*> \brief \b SSPGVD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SSPGVD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, * LWORK, IWORK, LIWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBZ, UPLO * INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N * .. * .. Array Arguments .. * INTEGER IWORK( * ) * REAL AP( * ), BP( * ), W( * ), WORK( * ), * \$ Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SSPGVD computes all the eigenvalues, and optionally, the eigenvectors *> of a real generalized symmetric-definite eigenproblem, of the form *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and *> B are assumed to be symmetric, stored in packed format, and B is also *> positive definite. *> If eigenvectors are desired, it uses a divide and conquer algorithm. *> *> The divide and conquer algorithm makes very mild assumptions about *> floating point arithmetic. It will work on machines with a guard *> digit in add/subtract, or on those binary machines without guard *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or *> Cray-2. It could conceivably fail on hexadecimal or decimal machines *> without guard digits, but we know of none. *> \endverbatim * * Arguments: * ========== * *> \param[in] ITYPE *> \verbatim *> ITYPE is INTEGER *> Specifies the problem type to be solved: *> = 1: A*x = (lambda)*B*x *> = 2: A*B*x = (lambda)*x *> = 3: B*A*x = (lambda)*x *> \endverbatim *> *> \param[in] JOBZ *> \verbatim *> JOBZ is CHARACTER*1 *> = 'N': Compute eigenvalues only; *> = 'V': Compute eigenvalues and eigenvectors. *> \endverbatim *> *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> = 'U': Upper triangles of A and B are stored; *> = 'L': Lower triangles of A and B are stored. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrices A and B. N >= 0. *> \endverbatim *> *> \param[in,out] AP *> \verbatim *> AP is REAL array, dimension (N*(N+1)/2) *> On entry, the upper or lower triangle of the symmetric matrix *> A, packed columnwise in a linear array. The j-th column of A *> is stored in the array AP as follows: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. *> *> On exit, the contents of AP are destroyed. *> \endverbatim *> *> \param[in,out] BP *> \verbatim *> BP is REAL array, dimension (N*(N+1)/2) *> On entry, the upper or lower triangle of the symmetric matrix *> B, packed columnwise in a linear array. The j-th column of B *> is stored in the array BP as follows: *> if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; *> if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. *> *> On exit, the triangular factor U or L from the Cholesky *> factorization B = U**T*U or B = L*L**T, in the same storage *> format as B. *> \endverbatim *> *> \param[out] W *> \verbatim *> W is REAL array, dimension (N) *> If INFO = 0, the eigenvalues in ascending order. *> \endverbatim *> *> \param[out] Z *> \verbatim *> Z is REAL array, dimension (LDZ, N) *> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of *> eigenvectors. The eigenvectors are normalized as follows: *> if ITYPE = 1 or 2, Z**T*B*Z = I; *> if ITYPE = 3, Z**T*inv(B)*Z = I. *> If JOBZ = 'N', then Z is not referenced. *> \endverbatim *> *> \param[in] LDZ *> \verbatim *> LDZ is INTEGER *> The leading dimension of the array Z. LDZ >= 1, and if *> JOBZ = 'V', LDZ >= max(1,N). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the required LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> If N <= 1, LWORK >= 1. *> If JOBZ = 'N' and N > 1, LWORK >= 2*N. *> If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the required sizes of the WORK and IWORK *> arrays, returns these values as the first entries of the WORK *> and IWORK arrays, and no error message related to LWORK or *> LIWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) *> On exit, if INFO = 0, IWORK(1) returns the required LIWORK. *> \endverbatim *> *> \param[in] LIWORK *> \verbatim *> LIWORK is INTEGER *> The dimension of the array IWORK. *> If JOBZ = 'N' or N <= 1, LIWORK >= 1. *> If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. *> *> If LIWORK = -1, then a workspace query is assumed; the *> routine only calculates the required sizes of the WORK and *> IWORK arrays, returns these values as the first entries of *> the WORK and IWORK arrays, and no error message related to *> LWORK or LIWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> > 0: SPPTRF or SSPEVD returned an error code: *> <= N: if INFO = i, SSPEVD failed to converge; *> i off-diagonal elements of an intermediate *> tridiagonal form did not converge to zero; *> > N: if INFO = N + i, for 1 <= i <= N, then the leading *> minor of order i of B is not positive definite. *> The factorization of B could not be completed and *> no eigenvalues or eigenvectors were computed. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup realOTHEReigen * *> \par Contributors: * ================== *> *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA * * ===================================================================== SUBROUTINE SSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, \$ LWORK, IWORK, LIWORK, INFO ) * * -- LAPACK driver routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL AP( * ), BP( * ), W( * ), WORK( * ), \$ Z( LDZ, * ) * .. * * ===================================================================== * * .. Local Scalars .. LOGICAL LQUERY, UPPER, WANTZ CHARACTER TRANS INTEGER J, LIWMIN, LWMIN, NEIG * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL SPPTRF, SSPEVD, SSPGST, STPMV, STPSV, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, REAL * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * INFO = 0 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -9 END IF * IF( INFO.EQ.0 ) THEN IF( N.LE.1 ) THEN LIWMIN = 1 LWMIN = 1 ELSE IF( WANTZ ) THEN LIWMIN = 3 + 5*N LWMIN = 1 + 6*N + 2*N**2 ELSE LIWMIN = 1 LWMIN = 2*N END IF END IF WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -11 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -13 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SSPGVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * Form a Cholesky factorization of BP. * CALL SPPTRF( UPLO, N, BP, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF * * Transform problem to standard eigenvalue problem and solve. * CALL SSPGST( ITYPE, UPLO, N, AP, BP, INFO ) CALL SSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, \$ LIWORK, INFO ) LWMIN = MAX( REAL( LWMIN ), REAL( WORK( 1 ) ) ) LIWMIN = MAX( REAL( LIWMIN ), REAL( IWORK( 1 ) ) ) * IF( WANTZ ) THEN * * Backtransform eigenvectors to the original problem. * NEIG = N IF( INFO.GT.0 ) \$ NEIG = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN * * For A*x=(lambda)*B*x and A*B*x=(lambda)*x; * backtransform eigenvectors: x = inv(L)**T *y or inv(U)*y * IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF * DO 10 J = 1, NEIG CALL STPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), \$ 1 ) 10 CONTINUE * ELSE IF( ITYPE.EQ.3 ) THEN * * For B*A*x=(lambda)*x; * backtransform eigenvectors: x = L*y or U**T *y * IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF * DO 20 J = 1, NEIG CALL STPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), \$ 1 ) 20 CONTINUE END IF END IF * WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN * RETURN * * End of SSPGVD * END