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|
*> \brief \b CLAROR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE CLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO )
*
* .. Scalar Arguments ..
* CHARACTER INIT, SIDE
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER ISEED( 4 )
* COMPLEX A( LDA, * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CLAROR 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)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] INIT
*> \verbatim
*> 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 CDOTC):
*>
*> 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.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> Number of rows of A. Not modified.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> Number of columns of A. Not modified.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> Leading dimension of A. Must be at least MAX ( 1, M ).
*> Not modified.
*> \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 CLAROR to continue the same random number
*> sequence.
*> Modified.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is COMPLEX 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> An error flag. It is set to:
*> 0 if no error.
*> 1 if CLARND 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex_matgen
*
* =====================================================================
SUBROUTINE CLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER INIT, SIDE
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
COMPLEX A( LDA, * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TOOSML
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0,
$ TOOSML = 1.0E-20 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
INTEGER IROW, ITYPE, IXFRM, J, JCOL, KBEG, NXFRM
REAL FACTOR, XABS, XNORM
COMPLEX CSIGN, XNORMS
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SCNRM2
COMPLEX CLARND
EXTERNAL LSAME, SCNRM2, CLARND
* ..
* .. External Subroutines ..
EXTERNAL CGEMV, CGERC, CLACGV, CLASET, CSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CMPLX, CONJG
* ..
* .. 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' ) ) THEN
ITYPE = 3
ELSE IF( LSAME( SIDE, 'T' ) ) THEN
ITYPE = 4
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( 'CLAROR', -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 CLASET( 'Full', M, N, CZERO, CONE, A, LDA )
*
* If no rotation possible, still multiply by
* a random complex number from the circle |x| = 1
*
* 2) Compute Rotation by computing Householder
* Transformations H(2), H(3), ..., H(n). Note that the
* order in which they are computed is irrelevant.
*
DO 40 J = 1, NXFRM
X( J ) = CZERO
40 CONTINUE
*
DO 60 IXFRM = 2, NXFRM
KBEG = NXFRM - IXFRM + 1
*
* Generate independent normal( 0, 1 ) random numbers
*
DO 50 J = KBEG, NXFRM
X( J ) = CLARND( 3, ISEED )
50 CONTINUE
*
* Generate a Householder transformation from the random vector X
*
XNORM = SCNRM2( IXFRM, X( KBEG ), 1 )
XABS = ABS( X( KBEG ) )
IF( XABS.NE.CZERO ) THEN
CSIGN = X( KBEG ) / XABS
ELSE
CSIGN = CONE
END IF
XNORMS = CSIGN*XNORM
X( NXFRM+KBEG ) = -CSIGN
FACTOR = XNORM*( XNORM+XABS )
IF( ABS( FACTOR ).LT.TOOSML ) THEN
INFO = 1
CALL XERBLA( 'CLAROR', -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 .OR. ITYPE.EQ.4 ) THEN
*
* Apply H(k) on the left of A
*
CALL CGEMV( 'C', IXFRM, N, CONE, A( KBEG, 1 ), LDA,
$ X( KBEG ), 1, CZERO, X( 2*NXFRM+1 ), 1 )
CALL CGERC( IXFRM, N, -CMPLX( FACTOR ), X( KBEG ), 1,
$ X( 2*NXFRM+1 ), 1, A( KBEG, 1 ), LDA )
*
END IF
*
IF( ITYPE.GE.2 .AND. ITYPE.LE.4 ) THEN
*
* Apply H(k)* (or H(k)') on the right of A
*
IF( ITYPE.EQ.4 ) THEN
CALL CLACGV( IXFRM, X( KBEG ), 1 )
END IF
*
CALL CGEMV( 'N', M, IXFRM, CONE, A( 1, KBEG ), LDA,
$ X( KBEG ), 1, CZERO, X( 2*NXFRM+1 ), 1 )
CALL CGERC( M, IXFRM, -CMPLX( FACTOR ), X( 2*NXFRM+1 ), 1,
$ X( KBEG ), 1, A( 1, KBEG ), LDA )
*
END IF
60 CONTINUE
*
X( 1 ) = CLARND( 3, ISEED )
XABS = ABS( X( 1 ) )
IF( XABS.NE.ZERO ) THEN
CSIGN = X( 1 ) / XABS
ELSE
CSIGN = CONE
END IF
X( 2*NXFRM ) = CSIGN
*
* Scale the matrix A by D.
*
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.3 .OR. ITYPE.EQ.4 ) THEN
DO 70 IROW = 1, M
CALL CSCAL( N, CONJG( X( NXFRM+IROW ) ), A( IROW, 1 ), LDA )
70 CONTINUE
END IF
*
IF( ITYPE.EQ.2 .OR. ITYPE.EQ.3 ) THEN
DO 80 JCOL = 1, N
CALL CSCAL( M, X( NXFRM+JCOL ), A( 1, JCOL ), 1 )
80 CONTINUE
END IF
*
IF( ITYPE.EQ.4 ) THEN
DO 90 JCOL = 1, N
CALL CSCAL( M, CONJG( X( NXFRM+JCOL ) ), A( 1, JCOL ), 1 )
90 CONTINUE
END IF
RETURN
*
* End of CLAROR
*
END
|
