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

 subroutine zlaror ( character side, character init, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( 4 ) iseed, complex*16, dimension( * ) x, integer info )

ZLAROR

Purpose:
```    ZLAROR pre- or post-multiplies an M by N matrix A by a random
unitary matrix U, overwriting A. A may optionally be
initialized to the identity matrix before multiplying by U.
U is generated using the method of G.W. Stewart
( SIAM J. Numer. Anal. 17, 1980, pp. 403-409 ).
(BLAS-2 version)```
Parameters
 [in] SIDE ``` SIDE is CHARACTER*1 SIDE specifies whether A is multiplied on the left or right by U. SIDE = 'L' Multiply A on the left (premultiply) by U SIDE = 'R' Multiply A on the right (postmultiply) by UC> SIDE = 'C' Multiply A on the left by U and the right by UC> SIDE = 'T' Multiply A on the left by U and the right by U' Not modified.``` [in] INIT ``` INIT is CHARACTER*1 INIT specifies whether or not A should be initialized to the identity matrix. INIT = 'I' Initialize A to (a section of) the identity matrix before applying U. INIT = 'N' No initialization. Apply U to the input matrix A. INIT = 'I' may be used to generate square (i.e., unitary) or rectangular orthogonal matrices (orthogonality being in the sense of ZDOTC): For square matrices, M=N, and SIDE many be either 'L' or 'R'; the rows will be orthogonal to each other, as will the columns. For rectangular matrices where M < N, SIDE = 'R' will produce a dense matrix whose rows will be orthogonal and whose columns will not, while SIDE = 'L' will produce a matrix whose rows will be orthogonal, and whose first M columns will be orthogonal, the remaining columns being zero. For matrices where M > N, just use the previous explanation, interchanging 'L' and 'R' and "rows" and "columns". Not modified.``` [in] M ``` M is INTEGER Number of rows of A. Not modified.``` [in] N ``` N is INTEGER Number of columns of A. Not modified.``` [in,out] A ``` A is COMPLEX*16 array, dimension ( LDA, N ) Input and output array. Overwritten by U A ( if SIDE = 'L' ) or by A U ( if SIDE = 'R' ) or by U A U* ( if SIDE = 'C') or by U A U' ( if SIDE = 'T') on exit.``` [in] LDA ``` LDA is INTEGER Leading dimension of A. Must be at least MAX ( 1, M ). Not modified.``` [in,out] ISEED ``` ISEED is INTEGER array, dimension ( 4 ) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZLAROR to continue the same random number sequence. Modified.``` [out] X ``` X is COMPLEX*16 array, dimension ( 3*MAX( M, N ) ) Workspace. Of length: 2*M + N if SIDE = 'L', 2*N + M if SIDE = 'R', 3*N if SIDE = 'C' or 'T'. Modified.``` [out] INFO ``` INFO is INTEGER An error flag. It is set to: 0 if no error. 1 if ZLARND returned a bad random number (installation problem) -1 if SIDE is not L, R, C, or T. -3 if M is negative. -4 if N is negative or if SIDE is C or T and N is not equal to M. -6 if LDA is less than M.```

Definition at line 157 of file zlaror.f.

158*
159* -- LAPACK auxiliary routine --
160* -- LAPACK is a software package provided by Univ. of Tennessee, --
161* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
162*
163* .. Scalar Arguments ..
164 CHARACTER INIT, SIDE
165 INTEGER INFO, LDA, M, N
166* ..
167* .. Array Arguments ..
168 INTEGER ISEED( 4 )
169 COMPLEX*16 A( LDA, * ), X( * )
170* ..
171*
172* =====================================================================
173*
174* .. Parameters ..
175 DOUBLE PRECISION ZERO, ONE, TOOSML
176 parameter( zero = 0.0d+0, one = 1.0d+0,
177 \$ toosml = 1.0d-20 )
178 COMPLEX*16 CZERO, CONE
179 parameter( czero = ( 0.0d+0, 0.0d+0 ),
180 \$ cone = ( 1.0d+0, 0.0d+0 ) )
181* ..
182* .. Local Scalars ..
183 INTEGER IROW, ITYPE, IXFRM, J, JCOL, KBEG, NXFRM
184 DOUBLE PRECISION FACTOR, XABS, XNORM
185 COMPLEX*16 CSIGN, XNORMS
186* ..
187* .. External Functions ..
188 LOGICAL LSAME
189 DOUBLE PRECISION DZNRM2
190 COMPLEX*16 ZLARND
191 EXTERNAL lsame, dznrm2, zlarnd
192* ..
193* .. External Subroutines ..
194 EXTERNAL xerbla, zgemv, zgerc, zlacgv, zlaset, zscal
195* ..
196* .. Intrinsic Functions ..
197 INTRINSIC abs, dcmplx, dconjg
198* ..
