 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dlaror()

 subroutine dlaror ( character SIDE, character INIT, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, double precision, dimension( * ) X, integer INFO )

DLAROR

Purpose:
``` DLAROR pre- or post-multiplies an M by N matrix A by a random
orthogonal 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, 403-409).```
Parameters
 [in] SIDE ``` SIDE is CHARACTER*1 Specifies whether A is multiplied on the left or right by U. = 'L': Multiply A on the left (premultiply) by U = 'R': Multiply A on the right (postmultiply) by U' = 'C' or 'T': Multiply A on the left by U and the right by U' (Here, U' means U-transpose.)``` [in] INIT ``` INIT is CHARACTER*1 Specifies whether or not A should be initialized to the identity matrix. = 'I': Initialize A to (a section of) the identity matrix before applying U. = 'N': No initialization. Apply U to the input matrix A. INIT = 'I' may be used to generate square or rectangular orthogonal matrices: For M = N and SIDE = 'L' or 'R', the rows will be orthogonal to each other, as will the columns. If M < N, SIDE = 'R' produces a dense matrix whose rows are orthogonal and whose columns are not, while SIDE = 'L' produces a matrix whose rows are orthogonal, and whose first M columns are orthogonal, and whose remaining columns are zero. If M > N, SIDE = 'L' produces a dense matrix whose columns are orthogonal and whose rows are not, while SIDE = 'R' produces a matrix whose columns are orthogonal, and whose first M rows are orthogonal, and whose remaining rows are zero.``` [in] M ``` M is INTEGER The number of rows of A.``` [in] N ``` N is INTEGER The number of columns of A.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA, N) On entry, the array A. On exit, overwritten by U A ( if SIDE = 'L' ), or by A U ( if SIDE = 'R' ), or by U A U' ( if SIDE = 'C' or 'T').``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [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 DLAROR to continue the same random number sequence.``` [out] X ``` X is DOUBLE PRECISION 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'.``` [out] INFO ``` INFO is INTEGER An error flag. It is set to: = 0: normal return < 0: if INFO = -k, the k-th argument had an illegal value = 1: if the random numbers generated by DLARND are bad.```

Definition at line 145 of file dlaror.f.

146 *
147 * -- LAPACK auxiliary routine --
148 * -- LAPACK is a software package provided by Univ. of Tennessee, --
149 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
150 *
151 * .. Scalar Arguments ..
152  CHARACTER INIT, SIDE
153  INTEGER INFO, LDA, M, N
154 * ..
155 * .. Array Arguments ..
156  INTEGER ISEED( 4 )
157  DOUBLE PRECISION A( LDA, * ), X( * )
158 * ..
159 *
160 * =====================================================================
161 *
162 * .. Parameters ..
163  DOUBLE PRECISION ZERO, ONE, TOOSML
164  parameter( zero = 0.0d+0, one = 1.0d+0,
165  \$ toosml = 1.0d-20 )
166 * ..
167 * .. Local Scalars ..
168  INTEGER IROW, ITYPE, IXFRM, J, JCOL, KBEG, NXFRM
169  DOUBLE PRECISION FACTOR, XNORM, XNORMS
170 * ..
171 * .. External Functions ..
172  LOGICAL LSAME
173  DOUBLE PRECISION DLARND, DNRM2
174  EXTERNAL lsame, dlarnd, dnrm2
175 * ..
176 * .. External Subroutines ..
177  EXTERNAL dgemv, dger, dlaset, dscal, xerbla
