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|
*> \brief \b CLATME
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE CLATME( N, DIST, ISEED, D, MODE, COND, DMAX,
* RSIGN,
* UPPER, SIM, DS, MODES, CONDS, KL, KU, ANORM,
* A,
* LDA, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIST, RSIGN, SIM, UPPER
* INTEGER INFO, KL, KU, LDA, MODE, MODES, N
* REAL ANORM, COND, CONDS
* COMPLEX DMAX
* ..
* .. Array Arguments ..
* INTEGER ISEED( 4 )
* REAL DS( * )
* COMPLEX A( LDA, * ), D( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CLATME generates random non-symmetric square matrices with
*> specified eigenvalues for testing LAPACK programs.
*>
*> CLATME operates by applying the following sequence of
*> operations:
*>
*> 1. Set the diagonal to D, where D may be input or
*> computed according to MODE, COND, DMAX, and RSIGN
*> as described below.
*>
*> 2. If UPPER='T', the upper triangle of A is set to random values
*> out of distribution DIST.
*>
*> 3. If SIM='T', A is multiplied on the left by a random matrix
*> X, whose singular values are specified by DS, MODES, and
*> CONDS, and on the right by X inverse.
*>
*> 4. If KL < N-1, the lower bandwidth is reduced to KL using
*> Householder transformations. If KU < N-1, the upper
*> bandwidth is reduced to KU.
*>
*> 5. If ANORM is not negative, the matrix is scaled to have
*> maximum-element-norm ANORM.
*>
*> (Note: since the matrix cannot be reduced beyond Hessenberg form,
*> no packing options are available.)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns (or rows) of A. Not modified.
*> \endverbatim
*>
*> \param[in] DIST
*> \verbatim
*> DIST is CHARACTER*1
*> On entry, DIST specifies the type of distribution to be used
*> to generate the random eigen-/singular values, and on the
*> upper triangle (see UPPER).
*> 'U' => UNIFORM( 0, 1 ) ( 'U' for uniform )
*> 'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
*> 'N' => NORMAL( 0, 1 ) ( 'N' for normal )
*> 'D' => uniform on the complex disc |z| < 1.
*> 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. They should lie between 0 and 4095 inclusive,
*> and ISEED(4) should 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 CLATME
*> to continue the same random number sequence.
*> Changed on exit.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is COMPLEX array, dimension ( N )
*> This array is used to specify the eigenvalues of A. If
*> MODE=0, then D is assumed to contain the eigenvalues
*> otherwise they will be computed according to MODE, COND,
*> DMAX, and RSIGN and placed in D.
*> Modified if MODE is nonzero.
*> \endverbatim
*>
*> \param[in] MODE
*> \verbatim
*> MODE is INTEGER
*> On entry this describes how the eigenvalues are to
*> be specified:
*> MODE = 0 means use D as input
*> MODE = 1 sets D(1)=1 and D(2:N)=1.0/COND
*> MODE = 2 sets D(1:N-1)=1 and D(N)=1.0/COND
*> MODE = 3 sets D(I)=COND**(-(I-1)/(N-1))
*> MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
*> MODE = 5 sets D to random numbers in the range
*> ( 1/COND , 1 ) such that their logarithms
*> are uniformly distributed.
*> MODE = 6 set D to random numbers from same distribution
*> as the rest of the matrix.
*> MODE < 0 has the same meaning as ABS(MODE), except that
*> the order of the elements of D is reversed.
*> Thus if MODE is between 1 and 4, D has entries ranging
*> from 1 to 1/COND, if between -1 and -4, D has entries
*> ranging from 1/COND to 1,
*> Not modified.
*> \endverbatim
*>
*> \param[in] COND
*> \verbatim
*> COND is REAL
*> On entry, this is used as described under MODE above.
*> If used, it must be >= 1. Not modified.
*> \endverbatim
*>
*> \param[in] DMAX
*> \verbatim
*> DMAX is COMPLEX
*> If MODE is neither -6, 0 nor 6, the contents of D, as
*> computed according to MODE and COND, will be scaled by
*> DMAX / max(abs(D(i))). Note that DMAX need not be
*> positive or real: if DMAX is negative or complex (or zero),
*> D will be scaled by a negative or complex number (or zero).
*> If RSIGN='F' then the largest (absolute) eigenvalue will be
*> equal to DMAX.
*> Not modified.
*> \endverbatim
*>
*> \param[in] RSIGN
*> \verbatim
*> RSIGN is CHARACTER*1
*> If MODE is not 0, 6, or -6, and RSIGN='T', then the
*> elements of D, as computed according to MODE and COND, will
*> be multiplied by a random complex number from the unit
*> circle |z| = 1. If RSIGN='F', they will not be. RSIGN may
*> only have the values 'T' or 'F'.
*> Not modified.
