MAGMA  1.7.0
Matrix Algebra for GPU and Multicore Architectures
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magma_int_t magma_zhegvd (magma_int_t itype, magma_vec_t jobz, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *B, magma_int_t ldb, double *w, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
 ZHEGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. More...
 
magma_int_t magma_zhegvd_m (magma_int_t ngpu, magma_int_t itype, magma_vec_t jobz, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *B, magma_int_t ldb, double *w, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
 ZHEGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. More...
 
magma_int_t magma_zhegvdx (magma_int_t itype, magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *B, magma_int_t ldb, double vl, double vu, magma_int_t il, magma_int_t iu, magma_int_t *mout, double *w, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
 ZHEGVDX computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. More...
 
magma_int_t magma_zhegvdx_2stage (magma_int_t itype, magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *B, magma_int_t ldb, double vl, double vu, magma_int_t il, magma_int_t iu, magma_int_t *m, double *w, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
 ZHEGVDX_2STAGE computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. More...
 
magma_int_t magma_zhegvdx_2stage_m (magma_int_t ngpu, magma_int_t itype, magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *B, magma_int_t ldb, double vl, double vu, magma_int_t il, magma_int_t iu, magma_int_t *mout, double *w, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
 ZHEGVDX_2STAGE computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. More...
 
magma_int_t magma_zhegvdx_m (magma_int_t ngpu, magma_int_t itype, magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *B, magma_int_t ldb, double vl, double vu, magma_int_t il, magma_int_t iu, magma_int_t *m, double *w, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
 ZHEGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. More...
 
magma_int_t magma_zhegvr (magma_int_t itype, magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *B, magma_int_t ldb, double vl, double vu, magma_int_t il, magma_int_t iu, double abstol, magma_int_t *m, double *w, magmaDoubleComplex *Z, magma_int_t ldz, magma_int_t *isuppz, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
 ZHEGVR computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. More...
 
magma_int_t magma_zhegvx (magma_int_t itype, magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *B, magma_int_t ldb, double vl, double vu, magma_int_t il, magma_int_t iu, double abstol, magma_int_t *m, double *w, magmaDoubleComplex *Z, magma_int_t ldz, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t *iwork, magma_int_t *ifail, magma_int_t *info)
 ZHEGVX computes selected eigenvalues, and optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. More...
 

Detailed Description

Function Documentation

magma_int_t magma_zhegvd ( magma_int_t  itype,
magma_vec_t  jobz,
magma_uplo_t  uplo,
magma_int_t  n,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  B,
magma_int_t  ldb,
double *  w,
magmaDoubleComplex *  work,
magma_int_t  lwork,
double *  rwork,
magma_int_t  lrwork,
magma_int_t *  iwork,
magma_int_t  liwork,
magma_int_t *  info 
)

ZHEGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-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 Hermitian 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.

Parameters
[in]itypeINTEGER 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
[in]jobzmagma_vec_t
  • = MagmaNoVec: Compute eigenvalues only;
  • = MagmaVec: Compute eigenvalues and eigenvectors.
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A and B are stored;
  • = MagmaLower: Lower triangles of A and B are stored.
[in]nINTEGER The order of the matrices A and B. N >= 0.
[in,out]ACOMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, if JOBZ = MagmaVec, then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = MagmaNoVec, then on exit the upper triangle (if UPLO=MagmaUpper) or the lower triangle (if UPLO=MagmaLower) of A, including the diagonal, is destroyed.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out]BCOMPLEX_16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = MagmaLower, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
[in]ldbINTEGER The leading dimension of the array B. LDB >= max(1,N).
[out]wDOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LWORK >= N + N*NB. If JOBZ = MagmaVec and N > 1, LWORK >= max( N + N*NB, 2*N + N**2 ). NB can be obtained through magma_get_zhetrd_nb(N).
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]rwork(workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK[0] returns the optimal LRWORK.
[in]lrworkINTEGER The dimension of the array RWORK. If N <= 1, LRWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LRWORK >= N. If JOBZ = MagmaVec and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]iwork(workspace) INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK[0] returns the optimal LIWORK.
[in]liworkINTEGER The dimension of the array IWORK. If N <= 1, LIWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LIWORK >= 1. If JOBZ = MagmaVec and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: ZPOTRF or ZHEEVD returned an error code: <= N: if INFO = i and JOBZ = MagmaNoVec, then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = MagmaVec, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1); > 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.

