hetrd_2stage#

chetrd_2stage

Functions

void chetrd_2stage(
    const char* vect,
    const char* uplo,
    const INT   n,
          c64*  A,
    const INT   lda,
          f32*  D,
          f32*  E,
          c64*  tau,
          c64*  hous2,
    const INT   lhous2,
          c64*  work,
    const INT   lwork,
          INT*  info
);

void chetrd_2stage(const char *vect, const char *uplo, const INT n, c64 *A, const INT lda, f32 *D, f32 *E, c64 *tau, c64 *hous2, const INT lhous2, c64 *work, const INT lwork, INT *info)#

CHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q1**H Q2**H * A * Q2 * Q1 = T.

Further Details:

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC ‘11), New York, NY, USA, Article 8 , 11 pages. https://doi.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC ‘13). Denver, Colorado, USA, 2013. Article 90, 12 pages. https://doi.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. https://doi.org/10.1177/1094342013502097

Parameters

in

vect

'N': No need for the Householder representation, in particular for the second stage (Band to tridiagonal) and thus lhous2 is of size max(1,4*n); 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then lhous2 is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).

in

uplo

'U': Upper triangle of A is stored; 'L': Lower triangle of A is stored.

in

n

The order of the matrix A. n>=0.

inout

A

On entry, the Hermitian matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if uplo='U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array tau, represent the unitary matrix Q1 as a product of elementary reflectors; if uplo='L', the diagonal and band subdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array tau, represent the unitary matrix Q1 as a product of elementary reflectors. See Further Details.

in

lda

The leading dimension of the array A. lda>=max(1,n).

out

D

Array of dimension (n). The diagonal elements of the tridiagonal matrix T.

out

E

Array of dimension (n-1). The off-diagonal elements of the tridiagonal matrix T.

out

tau

Array of dimension (n-kd). The scalar factors of the elementary reflectors of the first stage (see Further Details).

out

hous2

Array of dimension (max(1,lhous2)). Stores the Householder representation of the stage2 band to tridiagonal.

in

lhous2

The dimension of the array hous2. lhous2>=1. If lwork=-1, or lhous2=-1, then a query is assumed; the routine only calculates the optimal size of the hous2 array, returns this value as the first entry of the hous2 array. If vect='N', lhous2 = max(1,4*n); if vect='V', option not yet available.

out

work

Array of dimension (max(1,lwork)). On exit, if info=0, work[0] returns the optimal lwork.

in

lwork

The dimension of the array work. If n=0, lwork>=1, else lwork = max(1, dimension). If lwork=-1, or lhous2=-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.

lwork = MAX(1, dimension) where
dimension = max(stage1,stage2) + (kd+1)*n
          = n*kd + n*max(kd+1,FACTOPTNB)
            + max(2*kd*kd, kd*NTHREADS)
            + (kd+1)*n

where kd is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice, NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.

out

info

info=0: successful exit
info<0: if info=-i, the i-th argument had an illegal value

zhetrd_2stage

Functions

void zhetrd_2stage(
    const char* vect,
    const char* uplo,
    const INT   n,
          c128* A,
    const INT   lda,
          f64*  D,
          f64*  E,
          c128* tau,
          c128* hous2,
    const INT   lhous2,
          c128* work,
    const INT   lwork,
          INT*  info
);

void zhetrd_2stage(const char *vect, const char *uplo, const INT n, c128 *A, const INT lda, f64 *D, f64 *E, c128 *tau, c128 *hous2, const INT lhous2, c128 *work, const INT lwork, INT *info)#

ZHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q1**H Q2**H * A * Q2 * Q1 = T.

Further Details:

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC ‘11), New York, NY, USA, Article 8 , 11 pages. https://doi.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC ‘13). Denver, Colorado, USA, 2013. Article 90, 12 pages. https://doi.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. https://doi.org/10.1177/1094342013502097

Parameters

in

vect

'N': No need for the Householder representation, in particular for the second stage (Band to tridiagonal) and thus lhous2 is of size max(1,4*n); 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then lhous2 is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).

in

uplo

'U': Upper triangle of A is stored; 'L': Lower triangle of A is stored.

in

n

The order of the matrix A. n>=0.

inout

A

On entry, the Hermitian matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if uplo='U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array tau, represent the unitary matrix Q1 as a product of elementary reflectors; if uplo='L', the diagonal and band subdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array tau, represent the unitary matrix Q1 as a product of elementary reflectors. See Further Details.

in

lda

The leading dimension of the array A. lda>=max(1,n).

out

D

Array of dimension (n). The diagonal elements of the tridiagonal matrix T.

out

E

Array of dimension (n-1). The off-diagonal elements of the tridiagonal matrix T.

out

tau

Array of dimension (n-kd). The scalar factors of the elementary reflectors of the first stage (see Further Details).

out

hous2

Array of dimension (max(1,lhous2)). Stores the Householder representation of the stage2 band to tridiagonal.

in

lhous2

The dimension of the array hous2. lhous2>=1. If lwork=-1, or lhous2=-1, then a query is assumed; the routine only calculates the optimal size of the hous2 array, returns this value as the first entry of the hous2 array. If vect='N', lhous2 = max(1,4*n); if vect='V', option not yet available.

out

work

Array of dimension (max(1,lwork)). On exit, if info=0, work[0] returns the optimal lwork.

in

lwork

The dimension of the array work. If n=0, lwork>=1, else lwork = max(1, dimension). If lwork=-1, or lhous2=-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.

lwork = MAX(1, dimension) where
dimension = max(stage1,stage2) + (kd+1)*n
          = n*kd + n*max(kd+1,FACTOPTNB)
            + max(2*kd*kd, kd*NTHREADS)
            + (kd+1)*n

where kd is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice, NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.

out

info

info=0: successful exit
info<0: if info=-i, the i-th argument had an illegal value