= 0.

inout
A

Double 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.

in
lda

The leading dimension of the array A. lda >= max(1, m).

out
D

Double complex array, dimension min(m, n). The diagonal elements of the diagonal M-by-N sign matrix S, D(i) = S(i,i), where 0 <= i < min(m, n). The elements can be only ( +1.0, 0.0 ) or (-1.0, 0.0 ).

out
info

= 0: successful exit < 0: if info = -i, the i-th argument had an illegal value