Further Details

Based on contributions by Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Modified so that no backsubstitution is performed if ZHEEVD fails to converge (NEIG in old code could be greater than N causing out of bounds reference to A - reported by Ralf Meyer). Also corrected the description of INFO and the test on ITYPE. Sven, 16 Feb 05.

magma_int_t magma_zhegvd_m ( magma_int_t  ngpu,
magma_int_t  itype,
magma_vec_t  jobz,
magma_uplo_t  uplo,
magma_int_t  n,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  B,
magma_int_t  ldb,
double *  w,
magmaDoubleComplex *  work,
magma_int_t  lwork,
double *  rwork,
magma_int_t  lrwork,
magma_int_t *  iwork,
magma_int_t  liwork,
magma_int_t *  info 
)

ZHEGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-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 Hermitian 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.

Parameters
[in]ngpuINTEGER Number of GPUs to use. ngpu > 0.
[in]itypeINTEGER 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
[in]jobzmagma_vec_t
  • = MagmaNoVec: Compute eigenvalues only;
  • = MagmaVec: Compute eigenvalues and eigenvectors.
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A and B are stored;
  • = MagmaLower: Lower triangles of A and B are stored.
[in]nINTEGER The order of the matrices A and B. N >= 0.
[in,out]ACOMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, if JOBZ = MagmaVec, then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = MagmaNoVec, then on exit the upper triangle (if UPLO=MagmaUpper) or the lower triangle (if UPLO=MagmaLower) of A, including the diagonal, is destroyed.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out]BCOMPLEX_16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = MagmaLower, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
[in]ldbINTEGER The leading dimension of the array B. LDB >= max(1,N).
[out]wDOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LWORK >= N + N*NB. If JOBZ = MagmaVec and N > 1, LWORK >= max( N + N*NB, 2*N + N**2 ). NB can be obtained through magma_get_zhetrd_nb(N).
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]rwork(workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK[0] returns the optimal LRWORK.
[in]lrworkINTEGER The dimension of the array RWORK. If N <= 1, LRWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LRWORK >= N. If JOBZ = MagmaVec and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]iwork(workspace) INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK[0] returns the optimal LIWORK.
[in]liworkINTEGER The dimension of the array IWORK. If N <= 1, LIWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LIWORK >= 1. If JOBZ = MagmaVec and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: ZPOTRF or ZHEEVD returned an error code: <= N: if INFO = i and JOBZ = MagmaNoVec, then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = MagmaVec, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1); > 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.

Further Details

Based on contributions by Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Modified so that no backsubstitution is performed if ZHEEVD fails to converge (NEIG in old code could be greater than N causing out of bounds reference to A - reported by Ralf Meyer). Also corrected the description of INFO and the test on ITYPE. Sven, 16 Feb 05.

magma_int_t magma_zhegvdx ( magma_int_t  itype,
magma_vec_t  jobz,
magma_range_t  range,
magma_uplo_t  uplo,
magma_int_t  n,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  B,
magma_int_t  ldb,
double  vl,
double  vu,
magma_int_t  il,
magma_int_t  iu,
magma_int_t *  mout,
double *  w,
magmaDoubleComplex *  work,
magma_int_t  lwork,
double *  rwork,
magma_int_t  lrwork,
magma_int_t *  iwork,
magma_int_t  liwork,
magma_int_t *  info 
)

ZHEGVDX computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian-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 Hermitian and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. 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.

