*> \brief \b ZLATM6 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE ZLATM6( TYPE, N, A, LDA, B, X, LDX, Y, LDY, ALPHA, * BETA, WX, WY, S, DIF ) * * .. Scalar Arguments .. * INTEGER LDA, LDX, LDY, N, TYPE * COMPLEX*16 ALPHA, BETA, WX, WY * .. * .. Array Arguments .. * DOUBLE PRECISION DIF( * ), S( * ) * COMPLEX*16 A( LDA, * ), B( LDA, * ), X( LDX, * ), * $ Y( LDY, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLATM6 generates test matrices for the generalized eigenvalue *> problem, their corresponding right and left eigenvector matrices, *> and also reciprocal condition numbers for all eigenvalues and *> the reciprocal condition numbers of eigenvectors corresponding to *> the 1th and 5th eigenvalues. *> *> Test Matrices *> ============= *> *> Two kinds of test matrix pairs *> (A, B) = inverse(YH) * (Da, Db) * inverse(X) *> are used in the tests: *> *> Type 1: *> Da = 1+a 0 0 0 0 Db = 1 0 0 0 0 *> 0 2+a 0 0 0 0 1 0 0 0 *> 0 0 3+a 0 0 0 0 1 0 0 *> 0 0 0 4+a 0 0 0 0 1 0 *> 0 0 0 0 5+a , 0 0 0 0 1 *> and Type 2: *> Da = 1+i 0 0 0 0 Db = 1 0 0 0 0 *> 0 1-i 0 0 0 0 1 0 0 0 *> 0 0 1 0 0 0 0 1 0 0 *> 0 0 0 (1+a)+(1+b)i 0 0 0 0 1 0 *> 0 0 0 0 (1+a)-(1+b)i, 0 0 0 0 1 . *> *> In both cases the same inverse(YH) and inverse(X) are used to compute *> (A, B), giving the exact eigenvectors to (A,B) as (YH, X): *> *> YH: = 1 0 -y y -y X = 1 0 -x -x x *> 0 1 -y y -y 0 1 x -x -x *> 0 0 1 0 0 0 0 1 0 0 *> 0 0 0 1 0 0 0 0 1 0 *> 0 0 0 0 1, 0 0 0 0 1 , where *> *> a, b, x and y will have all values independently of each other. *> \endverbatim * * Arguments: * ========== * *> \param[in] TYPE *> \verbatim *> TYPE is INTEGER *> Specifies the problem type (see futher details). *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> Size of the matrices A and B. *> \endverbatim *> *> \param[out] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA, N). *> On exit A N-by-N is initialized according to TYPE. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of A and of B. *> \endverbatim *> *> \param[out] B *> \verbatim *> B is COMPLEX*16 array, dimension (LDA, N). *> On exit B N-by-N is initialized according to TYPE. *> \endverbatim *> *> \param[out] X *> \verbatim *> X is COMPLEX*16 array, dimension (LDX, N). *> On exit X is the N-by-N matrix of right eigenvectors. *> \endverbatim *> *> \param[in] LDX *> \verbatim *> LDX is INTEGER *> The leading dimension of X. *> \endverbatim *> *> \param[out] Y *> \verbatim *> Y is COMPLEX*16 array, dimension (LDY, N). *> On exit Y is the N-by-N matrix of left eigenvectors. *> \endverbatim *> *> \param[in] LDY *> \verbatim *> LDY is INTEGER *> The leading dimension of Y. *> \endverbatim *> *> \param[in] ALPHA *> \verbatim *> ALPHA is COMPLEX*16 *> \endverbatim *> *> \param[in] BETA *> \verbatim *> BETA is COMPLEX*16 *> \verbatim *> Weighting constants for matrix A. *> \endverbatim *> *> \param[in] WX *> \verbatim *> WX is COMPLEX*16 *> Constant for right eigenvector matrix. *> \endverbatim *> *> \param[in] WY *> \verbatim *> WY is COMPLEX*16 *> Constant for left eigenvector matrix. *> \endverbatim *> *> \param[out] S *> \verbatim *> S is DOUBLE PRECISION array, dimension (N) *> S(i) is the reciprocal condition number for eigenvalue i. *> \endverbatim *> *> \param[out] DIF *> \verbatim *> DIF is DOUBLE PRECISION array, dimension (N) *> DIF(i) is the reciprocal condition number for eigenvector i. