LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ slaorhr_col_getrfnp()

 subroutine slaorhr_col_getrfnp ( integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, integer INFO )

SLAORHR_COL_GETRFNP

Purpose:
``` 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.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in,out] 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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] 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.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Contributors:
``` November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 145 of file slaorhr_col_getrfnp.f.

146  IMPLICIT NONE
147 *
148 * -- LAPACK computational routine --
149 * -- LAPACK is a software package provided by Univ. of Tennessee, --
150 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
151 *
152 * .. Scalar Arguments ..
153  INTEGER INFO, LDA, M, N
154 * ..
155 * .. Array Arguments ..
156  REAL A( LDA, * ), D( * )
157 * ..
158 *
159 * =====================================================================
160 *
161 * .. Parameters ..
162  REAL ONE
163  parameter( one = 1.0e+0 )
164 * ..
165 * .. Local Scalars ..
166  INTEGER IINFO, J, JB, NB
167 * ..
168 * .. External Subroutines ..
170 * ..
171 * .. External Functions ..
172  INTEGER ILAENV
173  EXTERNAL ilaenv
174 * ..
175 * .. Intrinsic Functions ..
176  INTRINSIC max, min
177 * ..
178 * .. Executable Statements ..
179 *
180 * Test the input parameters.
181 *
182  info = 0
183  IF( m.LT.0 ) THEN
184  info = -1
185  ELSE IF( n.LT.0 ) THEN
186  info = -2
187  ELSE IF( lda.LT.max( 1, m ) ) THEN
188  info = -4
189  END IF
190  IF( info.NE.0 ) THEN
191  CALL xerbla( 'SLAORHR_COL_GETRFNP', -info )
192  RETURN
193  END IF
194 *
195 * Quick return if possible
196 *
197  IF( min( m, n ).EQ.0 )
198  \$ RETURN
199 *
200 * Determine the block size for this environment.
201 *
202
203  nb = ilaenv( 1, 'SLAORHR_COL_GETRFNP', ' ', m, n, -1, -1 )
204
205  IF( nb.LE.1 .OR. nb.GE.min( m, n ) ) THEN
206 *
207 * Use unblocked code.
208 *
209  CALL slaorhr_col_getrfnp2( m, n, a, lda, d, info )
210  ELSE
211 *
212 * Use blocked code.
213 *
214  DO j = 1, min( m, n ), nb
215  jb = min( min( m, n )-j+1, nb )
216 *
217 * Factor diagonal and subdiagonal blocks.
218 *
219  CALL slaorhr_col_getrfnp2( m-j+1, jb, a( j, j ), lda,
220  \$ d( j ), iinfo )
221 *
222  IF( j+jb.LE.n ) THEN
223 *
224 * Compute block row of U.
225 *
226  CALL strsm( 'Left', 'Lower', 'No transpose', 'Unit', jb,
227  \$ n-j-jb+1, one, a( j, j ), lda, a( j, j+jb ),
228  \$ lda )
229  IF( j+jb.LE.m ) THEN
230 *
231 * Update trailing submatrix.
232 *
233  CALL sgemm( 'No transpose', 'No transpose', m-j-jb+1,
234  \$ n-j-jb+1, jb, -one, a( j+jb, j ), lda,
235  \$ a( j, j+jb ), lda, one, a( j+jb, j+jb ),
236  \$ lda )
237  END IF
238  END IF
239  END DO
240  END IF
241  RETURN
242 *
243 * End of SLAORHR_COL_GETRFNP
244 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
recursive subroutine slaorhr_col_getrfnp2(M, N, A, LDA, D, INFO)
SLAORHR_COL_GETRFNP2
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:181
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
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