*> \endverbatim
*>
*> \param[in] UPPER
*> \verbatim
*> UPPER is CHARACTER*1
*> If UPPER='T', then the elements of A above the diagonal
*> will be set to random numbers out of DIST. If UPPER='F',
*> they will not. UPPER may only have the values 'T' or 'F'.
*> Not modified.
*> \endverbatim
*>
*> \param[in] SIM
*> \verbatim
*> SIM is CHARACTER*1
*> If SIM='T', then A will be operated on by a "similarity
*> transform", i.e., multiplied on the left by a matrix X and
*> on the right by X inverse. X = U S V, where U and V are
*> random unitary matrices and S is a (diagonal) matrix of
*> singular values specified by DS, MODES, and CONDS. If
*> SIM='F', then A will not be transformed.
*> Not modified.
*> \endverbatim
*>
*> \param[in,out] DS
*> \verbatim
*> DS is REAL array, dimension ( N )
*> This array is used to specify the singular values of X,
*> in the same way that D specifies the eigenvalues of A.
*> If MODE=0, the DS contains the singular values, which
*> may not be zero.
*> Modified if MODE is nonzero.
*> \endverbatim
*>
*> \param[in] MODES
*> \verbatim
*> MODES is INTEGER
*> \endverbatim
*>
*> \param[in] CONDS
*> \verbatim
*> CONDS is REAL
*> Similar to MODE and COND, but for specifying the diagonal
*> of S. MODES=-6 and +6 are not allowed (since they would
*> result in randomly ill-conditioned eigenvalues.)
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> This specifies the lower bandwidth of the matrix. KL=1
*> specifies upper Hessenberg form. If KL is at least N-1,
*> then A will have full lower bandwidth.
*> Not modified.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> This specifies the upper bandwidth of the matrix. KU=1
*> specifies lower Hessenberg form. If KU is at least N-1,
*> then A will have full upper bandwidth; if KU and KL
*> are both at least N-1, then A will be dense. Only one of
*> KU and KL may be less than N-1.
*> Not modified.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is REAL
*> If ANORM is not negative, then A will be scaled by a non-
*> negative real number to make the maximum-element-norm of A
*> to be ANORM.
*> Not modified.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*> A is COMPLEX array, dimension ( LDA, N )
*> On exit A is the desired test matrix.
*> Modified.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> LDA specifies the first dimension of A as declared in the
*> calling program. LDA must be at least M.
*> Not modified.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension ( 3*N )
*> Workspace.
*> Modified.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> Error code. On exit, INFO will be set to one of the
*> following values:
*> 0 => normal return
*> -1 => N negative
*> -2 => DIST illegal string
*> -5 => MODE not in range -6 to 6
*> -6 => COND less than 1.0, and MODE neither -6, 0 nor 6
*> -9 => RSIGN is not 'T' or 'F'
*> -10 => UPPER is not 'T' or 'F'
*> -11 => SIM is not 'T' or 'F'
*> -12 => MODES=0 and DS has a zero singular value.
*> -13 => MODES is not in the range -5 to 5.
*> -14 => MODES is nonzero and CONDS is less than 1.
*> -15 => KL is less than 1.
*> -16 => KU is less than 1, or KL and KU are both less than
*> N-1.
*> -19 => LDA is less than M.
*> 1 => Error return from CLATM1 (computing D)
*> 2 => Cannot scale to DMAX (max. eigenvalue is 0)
*> 3 => Error return from SLATM1 (computing DS)
*> 4 => Error return from CLARGE
*> 5 => Zero singular value from SLATM1.
*> \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 CLATME( N, DIST, ISEED, D, MODE, COND, DMAX,
$ RSIGN,
$ UPPER, SIM, DS, MODES, CONDS, KL, KU, ANORM,
$ A,
$ LDA, WORK, INFO )
*
* -- LAPACK computational 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 DIST, RSIGN, SIM, UPPER
INTEGER INFO, KL, KU, LDA, MODE, MODES, N
REAL ANORM, COND, CONDS
COMPLEX DMAX
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
REAL DS( * )
COMPLEX A( LDA, * ), D( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
REAL ONE
PARAMETER ( ONE = 1.0E+0 )
COMPLEX CZERO
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ) )
COMPLEX CONE
PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL BADS
INTEGER I, IC, ICOLS, IDIST, IINFO, IR, IROWS, IRSIGN,
$ ISIM, IUPPER, J, JC, JCR
REAL RALPHA, TEMP
COMPLEX ALPHA, TAU, XNORMS
* ..
* .. Local Arrays ..
REAL TEMPA( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
REAL CLANGE
COMPLEX CLARND
EXTERNAL LSAME, CLANGE, CLARND
* ..
* .. External Subroutines ..
EXTERNAL CCOPY, CGEMV, CGERC, CLACGV, CLARFG, CLARGE,
$ CLARNV, CLATM1, CLASET, CSCAL, CSSCAL, SLATM1,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX, MOD
* ..