Parameters
[in]itypeINTEGER 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
[in]jobzmagma_vec_t
  • = MagmaNoVec: Compute eigenvalues only;
  • = MagmaVec: Compute eigenvalues and eigenvectors.
[in]rangemagma_range_t
  • = MagmaRangeAll: all eigenvalues will be found.
  • = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found.
  • = MagmaRangeI: the IL-th through IU-th eigenvalues will be found.
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A and B are stored;
  • = MagmaLower: Lower triangles of A and B are stored.
[in]nINTEGER The order of the matrices A and B. N >= 0.
[in,out]ACOMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, if JOBZ = MagmaVec, then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = MagmaNoVec, then on exit the upper triangle (if UPLO=MagmaUpper) or the lower triangle (if UPLO=MagmaLower) of A, including the diagonal, is destroyed.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out]BCOMPLEX_16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = MagmaLower, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
[in]ldbINTEGER The leading dimension of the array B. LDB >= max(1,N).
[in]vlDOUBLE PRECISION
[in]vuDOUBLE PRECISION If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI.
[in]ilINTEGER
[in]iuINTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV.
[out]moutINTEGER The total number of eigenvalues found. 0 <= MOUT <= N. If RANGE = MagmaRangeAll, MOUT = N, and if RANGE = MagmaRangeI, MOUT = IU-IL+1.
[out]wDOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LWORK >= N + N*NB. If JOBZ = MagmaVec and N > 1, LWORK >= max( N + N*NB, 2*N + N**2 ). NB can be obtained through magma_get_zhetrd_nb(N).
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]rwork(workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK[0] returns the optimal LRWORK.
[in]lrworkINTEGER The dimension of the array RWORK. If N <= 1, LRWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LRWORK >= N. If JOBZ = MagmaVec and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]iwork(workspace) INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK[0] returns the optimal LIWORK.
[in]liworkINTEGER The dimension of the array IWORK. If N <= 1, LIWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LIWORK >= 1. If JOBZ = MagmaVec and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: ZPOTRF or ZHEEVD returned an error code: <= N: if INFO = i and JOBZ = MagmaNoVec, then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = MagmaVec, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1); > 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.

Further Details

Based on contributions by Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Modified so that no backsubstitution is performed if ZHEEVD fails to converge (NEIG in old code could be greater than N causing out of bounds reference to A - reported by Ralf Meyer). Also corrected the description of INFO and the test on ITYPE. Sven, 16 Feb 05.

magma_int_t magma_zhegvdx_2stage ( magma_int_t  itype,
magma_vec_t  jobz,
magma_range_t  range,
magma_uplo_t  uplo,
magma_int_t  n,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  B,
magma_int_t  ldb,
double  vl,
double  vu,
magma_int_t  il,
magma_int_t  iu,
magma_int_t *  m,
double *  w,
magmaDoubleComplex *  work,
magma_int_t  lwork,
double *  rwork,
magma_int_t  lrwork,
magma_int_t *  iwork,
magma_int_t  liwork,
magma_int_t *  info 
)

ZHEGVDX_2STAGE computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-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 Hermitian and B is also positive definite. It uses a two-stage algorithm for the tridiagonalization. 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.

Parameters
[in]itypeINTEGER 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
[in]jobzmagma_vec_t
  • = MagmaNoVec: Compute eigenvalues only;
  • = MagmaVec: Compute eigenvalues and eigenvectors.
[in]rangemagma_range_t
  • = MagmaRangeAll: all eigenvalues will be found.
  • = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found.
  • = MagmaRangeI: the IL-th through IU-th eigenvalues will be found.
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A and B are stored;
  • = MagmaLower: Lower triangles of A and B are stored.
[in]nINTEGER The order of the matrices A and B. N >= 0.
[in,out]ACOMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, if JOBZ = MagmaVec, then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = MagmaNoVec, then on exit the upper triangle (if UPLO=MagmaUpper) or the lower triangle (if UPLO=MagmaLower) of A, including the diagonal, is destroyed.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out]BCOMPLEX_16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = MagmaLower, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
[in]ldbINTEGER The leading dimension of the array B. LDB >= max(1,N).
[in]vlDOUBLE PRECISION
[in]vuDOUBLE PRECISION If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI.
[in]ilINTEGER
[in]iuINTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV.
[out]mINTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1.
[out]wDOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LWORK >= LQ2 + N + N*NB. If JOBZ = MagmaVec and N > 1, LWORK >= LQ2 + 2*N + N**2. where LQ2 is the size needed to store the Q2 matrix and is returned by magma_bulge_get_lq2.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]rwork(workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK[0] returns the optimal LRWORK.
[in]lrworkINTEGER The dimension of the array RWORK. If N <= 1, LRWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LRWORK >= N. If JOBZ = MagmaVec and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]iwork(workspace) INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK[0] returns the optimal LIWORK.
[in]liworkINTEGER The dimension of the array IWORK. If N <= 1, LIWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LIWORK >= 1. If JOBZ = MagmaVec and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: ZPOTRF or ZHEEVD returned an error code: <= N: if INFO = i and JOBZ = MagmaNoVec, then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = MagmaVec, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1); > 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.