199* .. Executable Statements ..
200*
201 info = 0
202 IF( n.EQ.0 .OR. m.EQ.0 )
203 \$ RETURN
204*
205 itype = 0
206 IF( lsame( side, 'L' ) ) THEN
207 itype = 1
208 ELSE IF( lsame( side, 'R' ) ) THEN
209 itype = 2
210 ELSE IF( lsame( side, 'C' ) ) THEN
211 itype = 3
212 ELSE IF( lsame( side, 'T' ) ) THEN
213 itype = 4
214 END IF
215*
216* Check for argument errors.
217*
218 IF( itype.EQ.0 ) THEN
219 info = -1
220 ELSE IF( m.LT.0 ) THEN
221 info = -3
222 ELSE IF( n.LT.0 .OR. ( itype.EQ.3 .AND. n.NE.m ) ) THEN
223 info = -4
224 ELSE IF( lda.LT.m ) THEN
225 info = -6
226 END IF
227 IF( info.NE.0 ) THEN
228 CALL xerbla( 'ZLAROR', -info )
229 RETURN
230 END IF
231*
232 IF( itype.EQ.1 ) THEN
233 nxfrm = m
234 ELSE
235 nxfrm = n
236 END IF
237*
238* Initialize A to the identity matrix if desired
239*
240 IF( lsame( init, 'I' ) )
241 \$ CALL zlaset( 'Full', m, n, czero, cone, a, lda )
242*
243* If no rotation possible, still multiply by
244* a random complex number from the circle |x| = 1
245*
246* 2) Compute Rotation by computing Householder
247* Transformations H(2), H(3), ..., H(n). Note that the
248* order in which they are computed is irrelevant.
249*
250 DO 10 j = 1, nxfrm
251 x( j ) = czero
252 10 CONTINUE
253*
254 DO 30 ixfrm = 2, nxfrm
255 kbeg = nxfrm - ixfrm + 1
256*
257* Generate independent normal( 0, 1 ) random numbers
258*
259 DO 20 j = kbeg, nxfrm
260 x( j ) = zlarnd( 3, iseed )
261 20 CONTINUE
262*
263* Generate a Householder transformation from the random vector X
264*
265 xnorm = dznrm2( ixfrm, x( kbeg ), 1 )
266 xabs = abs( x( kbeg ) )
267 IF( xabs.NE.czero ) THEN
268 csign = x( kbeg ) / xabs
269 ELSE
270 csign = cone
271 END IF
272 xnorms = csign*xnorm
273 x( nxfrm+kbeg ) = -csign
274 factor = xnorm*( xnorm+xabs )
275 IF( abs( factor ).LT.toosml ) THEN
276 info = 1
277 CALL xerbla( 'ZLAROR', -info )
278 RETURN
279 ELSE
280 factor = one / factor
281 END IF
282 x( kbeg ) = x( kbeg ) + xnorms
283*
284* Apply Householder transformation to A
285*
286 IF( itype.EQ.1 .OR. itype.EQ.3 .OR. itype.EQ.4 ) THEN
287*
288* Apply H(k) on the left of A
289*
290 CALL zgemv( 'C', ixfrm, n, cone, a( kbeg, 1 ), lda,
291 \$ x( kbeg ), 1, czero, x( 2*nxfrm+1 ), 1 )
292 CALL zgerc( ixfrm, n, -dcmplx( factor ), x( kbeg ), 1,
293 \$ x( 2*nxfrm+1 ), 1, a( kbeg, 1 ), lda )
294*
295 END IF
296*
297 IF( itype.GE.2 .AND. itype.LE.4 ) THEN
298*
299* Apply H(k)* (or H(k)') on the right of A
300*
301 IF( itype.EQ.4 ) THEN
302 CALL zlacgv( ixfrm, x( kbeg ), 1 )
303 END IF
304*
305 CALL zgemv( 'N', m, ixfrm, cone, a( 1, kbeg ), lda,
306 \$ x( kbeg ), 1, czero, x( 2*nxfrm+1 ), 1 )
307 CALL zgerc( m, ixfrm, -dcmplx( factor ), x( 2*nxfrm+1 ), 1,
308 \$ x( kbeg ), 1, a( 1, kbeg ), lda )
309*
310 END IF
311 30 CONTINUE
312*
313 x( 1 ) = zlarnd( 3, iseed )
314 xabs = abs( x( 1 ) )
315 IF( xabs.NE.zero ) THEN
316 csign = x( 1 ) / xabs
317 ELSE
318 csign = cone
319 END IF
320 x( 2*nxfrm ) = csign
321*
322* Scale the matrix A by D.
323*
324 IF( itype.EQ.1 .OR. itype.EQ.3 .OR. itype.EQ.4 ) THEN
325 DO 40 irow = 1, m
326 CALL zscal( n, dconjg( x( nxfrm+irow ) ), a( irow, 1 ),
327 \$ lda )
328 40 CONTINUE
329 END IF
330*
331 IF( itype.EQ.2 .OR. itype.EQ.3 ) THEN
332 DO 50 jcol = 1, n
333 CALL zscal( m, x( nxfrm+jcol ), a( 1, jcol ), 1 )
334 50 CONTINUE
335 END IF
336*
337 IF( itype.EQ.4 ) THEN
338 DO 60 jcol = 1, n
339 CALL zscal( m, dconjg( x( nxfrm+jcol ) ), a( 1, jcol ), 1 )
340 60 CONTINUE
341 END IF
342 RETURN
343*
344* End of ZLAROR
345*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
ZGEMV
Definition zgemv.f:160
subroutine zgerc(m, n, alpha, x, incx, y, incy, a, lda)
ZGERC
Definition zgerc.f:130
subroutine zlacgv(n, x, incx)
ZLACGV conjugates a complex vector.
Definition zlacgv.f:74
subroutine zlaset(uplo, m, n, alpha, beta, a, lda)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition zlaset.f:106
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition dznrm2.f90:90
subroutine zscal(n, za, zx, incx)
ZSCAL
Definition zscal.f:78
complex *16 function zlarnd(idist, iseed)
ZLARND
Definition zlarnd.f:75
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