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346 lines
10 KiB
346 lines
10 KiB
*> \brief \b ZLAROR
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE ZLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER INIT, SIDE
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* INTEGER INFO, LDA, M, N
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* ..
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* .. Array Arguments ..
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* INTEGER ISEED( 4 )
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* COMPLEX*16 A( LDA, * ), X( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> ZLAROR pre- or post-multiplies an M by N matrix A by a random
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*> unitary matrix U, overwriting A. A may optionally be
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*> initialized to the identity matrix before multiplying by U.
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*> U is generated using the method of G.W. Stewart
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*> ( SIAM J. Numer. Anal. 17, 1980, pp. 403-409 ).
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*> (BLAS-2 version)
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] SIDE
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*> \verbatim
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*> SIDE is CHARACTER*1
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*> SIDE specifies whether A is multiplied on the left or right
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*> by U.
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*> SIDE = 'L' Multiply A on the left (premultiply) by U
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*> 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'
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*> Not modified.
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*> \endverbatim
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*>
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*> \param[in] INIT
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*> \verbatim
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*> INIT is CHARACTER*1
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*> INIT specifies whether or not A should be initialized to
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*> the identity matrix.
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*> INIT = 'I' Initialize A to (a section of) the
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*> identity matrix before applying U.
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*> INIT = 'N' No initialization. Apply U to the
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*> input matrix A.
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*>
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*> INIT = 'I' may be used to generate square (i.e., unitary)
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*> or rectangular orthogonal matrices (orthogonality being
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*> in the sense of ZDOTC):
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*>
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*> For square matrices, M=N, and SIDE many be either 'L' or
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*> 'R'; the rows will be orthogonal to each other, as will the
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*> columns.
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*> For rectangular matrices where M < N, SIDE = 'R' will
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*> produce a dense matrix whose rows will be orthogonal and
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*> whose columns will not, while SIDE = 'L' will produce a
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*> matrix whose rows will be orthogonal, and whose first M
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*> columns will be orthogonal, the remaining columns being
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*> zero.
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*> For matrices where M > N, just use the previous
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*> explanation, interchanging 'L' and 'R' and "rows" and
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*> "columns".
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*>
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*> Not modified.
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*> \endverbatim
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*>
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*> \param[in] M
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*> \verbatim
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*> M is INTEGER
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*> Number of rows of A. Not modified.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> Number of columns of A. Not modified.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is COMPLEX*16 array, dimension ( LDA, N )
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*> Input and output array. Overwritten by U A ( if SIDE = 'L' )
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*> or by A U ( if SIDE = 'R' )
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*> or by U A U* ( if SIDE = 'C')
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*> or by U A U' ( if SIDE = 'T') on exit.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> Leading dimension of A. Must be at least MAX ( 1, M ).
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*> Not modified.
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*> \endverbatim
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*>
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*> \param[in,out] ISEED
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*> \verbatim
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*> ISEED is INTEGER array, dimension ( 4 )
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*> On entry ISEED specifies the seed of the random number
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*> generator. The array elements should be between 0 and 4095;
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*> if not they will be reduced mod 4096. Also, ISEED(4) must
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*> be odd. The random number generator uses a linear
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*> congruential sequence limited to small integers, and so
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*> should produce machine independent random numbers. The
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*> values of ISEED are changed on exit, and can be used in the
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*> next call to ZLAROR to continue the same random number
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*> sequence.
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*> Modified.
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*> \endverbatim
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*>
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*> \param[out] X
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*> \verbatim
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*> X is COMPLEX*16 array, dimension ( 3*MAX( M, N ) )
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*> Workspace. Of length:
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*> 2*M + N if SIDE = 'L',
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*> 2*N + M if SIDE = 'R',
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*> 3*N if SIDE = 'C' or 'T'.
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*> Modified.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> An error flag. It is set to:
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*> 0 if no error.
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*> 1 if ZLARND returned a bad random number (installation
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*> problem)
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*> -1 if SIDE is not L, R, C, or T.
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*> -3 if M is negative.
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*> -4 if N is negative or if SIDE is C or T and N is not equal
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*> to M.
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*> -6 if LDA is less than M.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup complex16_matgen
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*
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* =====================================================================
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SUBROUTINE ZLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO )
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*
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* -- LAPACK auxiliary routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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CHARACTER INIT, SIDE
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INTEGER INFO, LDA, M, N
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* ..
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* .. Array Arguments ..
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INTEGER ISEED( 4 )
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COMPLEX*16 A( LDA, * ), X( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE, TOOSML
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0,
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$ TOOSML = 1.0D-20 )
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COMPLEX*16 CZERO, CONE
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PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
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$ CONE = ( 1.0D+0, 0.0D+0 ) )
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* ..
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* .. Local Scalars ..
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INTEGER IROW, ITYPE, IXFRM, J, JCOL, KBEG, NXFRM
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DOUBLE PRECISION FACTOR, XABS, XNORM
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COMPLEX*16 CSIGN, XNORMS
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DZNRM2
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COMPLEX*16 ZLARND
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EXTERNAL LSAME, DZNRM2, ZLARND
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* ..
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* .. External Subroutines ..