Further Details

Based on contributions by Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Modified so that no backsubstitution is performed if ZHEEVD fails to converge (NEIG in old code could be greater than N causing out of bounds reference to A - reported by Ralf Meyer). Also corrected the description of INFO and the test on ITYPE. Sven, 16 Feb 05.

magma_int_t magma_zhegvdx_2stage_m ( magma_int_t  ngpu,
magma_int_t  itype,
magma_vec_t  jobz,
magma_range_t  range,
magma_uplo_t  uplo,
magma_int_t  n,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  B,
magma_int_t  ldb,
double  vl,
double  vu,
magma_int_t  il,
magma_int_t  iu,
magma_int_t *  mout,
double *  w,
magmaDoubleComplex *  work,
magma_int_t  lwork,
double *  rwork,
magma_int_t  lrwork,
magma_int_t *  iwork,
magma_int_t  liwork,
magma_int_t *  info 
)

ZHEGVDX_2STAGE computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-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 Hermitian and B is also positive definite. It uses a two-stage algorithm for the tridiagonalization. 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.

Parameters
[in]ngpuINTEGER Number of GPUs to use. ngpu > 0.
[in]itypeINTEGER 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
[in]jobzmagma_vec_t
  • = MagmaNoVec: Compute eigenvalues only;
  • = MagmaVec: Compute eigenvalues and eigenvectors.
[in]rangemagma_range_t
  • = MagmaRangeAll: all eigenvalues will be found.
  • = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found.
  • = MagmaRangeI: the IL-th through IU-th eigenvalues will be found.
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A and B are stored;
  • = MagmaLower: Lower triangles of A and B are stored.
[in]nINTEGER The order of the matrices A and B. N >= 0.
[in,out]ACOMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, if JOBZ = MagmaVec, then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = MagmaNoVec, then on exit the upper triangle (if UPLO=MagmaUpper) or the lower triangle (if UPLO=MagmaLower) of A, including the diagonal, is destroyed.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out]BCOMPLEX_16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = MagmaLower, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
[in]ldbINTEGER The leading dimension of the array B. LDB >= max(1,N).
[in]vlDOUBLE PRECISION
[in]vuDOUBLE PRECISION If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI.
[in]ilINTEGER
[in]iuINTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV.
[out]moutINTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1.
[out]wDOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LWORK >= LQ2 + N + N*NB. If JOBZ = MagmaVec and N > 1, LWORK >= LQ2 + 2*N + N**2. where LQ2 is the size needed to store the Q2 matrix and is returned by magma_bulge_get_lq2.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]rwork(workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK[0] returns the optimal LRWORK.
[in]lrworkINTEGER The dimension of the array RWORK. If N <= 1, LRWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LRWORK >= N. If JOBZ = MagmaVec and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]iwork(workspace) INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK[0] returns the optimal LIWORK.
[in]liworkINTEGER The dimension of the array IWORK. If N <= 1, LIWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LIWORK >= 1. If JOBZ = MagmaVec and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: ZPOTRF or ZHEEVD returned an error code: <= N: if INFO = i and JOBZ = MagmaNoVec, then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = MagmaVec, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1); > 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.

Further Details

Based on contributions by Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Modified so that no backsubstitution is performed if ZHEEVD fails to converge (NEIG in old code could be greater than N causing out of bounds reference to A - reported by Ralf Meyer). Also corrected the description of INFO and the test on ITYPE. Sven, 16 Feb 05.

magma_int_t magma_zhegvdx_m ( magma_int_t  ngpu,
magma_int_t  itype,
magma_vec_t  jobz,
magma_range_t  range,
magma_uplo_t  uplo,
magma_int_t  n,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  B,
magma_int_t  ldb,
double  vl,
double  vu,
magma_int_t  il,
magma_int_t  iu,
magma_int_t *  m,
double *  w,
magmaDoubleComplex *  work,
magma_int_t  lwork,
double *  rwork,
magma_int_t  lrwork,
magma_int_t *  iwork,
magma_int_t  liwork,
magma_int_t *  info 
)

ZHEGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-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 Hermitian 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.

