*> \brief \b SLATM6 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE SLATM6( TYPE, N, A, LDA, B, X, LDX, Y, LDY, ALPHA, * BETA, WX, WY, S, DIF ) * * .. Scalar Arguments .. * INTEGER LDA, LDX, LDY, N, TYPE * REAL ALPHA, BETA, WX, WY * .. * .. Array Arguments .. * REAL A( LDA, * ), B( LDA, * ), DIF( * ), S( * ), * $ X( LDX, * ), Y( LDY, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLATM6 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 -1 0 0 0 Db = 1 0 0 0 0 *> 1 1 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 0 0 0 1 0 *> 0 0 0 -1-b 1+a , 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 REAL 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 REAL array, dimension (LDA, N). *> On exit B N-by-N is initialized according to TYPE. *> \endverbatim *> *> \param[out] X *> \verbatim *> X is REAL 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 REAL 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 REAL *> \endverbatim *> *> \param[in] BETA *> \verbatim *> BETA is REAL *> *> Weighting constants for matrix A. *> \endverbatim *> *> \param[in] WX *> \verbatim *> WX is REAL *> Constant for right eigenvector matrix. *> \endverbatim *> *> \param[in] WY *> \verbatim *> WY is REAL *> Constant for left eigenvector matrix. *> \endverbatim *> *> \param[out] S *> \verbatim *> S is REAL array, dimension (N) *> S(i) is the reciprocal condition number for eigenvalue i. *> \endverbatim *> *> \param[out] DIF *> \verbatim *> DIF is REAL 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 real_matgen * * ===================================================================== SUBROUTINE SLATM6( 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 REAL ALPHA, BETA, WX, WY * .. * .. Array Arguments .. REAL A( LDA, * ), B( LDA, * ), DIF( * ), S( * ), $ X( LDX, * ), Y( LDY, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO, THREE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0, $ THREE = 3.0E+0 ) * .. * .. Local Scalars .. INTEGER I, INFO, J * .. * .. Local Arrays .. REAL WORK( 100 ), Z( 12, 12 ) * .. * .. Intrinsic Functions .. INTRINSIC REAL, SQRT * .. * .. External Subroutines .. EXTERNAL SGESVD, SLACPY, SLAKF2 * .. * .. 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 ) = REAL( I ) + ALPHA B( I, I ) = ONE ELSE A( I, J ) = ZERO B( I, J ) = ZERO END IF * 10 CONTINUE 20 CONTINUE * * Form X and Y * CALL SLACPY( 'F', N, N, B, LDA, Y, LDY ) Y( 3, 1 ) = -WY Y( 4, 1 ) = WY Y( 5, 1 ) = -WY Y( 3, 2 ) = -WY Y( 4, 2 ) = WY Y( 5, 2 ) = -WY * CALL SLACPY( '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 IF( TYPE.EQ.1 ) THEN 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 ) ELSE IF( TYPE.EQ.2 ) THEN A( 1, 3 ) = TWO*WX + WY A( 2, 3 ) = WY A( 1, 4 ) = -WY*( TWO+ALPHA+BETA ) A( 2, 4 ) = TWO*WX - WY*( TWO+ALPHA+BETA ) A( 1, 5 ) = -TWO*WX + WY*( ALPHA-BETA ) A( 2, 5 ) = WY*( ALPHA-BETA ) A( 1, 1 ) = ONE A( 1, 2 ) = -ONE A( 2, 1 ) = ONE A( 2, 2 ) = A( 1, 1 ) A( 3, 3 ) = ONE A( 4, 4 ) = ONE + ALPHA A( 4, 5 ) = ONE + BETA A( 5, 4 ) = -A( 4, 5 ) A( 5, 5 ) = A( 4, 4 ) END IF * * Compute condition numbers * IF( TYPE.EQ.1 ) THEN * S( 1 ) = ONE / SQRT( ( ONE+THREE*WY*WY ) / $ ( ONE+A( 1, 1 )*A( 1, 1 ) ) ) S( 2 ) = ONE / SQRT( ( ONE+THREE*WY*WY ) / $ ( ONE+A( 2, 2 )*A( 2, 2 ) ) ) S( 3 ) = ONE / SQRT( ( ONE+TWO*WX*WX ) / $ ( ONE+A( 3, 3 )*A( 3, 3 ) ) ) S( 4 ) = ONE / SQRT( ( ONE+TWO*WX*WX ) / $ ( ONE+A( 4, 4 )*A( 4, 4 ) ) ) S( 5 ) = ONE / SQRT( ( ONE+TWO*WX*WX ) / $ ( ONE+A( 5, 5 )*A( 5, 5 ) ) ) * CALL SLAKF2( 1, 4, A, LDA, A( 2, 2 ), B, B( 2, 2 ), Z, 12 ) CALL SGESVD( 'N', 'N', 8, 8, Z, 12, WORK, WORK( 9 ), 1, $ WORK( 10 ), 1, WORK( 11 ), 40, INFO ) DIF( 1 ) = WORK( 8 ) * CALL SLAKF2( 4, 1, A, LDA, A( 5, 5 ), B, B( 5, 5 ), Z, 12 ) CALL SGESVD( 'N', 'N', 8, 8, Z, 12, WORK, WORK( 9 ), 1, $ WORK( 10 ), 1, WORK( 11 ), 40, INFO ) DIF( 5 ) = WORK( 8 ) * ELSE IF( TYPE.EQ.2 ) THEN * S( 1 ) = ONE / SQRT( ONE / THREE+WY*WY ) S( 2 ) = S( 1 ) S( 3 ) = ONE / SQRT( ONE / TWO+WX*WX ) S( 4 ) = ONE / SQRT( ( ONE+TWO*WX*WX ) / $ ( ONE+( ONE+ALPHA )*( ONE+ALPHA )+( ONE+BETA )*( ONE+ $ BETA ) ) ) S( 5 ) = S( 4 ) * CALL SLAKF2( 2, 3, A, LDA, A( 3, 3 ), B, B( 3, 3 ), Z, 12 ) CALL SGESVD( 'N', 'N', 12, 12, Z, 12, WORK, WORK( 13 ), 1, $ WORK( 14 ), 1, WORK( 15 ), 60, INFO ) DIF( 1 ) = WORK( 12 ) * CALL SLAKF2( 3, 2, A, LDA, A( 4, 4 ), B, B( 4, 4 ), Z, 12 ) CALL SGESVD( 'N', 'N', 12, 12, Z, 12, WORK, WORK( 13 ), 1, $ WORK( 14 ), 1, WORK( 15 ), 60, INFO ) DIF( 5 ) = WORK( 12 ) * END IF * RETURN * * End of SLATM6 * END