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2011 * *> \ingroup complex16_matgen * * ===================================================================== SUBROUTINE ZLATM6( TYPE, N, A, LDA, B, X, LDX, Y, LDY, ALPHA, $ BETA, WX, WY, S, DIF ) * * -- 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 .. INTEGER LDA, LDX, LDY, N, TYPE COMPLEX*16 ALPHA, BETA, WX, WY * .. * .. Array Arguments .. DOUBLE PRECISION DIF( * ), S( * ) COMPLEX*16 A( LDA, * ), B( LDA, * ), X( LDX, * ), $ Y( LDY, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION RONE, TWO, THREE PARAMETER ( RONE = 1.0D+0, TWO = 2.0D+0, THREE = 3.0D+0 ) COMPLEX*16 ZERO, ONE PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), $ ONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. INTEGER I, INFO, J * .. * .. Local Arrays .. DOUBLE PRECISION RWORK( 50 ) COMPLEX*16 WORK( 26 ), Z( 8, 8 ) * .. * .. Intrinsic Functions .. INTRINSIC CDABS, DBLE, DCMPLX, DCONJG, SQRT * .. * .. External Subroutines .. EXTERNAL ZGESVD, ZLACPY, ZLAKF2 * .. * .. Executable Statements .. * * Generate test problem ... * (Da, Db) ... * DO 20 I = 1, N DO 10 J = 1, N * IF( I.EQ.J ) THEN A( I, I ) = DCMPLX( I ) + ALPHA B( I, I ) = ONE ELSE A( I, J ) = ZERO B( I, J ) = ZERO END IF * 10 CONTINUE 20 CONTINUE IF( TYPE.EQ.2 ) THEN A( 1, 1 ) = DCMPLX( RONE, RONE ) A( 2, 2 ) = DCONJG( A( 1, 1 ) ) A( 3, 3 ) = ONE A( 4, 4 ) = DCMPLX( DBLE( ONE+ALPHA ), DBLE( ONE+BETA ) ) A( 5, 5 ) = DCONJG( A( 4, 4 ) ) END IF * * Form X and Y * CALL ZLACPY( 'F', N, N, B, LDA, Y, LDY ) Y( 3, 1 ) = -DCONJG( WY ) Y( 4, 1 ) = DCONJG( WY ) Y( 5, 1 ) = -DCONJG( WY ) Y( 3, 2 ) = -DCONJG( WY ) Y( 4, 2 ) = DCONJG( WY ) Y( 5, 2 ) = -DCONJG( WY ) * CALL ZLACPY( 'F', N, N, B, LDA, X, LDX ) X( 1, 3 ) = -WX X( 1, 4 ) = -WX X( 1, 5 ) = WX X( 2, 3 ) = WX X( 2, 4 ) = -WX X( 2, 5 ) = -WX * * Form (A, B) * B( 1, 3 ) = WX + WY B( 2, 3 ) = -WX + WY B( 1, 4 ) = WX - WY B( 2, 4 ) = WX - WY B( 1, 5 ) = -WX + WY B( 2, 5 ) = WX + WY A( 1, 3 ) = WX*A( 1, 1 ) + WY*A( 3, 3 ) A( 2, 3 ) = -WX*A( 2, 2 ) + WY*A( 3, 3 ) A( 1, 4 ) = WX*A( 1, 1 ) - WY*A( 4, 4 ) A( 2, 4 ) = WX*A( 2, 2 ) - WY*A( 4, 4 ) A( 1, 5 ) = -WX*A( 1, 1 ) + WY*A( 5, 5 ) A( 2, 5 ) = WX*A( 2, 2 ) + WY*A( 5, 5 ) * * Compute condition numbers * S( 1 ) = RONE / SQRT( ( RONE+THREE*CDABS( WY )*CDABS( WY ) ) / $ ( RONE+CDABS( A( 1, 1 ) )*CDABS( A( 1, 1 ) ) ) ) S( 2 ) = RONE / SQRT( ( RONE+THREE*CDABS( WY )*CDABS( WY ) ) / $ ( RONE+CDABS( A( 2, 2 ) )*CDABS( A( 2, 2 ) ) ) ) S( 3 ) = RONE / SQRT( ( RONE+TWO*CDABS( WX )*CDABS( WX ) ) / $ ( RONE+CDABS( A( 3, 3 ) )*CDABS( A( 3, 3 ) ) ) ) S( 4 ) = RONE / SQRT( ( RONE+TWO*CDABS( WX )*CDABS( WX ) ) / $ ( RONE+CDABS( A( 4, 4 ) )*CDABS( A( 4, 4 ) ) ) ) S( 5 ) = RONE / SQRT( ( RONE+TWO*CDABS( WX )*CDABS( WX ) ) / $ ( RONE+CDABS( A( 5, 5 ) )*CDABS( A( 5, 5 ) ) ) ) * CALL ZLAKF2( 1, 4, A, LDA, A( 2, 2 ), B, B( 2, 2 ), Z, 8 ) CALL ZGESVD( 'N', 'N', 8, 8, Z, 8, RWORK, WORK, 1, WORK( 2 ), 1, $ WORK( 3 ), 24, RWORK( 9 ), INFO ) DIF( 1 ) = RWORK( 8 ) * CALL ZLAKF2( 4, 1, A, LDA, A( 5, 5 ), B, B( 5, 5 ), Z, 8 ) CALL ZGESVD( 'N', 'N', 8, 8, Z, 8, RWORK, WORK, 1, WORK( 2 ), 1, $ WORK( 3 ), 24, RWORK( 9 ), INFO ) DIF( 5 ) = RWORK( 8 ) * RETURN * * End of ZLATM6 * END