* .. Executable Statements ..
*
* 1) Decode and Test the input parameters.
* Initialize flags & seed.
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Decode DIST
*
IF( LSAME( DIST, 'U' ) ) THEN
IDIST = 1
ELSE IF( LSAME( DIST, 'S' ) ) THEN
IDIST = 2
ELSE IF( LSAME( DIST, 'N' ) ) THEN
IDIST = 3
ELSE IF( LSAME( DIST, 'D' ) ) THEN
IDIST = 4
ELSE
IDIST = -1
END IF
*
* Decode RSIGN
*
IF( LSAME( RSIGN, 'T' ) ) THEN
IRSIGN = 1
ELSE IF( LSAME( RSIGN, 'F' ) ) THEN
IRSIGN = 0
ELSE
IRSIGN = -1
END IF
*
* Decode UPPER
*
IF( LSAME( UPPER, 'T' ) ) THEN
IUPPER = 1
ELSE IF( LSAME( UPPER, 'F' ) ) THEN
IUPPER = 0
ELSE
IUPPER = -1
END IF
*
* Decode SIM
*
IF( LSAME( SIM, 'T' ) ) THEN
ISIM = 1
ELSE IF( LSAME( SIM, 'F' ) ) THEN
ISIM = 0
ELSE
ISIM = -1
END IF
*
* Check DS, if MODES=0 and ISIM=1
*
BADS = .FALSE.
IF( MODES.EQ.0 .AND. ISIM.EQ.1 ) THEN
DO 10 J = 1, N
IF( DS( J ).EQ.ZERO )
$ BADS = .TRUE.
10 CONTINUE
END IF
*
* Set INFO if an error
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( IDIST.EQ.-1 ) THEN
INFO = -2
ELSE IF( ABS( MODE ).GT.6 ) THEN
INFO = -5
ELSE IF( ( MODE.NE.0 .AND. ABS( MODE ).NE.6 ) .AND. COND.LT.ONE )
$ THEN
INFO = -6
ELSE IF( IRSIGN.EQ.-1 ) THEN
INFO = -9
ELSE IF( IUPPER.EQ.-1 ) THEN
INFO = -10
ELSE IF( ISIM.EQ.-1 ) THEN
INFO = -11
ELSE IF( BADS ) THEN
INFO = -12
ELSE IF( ISIM.EQ.1 .AND. ABS( MODES ).GT.5 ) THEN
INFO = -13
ELSE IF( ISIM.EQ.1 .AND. MODES.NE.0 .AND. CONDS.LT.ONE ) THEN
INFO = -14
ELSE IF( KL.LT.1 ) THEN
INFO = -15
ELSE IF( KU.LT.1 .OR. ( KU.LT.N-1 .AND. KL.LT.N-1 ) ) THEN
INFO = -16
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -19
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CLATME', -INFO )
RETURN
END IF
*
* Initialize random number generator
*
DO 20 I = 1, 4
ISEED( I ) = MOD( ABS( ISEED( I ) ), 4096 )
20 CONTINUE
*
IF( MOD( ISEED( 4 ), 2 ).NE.1 )
$ ISEED( 4 ) = ISEED( 4 ) + 1
*
* 2) Set up diagonal of A
*
* Compute D according to COND and MODE
*
CALL CLATM1( MODE, COND, IRSIGN, IDIST, ISEED, D, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 1
RETURN
END IF
IF( MODE.NE.0 .AND. ABS( MODE ).NE.6 ) THEN
*
* Scale by DMAX
*
TEMP = ABS( D( 1 ) )
DO 30 I = 2, N
TEMP = MAX( TEMP, ABS( D( I ) ) )
30 CONTINUE
*
IF( TEMP.GT.ZERO ) THEN
ALPHA = DMAX / TEMP
ELSE
INFO = 2
RETURN
END IF
*
CALL CSCAL( N, ALPHA, D, 1 )
*
END IF
*
CALL CLASET( 'Full', N, N, CZERO, CZERO, A, LDA )
CALL CCOPY( N, D, 1, A, LDA+1 )
*
* 3) If UPPER='T', set upper triangle of A to random numbers.
*
IF( IUPPER.NE.0 ) THEN
DO 40 JC = 2, N
CALL CLARNV( IDIST, ISEED, JC-1, A( 1, JC ) )
40 CONTINUE
END IF
*
* 4) If SIM='T', apply similarity transformation.
