LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ slaorhr_col_getrfnp2()

 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 "modified" 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] "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.

[2] "Recursion leads to automatic variable blocking for dense linear
algebra algorithms", F. Gustavson, IBM J. of Res. and Dev.,
vol. 41, no. 6, pp. 737-755, 1997.```
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 166 of file slaorhr_col_getrfnp2.f.

167 IMPLICIT NONE
168*
169* -- LAPACK computational routine --
170* -- LAPACK is a software package provided by Univ. of Tennessee, --
171* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
172*
173* .. Scalar Arguments ..
174 INTEGER INFO, LDA, M, N
175* ..
176* .. Array Arguments ..
177 REAL A( LDA, * ), D( * )
178* ..
179*
180* =====================================================================
181*
182* .. Parameters ..
183 REAL ONE
184 parameter( one = 1.0e+0 )
185* ..
186* .. Local Scalars ..
187 REAL SFMIN
188 INTEGER I, IINFO, N1, N2
189* ..
190* .. External Functions ..
191 REAL SLAMCH
192 EXTERNAL slamch
193* ..
194* .. External Subroutines ..
195 EXTERNAL sgemm, sscal, strsm, xerbla
196* ..
197* .. Intrinsic Functions ..
198 INTRINSIC abs, sign, max, min
199* ..
200* .. Executable Statements ..
201*
202* Test the input parameters
203*
204 info = 0
205 IF( m.LT.0 ) THEN
206 info = -1
207 ELSE IF( n.LT.0 ) THEN
208 info = -2
209 ELSE IF( lda.LT.max( 1, m ) ) THEN
210 info = -4
211 END IF
212 IF( info.NE.0 ) THEN
213 CALL xerbla( 'SLAORHR_COL_GETRFNP2', -info )
214 RETURN
215 END IF
216*
217* Quick return if possible
218*
219 IF( min( m, n ).EQ.0 )
220 \$ RETURN
221
222 IF ( m.EQ.1 ) THEN
223*
224* One row case, (also recursion termination case),
225* use unblocked code
226*
227* Transfer the sign
228*
229 d( 1 ) = -sign( one, a( 1, 1 ) )
230*
231* Construct the row of U
232*
233 a( 1, 1 ) = a( 1, 1 ) - d( 1 )
234*
235 ELSE IF( n.EQ.1 ) THEN
236*
237* One column case, (also recursion termination case),
238* use unblocked code
239*
240* Transfer the sign
241*
242 d( 1 ) = -sign( one, a( 1, 1 ) )
243*
244* Construct the row of U
245*
246 a( 1, 1 ) = a( 1, 1 ) - d( 1 )
247*
248* Scale the elements 2:M of the column
249*
250* Determine machine safe minimum
251*
252 sfmin = slamch('S')
253*
254* Construct the subdiagonal elements of L
255*
256 IF( abs( a( 1, 1 ) ) .GE. sfmin ) THEN
257 CALL sscal( m-1, one / a( 1, 1 ), a( 2, 1 ), 1 )
258 ELSE
259 DO i = 2, m
260 a( i, 1 ) = a( i, 1 ) / a( 1, 1 )
261 END DO
262 END IF
263*
264 ELSE
265*
266* Divide the matrix B into four submatrices
267*
268 n1 = min( m, n ) / 2
269 n2 = n-n1
270
271*
272* Factor B11, recursive call
273*
274 CALL slaorhr_col_getrfnp2( n1, n1, a, lda, d, iinfo )
275*
276* Solve for B21
277*
278 CALL strsm( 'R', 'U', 'N', 'N', m-n1, n1, one, a, lda,
279 \$ a( n1+1, 1 ), lda )
280*
281* Solve for B12
282*
283 CALL strsm( 'L', 'L', 'N', 'U', n1, n2, one, a, lda,
284 \$ a( 1, n1+1 ), lda )
285*
286* Update B22, i.e. compute the Schur complement
287* B22 := B22 - B21*B12
288*
289 CALL sgemm( 'N', 'N', m-n1, n2, n1, -one, a( n1+1, 1 ), lda,
290 \$ a( 1, n1+1 ), lda, one, a( n1+1, n1+1 ), lda )
291*
292* Factor B22, recursive call
293*
294 CALL slaorhr_col_getrfnp2( m-n1, n2, a( n1+1, n1+1 ), lda,
295 \$ d( n1+1 ), iinfo )
296*
297 END IF
298 RETURN
299*
300* End of SLAORHR_COL_GETRFNP2
301*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine sgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
SGEMM
Definition sgemm.f:188
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
recursive subroutine slaorhr_col_getrfnp2(m, n, a, lda, d, info)
SLAORHR_COL_GETRFNP2
subroutine sscal(n, sa, sx, incx)
SSCAL
Definition sscal.f:79
subroutine strsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb)
STRSM
Definition strsm.f:181
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