numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
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| lapack/TESTING/EIG/zhbt21.f | 8061B | -rw-r--r-- |
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*> \brief \b ZHBT21 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE ZHBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK, * RWORK, RESULT ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER KA, KS, LDA, LDU, N * .. * .. Array Arguments .. * DOUBLE PRECISION D( * ), E( * ), RESULT( 2 ), RWORK( * ) * COMPLEX*16 A( LDA, * ), U( LDU, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZHBT21 generally checks a decomposition of the form *> *> A = U S U**H *> *> where **H means conjugate transpose, A is hermitian banded, U is *> unitary, and S is diagonal (if KS=0) or symmetric *> tridiagonal (if KS=1). *> *> Specifically: *> *> RESULT(1) = | A - U S U**H | / ( |A| n ulp ) and *> RESULT(2) = | I - U U**H | / ( n ulp ) *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER *> If UPLO='U', the upper triangle of A and V will be used and *> the (strictly) lower triangle will not be referenced. *> If UPLO='L', the lower triangle of A and V will be used and *> the (strictly) upper triangle will not be referenced. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The size of the matrix. If it is zero, ZHBT21 does nothing. *> It must be at least zero. *> \endverbatim *> *> \param[in] KA *> \verbatim *> KA is INTEGER *> The bandwidth of the matrix A. It must be at least zero. If *> it is larger than N-1, then max( 0, N-1 ) will be used. *> \endverbatim *> *> \param[in] KS *> \verbatim *> KS is INTEGER *> The bandwidth of the matrix S. It may only be zero or one. *> If zero, then S is diagonal, and E is not referenced. If *> one, then S is symmetric tri-diagonal. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA, N) *> The original (unfactored) matrix. It is assumed to be *> hermitian, and only the upper (UPLO='U') or only the lower *> (UPLO='L') will be referenced. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of A. It must be at least 1 *> and at least min( KA, N-1 ). *> \endverbatim *> *> \param[in] D *> \verbatim *> D is DOUBLE PRECISION array, dimension (N) *> The diagonal of the (symmetric tri-) diagonal matrix S. *> \endverbatim *> *> \param[in] E *> \verbatim *> E is DOUBLE PRECISION array, dimension (N-1) *> The off-diagonal of the (symmetric tri-) diagonal matrix S. *> E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and *> (3,2) element, etc. *> Not referenced if KS=0. *> \endverbatim *> *> \param[in] U *> \verbatim *> U is COMPLEX*16 array, dimension (LDU, N) *> The unitary matrix in the decomposition, expressed as a *> dense matrix (i.e., not as a product of Householder *> transformations, Givens transformations, etc.) *> \endverbatim *> *> \param[in] LDU *> \verbatim *> LDU is INTEGER *> The leading dimension of U. LDU must be at least N and *> at least 1. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (N**2) *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[out] RESULT *> \verbatim *> RESULT is DOUBLE PRECISION array, dimension (2) *> The values computed by the two tests described above. The *> values are currently limited to 1/ulp, to avoid overflow. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complex16_eig * * ===================================================================== SUBROUTINE ZHBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK, $ RWORK, RESULT ) * * -- LAPACK test 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 UPLO INTEGER KA, KS, LDA, LDU, N * .. * .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ), RESULT( 2 ), RWORK( * ) COMPLEX*16 A( LDA, * ), U( LDU, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), $ CONE = ( 1.0D+0, 0.0D+0 ) ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL LOWER CHARACTER CUPLO INTEGER IKA, J, JC, JR DOUBLE PRECISION ANORM, ULP, UNFL, WNORM * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, ZLANGE, ZLANHB, ZLANHP EXTERNAL LSAME, DLAMCH, ZLANGE, ZLANHB, ZLANHP * .. * .. External Subroutines .. EXTERNAL ZGEMM, ZHPR, ZHPR2 * .. * .. Intrinsic Functions .. INTRINSIC DBLE, DCMPLX, MAX, MIN * .. * .. Executable Statements .. * * Constants * RESULT( 1 ) = ZERO RESULT( 2 ) = ZERO IF( N.LE.0 ) $ RETURN * IKA = MAX( 0, MIN( N-1, KA ) ) * IF( LSAME( UPLO, 'U' ) ) THEN LOWER = .FALSE. CUPLO = 'U' ELSE LOWER = .TRUE. CUPLO = 'L' END IF * UNFL = DLAMCH( 'Safe minimum' ) ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' ) * * Some Error Checks * * Do Test 1 * * Norm of A: * ANORM = MAX( ZLANHB( '1', CUPLO, N, IKA, A, LDA, RWORK ), UNFL ) * * Compute error matrix: Error = A - U S U**H * * Copy A from SB to SP storage format. * J = 0 DO 50 JC = 1, N IF( LOWER ) THEN DO 10 JR = 1, MIN( IKA+1, N+1-JC ) J = J + 1 WORK( J ) = A( JR, JC ) 10 CONTINUE DO 20 JR = IKA + 2, N + 1 - JC J = J + 1 WORK( J ) = ZERO 20 CONTINUE ELSE DO 30 JR = IKA + 2, JC J = J + 1 WORK( J ) = ZERO 30 CONTINUE DO 40 JR = MIN( IKA, JC-1 ), 0, -1 J = J + 1 WORK( J ) = A( IKA+1-JR, JC ) 40 CONTINUE END IF 50 CONTINUE * DO 60 J = 1, N CALL ZHPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK ) 60 CONTINUE * IF( N.GT.1 .AND. KS.EQ.1 ) THEN DO 70 J = 1, N - 1 CALL ZHPR2( CUPLO, N, -DCMPLX( E( J ) ), U( 1, J ), 1, $ U( 1, J+1 ), 1, WORK ) 70 CONTINUE END IF WNORM = ZLANHP( '1', CUPLO, N, WORK, RWORK ) * IF( ANORM.GT.WNORM ) THEN RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP ) ELSE IF( ANORM.LT.ONE ) THEN RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP ) ELSE RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP ) END IF END IF * * Do Test 2 * * Compute U U**H - I * CALL ZGEMM( 'N', 'C', N, N, N, CONE, U, LDU, U, LDU, CZERO, WORK, $ N ) * DO 80 J = 1, N WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - CONE 80 CONTINUE * RESULT( 2 ) = MIN( ZLANGE( '1', N, N, WORK, N, RWORK ), $ DBLE( N ) ) / ( N*ULP ) * RETURN * * End of ZHBT21 * END