*
* -1
* Transform is X A X , where X = U S V, thus
*
* it is U S V A V' (1/S) U'
*
IF( ISIM.NE.0 ) THEN
*
* Compute S (singular values of the eigenvector matrix)
* according to CONDS and MODES
*
CALL SLATM1( MODES, CONDS, 0, 0, ISEED, DS, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 3
RETURN
END IF
*
* Multiply by V and V'
*
CALL CLARGE( N, A, LDA, ISEED, WORK, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 4
RETURN
END IF
*
* Multiply by S and (1/S)
*
DO 50 J = 1, N
CALL CSSCAL( N, DS( J ), A( J, 1 ), LDA )
IF( DS( J ).NE.ZERO ) THEN
CALL CSSCAL( N, ONE / DS( J ), A( 1, J ), 1 )
ELSE
INFO = 5
RETURN
END IF
50 CONTINUE
*
* Multiply by U and U'
*
CALL CLARGE( N, A, LDA, ISEED, WORK, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 4
RETURN
END IF
END IF
*
* 5) Reduce the bandwidth.
*
IF( KL.LT.N-1 ) THEN
*
* Reduce bandwidth -- kill column
*
DO 60 JCR = KL + 1, N - 1
IC = JCR - KL
IROWS = N + 1 - JCR
ICOLS = N + KL - JCR
*
CALL CCOPY( IROWS, A( JCR, IC ), 1, WORK, 1 )
XNORMS = WORK( 1 )
CALL CLARFG( IROWS, XNORMS, WORK( 2 ), 1, TAU )
TAU = CONJG( TAU )
WORK( 1 ) = CONE
ALPHA = CLARND( 5, ISEED )
*
CALL CGEMV( 'C', IROWS, ICOLS, CONE, A( JCR, IC+1 ), LDA,
$ WORK, 1, CZERO, WORK( IROWS+1 ), 1 )
CALL CGERC( IROWS, ICOLS, -TAU, WORK, 1, WORK( IROWS+1 ), 1,
$ A( JCR, IC+1 ), LDA )
*
CALL CGEMV( 'N', N, IROWS, CONE, A( 1, JCR ), LDA, WORK, 1,
$ CZERO, WORK( IROWS+1 ), 1 )
CALL CGERC( N, IROWS, -CONJG( TAU ), WORK( IROWS+1 ), 1,
$ WORK, 1, A( 1, JCR ), LDA )
*
A( JCR, IC ) = XNORMS
CALL CLASET( 'Full', IROWS-1, 1, CZERO, CZERO,
$ A( JCR+1, IC ), LDA )
*
CALL CSCAL( ICOLS+1, ALPHA, A( JCR, IC ), LDA )
CALL CSCAL( N, CONJG( ALPHA ), A( 1, JCR ), 1 )
60 CONTINUE
ELSE IF( KU.LT.N-1 ) THEN
*
* Reduce upper bandwidth -- kill a row at a time.
*
DO 70 JCR = KU + 1, N - 1
IR = JCR - KU
IROWS = N + KU - JCR
ICOLS = N + 1 - JCR
*
CALL CCOPY( ICOLS, A( IR, JCR ), LDA, WORK, 1 )
XNORMS = WORK( 1 )
CALL CLARFG( ICOLS, XNORMS, WORK( 2 ), 1, TAU )
TAU = CONJG( TAU )
WORK( 1 ) = CONE
CALL CLACGV( ICOLS-1, WORK( 2 ), 1 )
ALPHA = CLARND( 5, ISEED )
*
CALL CGEMV( 'N', IROWS, ICOLS, CONE, A( IR+1, JCR ), LDA,
$ WORK, 1, CZERO, WORK( ICOLS+1 ), 1 )
CALL CGERC( IROWS, ICOLS, -TAU, WORK( ICOLS+1 ), 1, WORK, 1,
$ A( IR+1, JCR ), LDA )
*
CALL CGEMV( 'C', ICOLS, N, CONE, A( JCR, 1 ), LDA, WORK, 1,
$ CZERO, WORK( ICOLS+1 ), 1 )
CALL CGERC( ICOLS, N, -CONJG( TAU ), WORK, 1,
$ WORK( ICOLS+1 ), 1, A( JCR, 1 ), LDA )
*
A( IR, JCR ) = XNORMS
CALL CLASET( 'Full', 1, ICOLS-1, CZERO, CZERO,
$ A( IR, JCR+1 ), LDA )
*
CALL CSCAL( IROWS+1, ALPHA, A( IR, JCR ), 1 )
CALL CSCAL( N, CONJG( ALPHA ), A( JCR, 1 ), LDA )
70 CONTINUE
END IF
*
* Scale the matrix to have norm ANORM
*
IF( ANORM.GE.ZERO ) THEN
TEMP = CLANGE( 'M', N, N, A, LDA, TEMPA )
IF( TEMP.GT.ZERO ) THEN
RALPHA = ANORM / TEMP
DO 80 J = 1, N
CALL CSSCAL( N, RALPHA, A( 1, J ), 1 )
80 CONTINUE
END IF
END IF
*
RETURN
*
* End of CLATME
*
END
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