*> \brief \b SLAROR * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE SLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO ) * * .. Scalar Arguments .. * CHARACTER INIT, SIDE * INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. * INTEGER ISEED( 4 ) * REAL A( LDA, * ), X( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAROR 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 REAL 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 SLAROR to continue the same random number *> sequence. *> \endverbatim *> *> \param[out] X *> \verbatim *> X is REAL 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 SLARND are bad. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup real_matgen * * ===================================================================== SUBROUTINE SLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO ) * * -- LAPACK auxiliary routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER INIT, SIDE INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. INTEGER ISEED( 4 ) REAL A( LDA, * ), X( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TOOSML PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, $ TOOSML = 1.0E-20 ) * .. * .. Local Scalars .. INTEGER IROW, ITYPE, IXFRM, J, JCOL, KBEG, NXFRM REAL FACTOR, XNORM, XNORMS * .. * .. External Functions .. LOGICAL LSAME REAL SLARND, SNRM2 EXTERNAL LSAME, SLARND, SNRM2 * .. * .. External Subroutines .. EXTERNAL SGEMV, SGER, SLASET, SSCAL, 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( 'SLAROR', -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 SLASET( '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 ) = SLARND( 3, ISEED ) 20 CONTINUE * * Generate a Householder transformation from the random vector X * XNORM = SNRM2( 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( 'SLAROR', 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 SGEMV( 'T', IXFRM, N, ONE, A( KBEG, 1 ), LDA, $ X( KBEG ), 1, ZERO, X( 2*NXFRM+1 ), 1 ) CALL SGER( 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 SGEMV( 'N', M, IXFRM, ONE, A( 1, KBEG ), LDA, $ X( KBEG ), 1, ZERO, X( 2*NXFRM+1 ), 1 ) CALL SGER( M, IXFRM, -FACTOR, X( 2*NXFRM+1 ), 1, X( KBEG ), $ 1, A( 1, KBEG ), LDA ) * END IF 30 CONTINUE * X( 2*NXFRM ) = SIGN( ONE, SLARND( 3, ISEED ) ) * * Scale the matrix A by D. * IF( ITYPE.EQ.1 .OR. ITYPE.EQ.3 ) THEN DO 40 IROW = 1, M CALL SSCAL( 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 SSCAL( M, X( NXFRM+JCOL ), A( 1, JCOL ), 1 ) 50 CONTINUE END IF RETURN * * End of SLAROR * END