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|
*> \brief \b DLAROR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO )
*
* .. Scalar Arguments ..
* CHARACTER INIT, SIDE
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER ISEED( 4 )
* DOUBLE PRECISION A( LDA, * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> 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.)
*> \endverbatim
*>
*> \param[in] INIT
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of A.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> 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').
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] ISEED
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> 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'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup double_matgen
*
* =====================================================================
SUBROUTINE DLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER INIT, SIDE
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
DOUBLE PRECISION A( LDA, * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TOOSML
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0,
$ TOOSML = 1.0D-20 )
* ..
* .. Local Scalars ..
INTEGER IROW, ITYPE, IXFRM, J, JCOL, KBEG, NXFRM
DOUBLE PRECISION FACTOR, XNORM, XNORMS
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLARND, DNRM2
EXTERNAL LSAME, DLARND, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER, DLASET, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.EQ.0 .OR. M.EQ.0 )
$ RETURN
*
ITYPE = 0
IF( LSAME( SIDE, 'L' ) ) THEN
ITYPE = 1
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
ITYPE = 2
ELSE IF( LSAME( SIDE, 'C' ) .OR. LSAME( SIDE, 'T' ) ) THEN
ITYPE = 3
END IF
*
* Check for argument errors.
*
IF( ITYPE.EQ.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.3 .AND. N.NE.M ) ) THEN
INFO = -4
ELSE IF( LDA.LT.M ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAROR', -INFO )
RETURN
END IF
*
IF( ITYPE.EQ.1 ) THEN
NXFRM = M
ELSE
NXFRM = N
END IF
*
* Initialize A to the identity matrix if desired
*
IF( LSAME( INIT, 'I' ) )
$ CALL DLASET( 'Full', M, N, ZERO, ONE, A, LDA )
*
* If no rotation possible, multiply by random +/-1
*
* Compute rotation by computing Householder transformations
* H(2), H(3), ..., H(nhouse)
*
DO 10 J = 1, NXFRM
X( J ) = ZERO
10 CONTINUE
*
DO 30 IXFRM = 2, NXFRM
KBEG = NXFRM - IXFRM + 1
*
* Generate independent normal( 0, 1 ) random numbers
*
DO 20 J = KBEG, NXFRM
X( J ) = DLARND( 3, ISEED )
20 CONTINUE
*
* Generate a Householder transformation from the random vector X
*
XNORM = DNRM2( IXFRM, X( KBEG ), 1 )
XNORMS = SIGN( XNORM, X( KBEG ) )
X( KBEG+NXFRM ) = SIGN( ONE, -X( KBEG ) )
FACTOR = XNORMS*( XNORMS+X( KBEG ) )
IF( ABS( FACTOR ).LT.TOOSML ) THEN
INFO = 1
CALL XERBLA( 'DLAROR', INFO )
RETURN
ELSE
FACTOR = ONE / FACTOR
END IF
X( KBEG ) = X( KBEG ) + XNORMS
*
* Apply Householder transformation to A
*
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.3 ) THEN
*
* Apply H(k) from the left.
*
CALL DGEMV( 'T', IXFRM, N, ONE, A( KBEG, 1 ), LDA,
$ X( KBEG ), 1, ZERO, X( 2*NXFRM+1 ), 1 )
CALL DGER( IXFRM, N, -FACTOR, X( KBEG ), 1, X( 2*NXFRM+1 ),
$ 1, A( KBEG, 1 ), LDA )
*
END IF
*
IF( ITYPE.EQ.2 .OR. ITYPE.EQ.3 ) THEN
*
* Apply H(k) from the right.
*
CALL DGEMV( 'N', M, IXFRM, ONE, A( 1, KBEG ), LDA,
$ X( KBEG ), 1, ZERO, X( 2*NXFRM+1 ), 1 )
CALL DGER( M, IXFRM, -FACTOR, X( 2*NXFRM+1 ), 1, X( KBEG ),
$ 1, A( 1, KBEG ), LDA )
*
END IF
30 CONTINUE
*
X( 2*NXFRM ) = SIGN( ONE, DLARND( 3, ISEED ) )
*
* Scale the matrix A by D.
*
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.3 ) THEN
DO 40 IROW = 1, M
CALL DSCAL( N, X( NXFRM+IROW ), A( IROW, 1 ), LDA )
40 CONTINUE
END IF
*
IF( ITYPE.EQ.2 .OR. ITYPE.EQ.3 ) THEN
DO 50 JCOL = 1, N
CALL DSCAL( M, X( NXFRM+JCOL ), A( 1, JCOL ), 1 )
50 CONTINUE
END IF
RETURN
*
* End of DLAROR
*
END
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