Parameters
[in]ngpuINTEGER Number of GPUs to use. ngpu > 0.
[in]itypeINTEGER 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
[in]jobzmagma_vec_t
  • = MagmaNoVec: Compute eigenvalues only;
  • = MagmaVec: Compute eigenvalues and eigenvectors.
[in]rangemagma_range_t
  • = MagmaRangeAll: all eigenvalues will be found.
  • = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found.
  • = MagmaRangeI: the IL-th through IU-th eigenvalues will be found.
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A and B are stored;
  • = MagmaLower: Lower triangles of A and B are stored.
[in]nINTEGER The order of the matrices A and B. N >= 0.
[in,out]ACOMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, if JOBZ = MagmaVec, then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = MagmaNoVec, then on exit the upper triangle (if UPLO=MagmaUpper) or the lower triangle (if UPLO=MagmaLower) of A, including the diagonal, is destroyed.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out]BCOMPLEX_16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = MagmaLower, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
[in]ldbINTEGER The leading dimension of the array B. LDB >= max(1,N).
[in]vlDOUBLE PRECISION
[in]vuDOUBLE PRECISION If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI.
[in]ilINTEGER
[in]iuINTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV.
[out]mINTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1.
[out]wDOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LWORK >= N + N*NB. If JOBZ = MagmaVec and N > 1, LWORK >= max( N + N*NB, 2*N + N**2 ). NB can be obtained through magma_get_zhetrd_nb(N).
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]rwork(workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK[0] returns the optimal LRWORK.
[in]lrworkINTEGER The dimension of the array RWORK. If N <= 1, LRWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LRWORK >= N. If JOBZ = MagmaVec and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]iwork(workspace) INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK[0] returns the optimal LIWORK.
[in]liworkINTEGER The dimension of the array IWORK. If N <= 1, LIWORK >= 1. If JOBZ = MagmaNoVec and N > 1, LIWORK >= 1. If JOBZ = MagmaVec and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: ZPOTRF or ZHEEVD returned an error code: <= N: if INFO = i and JOBZ = MagmaNoVec, then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = MagmaVec, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1); > 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.

Further Details

Based on contributions by Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Modified so that no backsubstitution is performed if ZHEEVD fails to converge (NEIG in old code could be greater than N causing out of bounds reference to A - reported by Ralf Meyer). Also corrected the description of INFO and the test on ITYPE. Sven, 16 Feb 05.

magma_int_t magma_zhegvr ( magma_int_t  itype,
magma_vec_t  jobz,
magma_range_t  range,
magma_uplo_t  uplo,
magma_int_t  n,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  B,
magma_int_t  ldb,
double  vl,
double  vu,
magma_int_t  il,
magma_int_t  iu,
double  abstol,
magma_int_t *  m,
double *  w,
magmaDoubleComplex *  Z,
magma_int_t  ldz,
magma_int_t *  isuppz,
magmaDoubleComplex *  work,
magma_int_t  lwork,
double *  rwork,
magma_int_t  lrwork,
magma_int_t *  iwork,
magma_int_t  liwork,
magma_int_t *  info 
)

ZHEGVR computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-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 Hermitian and B is also positive definite.

Whenever possible, ZHEEVR calls ZSTEGR to compute the eigenspectrum using Relatively Robust Representations. ZSTEGR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T,

  1. Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation,
  2. Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm,
  3. If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a),
  4. Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997.

Note 1 : ZHEEVR calls ZSTEGR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and when partial spectrum requests are made.

Normal execution of ZSTEGR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

Parameters
[in]itypeINTEGER 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
[in]rangemagma_range_t
  • = MagmaRangeAll: all eigenvalues will be found.
  • = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found.
  • = MagmaRangeI: the IL-th through IU-th eigenvalues will be found.
[in]jobzmagma_vec_t
  • = MagmaNoVec: Compute eigenvalues only;
  • = MagmaVec: Compute eigenvalues and eigenvectors.
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A and B are stored;
  • = MagmaLower: Lower triangles of A and B are stored.
[in]nINTEGER The order of the matrices A and B. N >= 0.
[in,out]ACOMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, if JOBZ = MagmaVec, then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = MagmaNoVec, then on exit the upper triangle (if UPLO=MagmaUpper) or the lower triangle (if UPLO=MagmaLower) of A, including the diagonal, is destroyed.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out]BCOMPLEX_16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = MagmaLower, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
[in]ldbINTEGER The leading dimension of the array B. LDB >= max(1,N).
[in]vlDOUBLE PRECISION
[in]vuDOUBLE PRECISION If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI.
[in]ilINTEGER
[in]iuINTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV.
[in]abstolDOUBLE PRECISION The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
    ABSTOL + EPS * max( |a|,|b| ),