178 * ..
179 * .. Intrinsic Functions ..
180  INTRINSIC abs, sign
181 * ..
182 * .. Executable Statements ..
183 *
184  info = 0
185  IF( n.EQ.0 .OR. m.EQ.0 )
186  \$ RETURN
187 *
188  itype = 0
189  IF( lsame( side, 'L' ) ) THEN
190  itype = 1
191  ELSE IF( lsame( side, 'R' ) ) THEN
192  itype = 2
193  ELSE IF( lsame( side, 'C' ) .OR. lsame( side, 'T' ) ) THEN
194  itype = 3
195  END IF
196 *
197 * Check for argument errors.
198 *
199  IF( itype.EQ.0 ) THEN
200  info = -1
201  ELSE IF( m.LT.0 ) THEN
202  info = -3
203  ELSE IF( n.LT.0 .OR. ( itype.EQ.3 .AND. n.NE.m ) ) THEN
204  info = -4
205  ELSE IF( lda.LT.m ) THEN
206  info = -6
207  END IF
208  IF( info.NE.0 ) THEN
209  CALL xerbla( 'DLAROR', -info )
210  RETURN
211  END IF
212 *
213  IF( itype.EQ.1 ) THEN
214  nxfrm = m
215  ELSE
216  nxfrm = n
217  END IF
218 *
219 * Initialize A to the identity matrix if desired
220 *
221  IF( lsame( init, 'I' ) )
222  \$ CALL dlaset( 'Full', m, n, zero, one, a, lda )
223 *
224 * If no rotation possible, multiply by random +/-1
225 *
226 * Compute rotation by computing Householder transformations
227 * H(2), H(3), ..., H(nhouse)
228 *
229  DO 10 j = 1, nxfrm
230  x( j ) = zero
231  10 CONTINUE
232 *
233  DO 30 ixfrm = 2, nxfrm
234  kbeg = nxfrm - ixfrm + 1
235 *
236 * Generate independent normal( 0, 1 ) random numbers
237 *
238  DO 20 j = kbeg, nxfrm
239  x( j ) = dlarnd( 3, iseed )
240  20 CONTINUE
241 *
242 * Generate a Householder transformation from the random vector X
243 *
244  xnorm = dnrm2( ixfrm, x( kbeg ), 1 )
245  xnorms = sign( xnorm, x( kbeg ) )
246  x( kbeg+nxfrm ) = sign( one, -x( kbeg ) )
247  factor = xnorms*( xnorms+x( kbeg ) )
248  IF( abs( factor ).LT.toosml ) THEN
249  info = 1
250  CALL xerbla( 'DLAROR', info )
251  RETURN
252  ELSE
253  factor = one / factor
254  END IF
255  x( kbeg ) = x( kbeg ) + xnorms
256 *
257 * Apply Householder transformation to A
258 *
259  IF( itype.EQ.1 .OR. itype.EQ.3 ) THEN
260 *
261 * Apply H(k) from the left.
262 *
263  CALL dgemv( 'T', ixfrm, n, one, a( kbeg, 1 ), lda,
264  \$ x( kbeg ), 1, zero, x( 2*nxfrm+1 ), 1 )
265  CALL dger( ixfrm, n, -factor, x( kbeg ), 1, x( 2*nxfrm+1 ),
266  \$ 1, a( kbeg, 1 ), lda )
267 *
268  END IF
269 *
270  IF( itype.EQ.2 .OR. itype.EQ.3 ) THEN
271 *
272 * Apply H(k) from the right.
273 *
274  CALL dgemv( 'N', m, ixfrm, one, a( 1, kbeg ), lda,
275  \$ x( kbeg ), 1, zero, x( 2*nxfrm+1 ), 1 )
276  CALL dger( m, ixfrm, -factor, x( 2*nxfrm+1 ), 1, x( kbeg ),
277  \$ 1, a( 1, kbeg ), lda )
278 *
279  END IF
280  30 CONTINUE
281 *
282  x( 2*nxfrm ) = sign( one, dlarnd( 3, iseed ) )
283 *
284 * Scale the matrix A by D.
285 *
286  IF( itype.EQ.1 .OR. itype.EQ.3 ) THEN
287  DO 40 irow = 1, m
288  CALL dscal( n, x( nxfrm+irow ), a( irow, 1 ), lda )
289  40 CONTINUE
290  END IF
291 *
292  IF( itype.EQ.2 .OR. itype.EQ.3 ) THEN
293  DO 50 jcol = 1, n
294  CALL dscal( m, x( nxfrm+jcol ), a( 1, jcol ), 1 )
295  50 CONTINUE
296  END IF
297  RETURN
298 *
299 * End of DLAROR
300 *
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
DGER
Definition: dger.f:130
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:156
double precision function dlarnd(IDIST, ISEED)
DLARND
Definition: dlarnd.f:73
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition: dnrm2.f90:89
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