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EXTERNAL XERBLA, ZGEMV, ZGERC, ZLACGV, ZLASET, ZSCAL
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DCMPLX, DCONJG
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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IF( N.EQ.0 .OR. M.EQ.0 )
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$ RETURN
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*
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ITYPE = 0
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IF( LSAME( SIDE, 'L' ) ) THEN
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ITYPE = 1
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ELSE IF( LSAME( SIDE, 'R' ) ) THEN
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ITYPE = 2
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ELSE IF( LSAME( SIDE, 'C' ) ) THEN
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ITYPE = 3
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ELSE IF( LSAME( SIDE, 'T' ) ) THEN
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ITYPE = 4
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END IF
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*
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* Check for argument errors.
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*
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IF( ITYPE.EQ.0 ) THEN
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INFO = -1
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ELSE IF( M.LT.0 ) THEN
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INFO = -3
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ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.3 .AND. N.NE.M ) ) THEN
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INFO = -4
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ELSE IF( LDA.LT.M ) THEN
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INFO = -6
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZLAROR', -INFO )
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RETURN
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END IF
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*
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IF( ITYPE.EQ.1 ) THEN
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NXFRM = M
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ELSE
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NXFRM = N
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END IF
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*
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* Initialize A to the identity matrix if desired
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*
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IF( LSAME( INIT, 'I' ) )
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$ CALL ZLASET( 'Full', M, N, CZERO, CONE, A, LDA )
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*
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* If no rotation possible, still multiply by
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* a random complex number from the circle |x| = 1
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*
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* 2) Compute Rotation by computing Householder
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* Transformations H(2), H(3), ..., H(n). Note that the
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* order in which they are computed is irrelevant.
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*
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DO 10 J = 1, NXFRM
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X( J ) = CZERO
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10 CONTINUE
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*
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DO 30 IXFRM = 2, NXFRM
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KBEG = NXFRM - IXFRM + 1
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*
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* Generate independent normal( 0, 1 ) random numbers
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*
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DO 20 J = KBEG, NXFRM
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X( J ) = ZLARND( 3, ISEED )
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20 CONTINUE
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*
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* Generate a Householder transformation from the random vector X
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*
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XNORM = DZNRM2( IXFRM, X( KBEG ), 1 )
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XABS = ABS( X( KBEG ) )
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IF( XABS.NE.CZERO ) THEN
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CSIGN = X( KBEG ) / XABS
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ELSE
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CSIGN = CONE
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END IF
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XNORMS = CSIGN*XNORM
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X( NXFRM+KBEG ) = -CSIGN
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FACTOR = XNORM*( XNORM+XABS )
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IF( ABS( FACTOR ).LT.TOOSML ) THEN
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INFO = 1
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CALL XERBLA( 'ZLAROR', -INFO )
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RETURN
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ELSE
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FACTOR = ONE / FACTOR
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END IF
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X( KBEG ) = X( KBEG ) + XNORMS
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*
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* Apply Householder transformation to A
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*
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IF( ITYPE.EQ.1 .OR. ITYPE.EQ.3 .OR. ITYPE.EQ.4 ) THEN
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*
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* Apply H(k) on the left of A
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*
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CALL ZGEMV( 'C', IXFRM, N, CONE, A( KBEG, 1 ), LDA,
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$ X( KBEG ), 1, CZERO, X( 2*NXFRM+1 ), 1 )
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CALL ZGERC( IXFRM, N, -DCMPLX( FACTOR ), X( KBEG ), 1,
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$ X( 2*NXFRM+1 ), 1, A( KBEG, 1 ), LDA )
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*
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END IF
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*
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IF( ITYPE.GE.2 .AND. ITYPE.LE.4 ) THEN
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*
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* Apply H(k)* (or H(k)') on the right of A
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*
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IF( ITYPE.EQ.4 ) THEN
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CALL ZLACGV( IXFRM, X( KBEG ), 1 )
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END IF
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*
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CALL ZGEMV( 'N', M, IXFRM, CONE, A( 1, KBEG ), LDA,
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$ X( KBEG ), 1, CZERO, X( 2*NXFRM+1 ), 1 )
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CALL ZGERC( M, IXFRM, -DCMPLX( FACTOR ), X( 2*NXFRM+1 ), 1,
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$ X( KBEG ), 1, A( 1, KBEG ), LDA )
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*
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END IF
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30 CONTINUE
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*
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X( 1 ) = ZLARND( 3, ISEED )
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XABS = ABS( X( 1 ) )
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IF( XABS.NE.ZERO ) THEN
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CSIGN = X( 1 ) / XABS
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ELSE
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CSIGN = CONE
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END IF
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X( 2*NXFRM ) = CSIGN
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*
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* Scale the matrix A by D.
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*
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IF( ITYPE.EQ.1 .OR. ITYPE.EQ.3 .OR. ITYPE.EQ.4 ) THEN
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DO 40 IROW = 1, M
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CALL ZSCAL( N, DCONJG( X( NXFRM+IROW ) ), A( IROW, 1 ),
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$ LDA )
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40 CONTINUE
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END IF
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*
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IF( ITYPE.EQ.2 .OR. ITYPE.EQ.3 ) THEN
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DO 50 JCOL = 1, N
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CALL ZSCAL( M, X( NXFRM+JCOL ), A( 1, JCOL ), 1 )
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50 CONTINUE
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END IF
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*
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IF( ITYPE.EQ.4 ) THEN
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DO 60 JCOL = 1, N
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CALL ZSCAL( M, DCONJG( X( NXFRM+JCOL ) ), A( 1, JCOL ), 1 )
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60 CONTINUE
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END IF
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RETURN
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*
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* End of ZLAROR
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*
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END
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