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.
See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.
If high relative accuracy is important, set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but furutre releases will. See J. Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.
[out]mINTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1.
[out]wDOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
[out]ZCOMPLEX_16 array, dimension (LDZ, max(1,M)) If JOBZ = MagmaVec, then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = MagmaNoVec, then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = MagmaRangeV, the exact value of M is not known in advance and an upper bound must be used.
[in]ldzINTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = MagmaVec, LDZ >= max(1,N).
[out]isuppzINTEGER ARRAY, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). Implemented only for RANGE = MagmaRangeAll or MagmaRangeI and IU - IL = N - 1
[out]work(workspace) COMPLEX_16 array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. LWORK >= max(1,2*N). For optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the blocksize for ZHETRD and for ZUNMTR as returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
[out]rwork(workspace) DOUBLE PRECISION array, dimension (LRWORK) On exit, if INFO = 0, RWORK[0] returns the optimal (and minimal) LRWORK.
[in]lrworkINTEGER The length of the array RWORK. LRWORK >= max(1,24*N).
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA.
[out]iwork(workspace) INTEGER array, dimension (LIWORK) On exit, if INFO = 0, IWORK[0] returns the optimal (and minimal) LIWORK.
[in]liworkINTEGER The dimension of the array IWORK. LIWORK >= max(1,10*N).
If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: Internal error

Further Details

Based on contributions by Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USA

magma_int_t magma_zhegvx ( magma_int_t  itype,
magma_vec_t  jobz,
magma_range_t  range,
magma_uplo_t  uplo,
magma_int_t  n,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  B,
magma_int_t  ldb,
double  vl,
double  vu,
magma_int_t  il,
magma_int_t  iu,
double  abstol,
magma_int_t *  m,
double *  w,
magmaDoubleComplex *  Z,
magma_int_t  ldz,
magmaDoubleComplex *  work,
magma_int_t  lwork,
double *  rwork,
magma_int_t *  iwork,
magma_int_t *  ifail,
magma_int_t *  info 
)

ZHEGVX computes selected eigenvalues, and optionally, eigenvectors of a complex generalized Hermitian-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 Hermitian and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Parameters
[in]itypeINTEGER 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
[in]jobzmagma_vec_t
  • = MagmaNoVec: Compute eigenvalues only;
  • = MagmaVec: Compute eigenvalues and eigenvectors.
[in]rangemagma_range_t
  • = MagmaRangeAll: all eigenvalues will be found.
  • = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found.
  • = MagmaRangeI: the IL-th through IU-th eigenvalues will be found.
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A and B are stored;
  • = MagmaLower: Lower triangles of A and B are stored.
[in]nINTEGER The order of the matrices A and B. N >= 0.
[in,out]ACOMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO=MagmaLower) or the upper triangle (if UPLO=MagmaUpper) of A, including the diagonal, is destroyed.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out]BCOMPLEX_16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = MagmaLower, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.
[in]ldbINTEGER The leading dimension of the array B. LDB >= max(1,N).
[in]vlDOUBLE PRECISION
[in]vuDOUBLE PRECISION If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI.
[in]ilINTEGER
[in]iuINTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV.
[in]abstolDOUBLE PRECISION The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
ABSTOL + EPS * max( |a|,|b| ),
where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO > 0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
[out]mINTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1.
[out]wDOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.
[out]ZCOMPLEX_16 array, dimension (LDZ, max(1,M)) If JOBZ = MagmaNoVec, then Z is not referenced. If JOBZ = MagmaVec, then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = MagmaRangeV, the exact value of M is not known in advance and an upper bound must be used.
[in]ldzINTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = MagmaVec, LDZ >= max(1,N).
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. LWORK >= max(1,2*N). For optimal efficiency, LWORK >= (NB+1)*N, where NB is the blocksize for ZHETRD returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
rwork(workspace) DOUBLE PRECISION array, dimension (7*N)
iwork(workspace) INTEGER array, dimension (5*N)
[out]ifailINTEGER array, dimension (N) If JOBZ = MagmaVec, then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = MagmaNoVec, then IFAIL is not referenced.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: ZPOTRF or ZHEEVX returned an error code: <= N: if INFO = i, ZHEEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL. > 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.

Further Details

Based on contributions by Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA