LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ zlaunhr_col_getrfnp2()

 recursive subroutine zlaunhr_col_getrfnp2 ( integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) d, integer info )

ZLAUNHR_COL_GETRFNP2

Purpose:
ZLAUNHR_COL_GETRFNP2 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
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 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].

ZLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked
routine ZLAUNHR_COL_GETRFNP, which uses blocked code calling
Level 3 BLAS to update the submatrix. However, ZLAUNHR_COL_GETRFNP2
is self-sufficient and can be used without ZLAUNHR_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 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. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] 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 ). [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 zlaunhr_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 COMPLEX*16 A( LDA, * ), D( * )
178* ..
179*
180* =====================================================================
181*
182* .. Parameters ..
183 DOUBLE PRECISION ONE
184 parameter( one = 1.0d+0 )
185 COMPLEX*16 CONE
186 parameter( cone = ( 1.0d+0, 0.0d+0 ) )
187* ..
188* .. Local Scalars ..
189 DOUBLE PRECISION SFMIN
190 INTEGER I, IINFO, N1, N2
191 COMPLEX*16 Z
192* ..
193* .. External Functions ..
194 DOUBLE PRECISION DLAMCH
195 EXTERNAL dlamch
196* ..
197* .. External Subroutines ..
198 EXTERNAL zgemm, zscal, ztrsm, xerbla
199* ..
200* .. Intrinsic Functions ..
201 INTRINSIC abs, dble, dcmplx, dimag, dsign, max, min
202* ..
203* .. Statement Functions ..
204 DOUBLE PRECISION CABS1
205* ..
206* .. Statement Function definitions ..
207 cabs1( z ) = abs( dble( z ) ) + abs( dimag( z ) )
208* ..
209* .. Executable Statements ..
210*
211* Test the input parameters
212*
213 info = 0
214 IF( m.LT.0 ) THEN
215 info = -1
216 ELSE IF( n.LT.0 ) THEN
217 info = -2
218 ELSE IF( lda.LT.max( 1, m ) ) THEN
219 info = -4
220 END IF
221 IF( info.NE.0 ) THEN
222 CALL xerbla( 'ZLAUNHR_COL_GETRFNP2', -info )
223 RETURN
224 END IF
225*
226* Quick return if possible
227*
228 IF( min( m, n ).EQ.0 )
229 \$ RETURN
230
231 IF ( m.EQ.1 ) THEN
232*
233* One row case, (also recursion termination case),
234* use unblocked code
235*
236* Transfer the sign
237*
238 d( 1 ) = dcmplx( -dsign( one, dble( a( 1, 1 ) ) ) )
239*
240* Construct the row of U
241*
242 a( 1, 1 ) = a( 1, 1 ) - d( 1 )
243*
244 ELSE IF( n.EQ.1 ) THEN
245*
246* One column case, (also recursion termination case),
247* use unblocked code
248*
249* Transfer the sign
250*
251 d( 1 ) = dcmplx( -dsign( one, dble( a( 1, 1 ) ) ) )
252*
253* Construct the row of U
254*
255 a( 1, 1 ) = a( 1, 1 ) - d( 1 )
256*
257* Scale the elements 2:M of the column
258*
259* Determine machine safe minimum
260*
261 sfmin = dlamch('S')
262*
263* Construct the subdiagonal elements of L
264*
265 IF( cabs1( a( 1, 1 ) ) .GE. sfmin ) THEN
266 CALL zscal( m-1, cone / a( 1, 1 ), a( 2, 1 ), 1 )
267 ELSE
268 DO i = 2, m
269 a( i, 1 ) = a( i, 1 ) / a( 1, 1 )
270 END DO
271 END IF
272*
273 ELSE
274*
275* Divide the matrix B into four submatrices
276*
277 n1 = min( m, n ) / 2
278 n2 = n-n1
279
280*
281* Factor B11, recursive call
282*
283 CALL zlaunhr_col_getrfnp2( n1, n1, a, lda, d, iinfo )
284*
285* Solve for B21
286*
287 CALL ztrsm( 'R', 'U', 'N', 'N', m-n1, n1, cone, a, lda,
288 \$ a( n1+1, 1 ), lda )
289*
290* Solve for B12
291*
292 CALL ztrsm( 'L', 'L', 'N', 'U', n1, n2, cone, a, lda,
293 \$ a( 1, n1+1 ), lda )
294*
295* Update B22, i.e. compute the Schur complement
296* B22 := B22 - B21*B12
297*
298 CALL zgemm( 'N', 'N', m-n1, n2, n1, -cone, a( n1+1, 1 ), lda,
299 \$ a( 1, n1+1 ), lda, cone, a( n1+1, n1+1 ), lda )
300*
301* Factor B22, recursive call
302*
303 CALL zlaunhr_col_getrfnp2( m-n1, n2, a( n1+1, n1+1 ), lda,
304 \$ d( n1+1 ), iinfo )
305*
306 END IF
307 RETURN
308*
309* End of ZLAUNHR_COL_GETRFNP2
310*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
ZGEMM
Definition zgemm.f:188
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
recursive subroutine zlaunhr_col_getrfnp2(m, n, a, lda, d, info)
ZLAUNHR_COL_GETRFNP2
subroutine zscal(n, za, zx, incx)
ZSCAL
Definition zscal.f:78
subroutine ztrsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb)
ZTRSM
Definition ztrsm.f:180
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