hetrd_2stage - Man Page

{he,sy}trd_2stage: reduction to tridiagonal, 2-stage

Synopsis

Functions

subroutine chetrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
CHETRD_2STAGE
subroutine dsytrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
DSYTRD_2STAGE
subroutine ssytrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
SSYTRD_2STAGE
subroutine zhetrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
ZHETRD_2STAGE

Detailed Description

Function Documentation

subroutine chetrd_2stage (character vect, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tau, complex, dimension( * ) hous2, integer lhous2, complex, dimension( * ) work, integer lwork, integer info)

CHETRD_2STAGE  

Purpose:

 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.
Parameters

VECT

          VECT is CHARACTER*1
          = 'N':  No need for the Housholder 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).

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          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 over-
          written 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.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

D

          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

E

          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T.

TAU

          TAU is COMPLEX array, dimension (N-KD)
          The scalar factors of the elementary reflectors of
          the first stage (see Further Details).

HOUS2

          HOUS2 is COMPLEX array, dimension (LHOUS2)
          Stores the Householder representation of the stage2
          band to tridiagonal.

LHOUS2

          LHOUS2 is INTEGER
          The dimension of the array HOUS2.
          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, and no error
          message related to LHOUS2 is issued by XERBLA.
          If VECT='N', LHOUS2 = max(1, 4*n);
          if VECT='V', option not yet available.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. 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, and no error
          message related to LWORK is issued by XERBLA.
          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.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.
  http://doi.acm.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.
  http://doi.acm.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.
  http://hpc.sagepub.com/content/28/2/196

Definition at line 222 of file chetrd_2stage.f.

subroutine dsytrd_2stage (character vect, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tau, double precision, dimension( * ) hous2, integer lhous2, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRD_2STAGE  

Purpose:

 DSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric
 tridiagonal form T by a orthogonal similarity transformation:
 Q1**T Q2**T* A * Q2 * Q1 = T.
Parameters

VECT

          VECT is CHARACTER*1
          = 'N':  No need for the Housholder 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).

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric 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 orthogonal
          matrix Q1 as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and band subdiagonal of A are over-
          written by the corresponding elements of the internal band-diagonal
          matrix AB, and the elements below the KD subdiagonal, with
          the array TAU, represent the orthogonal matrix Q1 as a product
          of elementary reflectors. See Further Details.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

D

          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

E

          E is DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T.

TAU

          TAU is DOUBLE PRECISION array, dimension (N-KD)
          The scalar factors of the elementary reflectors of
          the first stage (see Further Details).

HOUS2

          HOUS2 is DOUBLE PRECISION array, dimension (LHOUS2)
          Stores the Householder representation of the stage2
          band to tridiagonal.

LHOUS2

          LHOUS2 is INTEGER
          The dimension of the array HOUS2.
          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, and no error
          message related to LHOUS2 is issued by XERBLA.
          If VECT='N', LHOUS2 = max(1, 4*n);
          if VECT='V', option not yet available.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. 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, and no error
          message related to LWORK is issued by XERBLA.
          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.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.
  http://doi.acm.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.
  http://doi.acm.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.
  http://hpc.sagepub.com/content/28/2/196

Definition at line 222 of file dsytrd_2stage.f.

subroutine ssytrd_2stage (character vect, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, real, dimension( * ) hous2, integer lhous2, real, dimension( * ) work, integer lwork, integer info)

SSYTRD_2STAGE  

Purpose:

 SSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric
 tridiagonal form T by a orthogonal similarity transformation:
 Q1**T Q2**T* A * Q2 * Q1 = T.
Parameters

VECT

          VECT is CHARACTER*1
          = 'N':  No need for the Housholder 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).

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is REAL array, dimension (LDA,N)
          On entry, the symmetric 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 orthogonal
          matrix Q1 as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and band subdiagonal of A are over-
          written by the corresponding elements of the internal band-diagonal
          matrix AB, and the elements below the KD subdiagonal, with
          the array TAU, represent the orthogonal matrix Q1 as a product
          of elementary reflectors. See Further Details.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

D

          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

E

          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T.

TAU

          TAU is REAL array, dimension (N-KD)
          The scalar factors of the elementary reflectors of
          the first stage (see Further Details).

HOUS2

          HOUS2 is REAL array, dimension (LHOUS2)
          Stores the Householder representation of the stage2
          band to tridiagonal.

LHOUS2

          LHOUS2 is INTEGER
          The dimension of the array HOUS2.
          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, and no error
          message related to LHOUS2 is issued by XERBLA.
          If VECT='N', LHOUS2 = max(1, 4*n);
          if VECT='V', option not yet available.

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. 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, and no error
          message related to LWORK is issued by XERBLA.
          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.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.
  http://doi.acm.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.
  http://doi.acm.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.
  http://hpc.sagepub.com/content/28/2/196

Definition at line 222 of file ssytrd_2stage.f.

subroutine zhetrd_2stage (character vect, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, complex*16, dimension( * ) hous2, integer lhous2, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD_2STAGE  

Purpose:

 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.
Parameters

VECT

          VECT is CHARACTER*1
          = 'N':  No need for the Housholder 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).

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX*16 array, dimension (LDA,N)
          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 over-
          written 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.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

D

          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

E

          E is DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T.

TAU

          TAU is COMPLEX*16 array, dimension (N-KD)
          The scalar factors of the elementary reflectors of
          the first stage (see Further Details).

HOUS2

          HOUS2 is COMPLEX*16 array, dimension (LHOUS2)
          Stores the Householder representation of the stage2
          band to tridiagonal.

LHOUS2

          LHOUS2 is INTEGER
          The dimension of the array HOUS2.
          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, and no error
          message related to LHOUS2 is issued by XERBLA.
          If VECT='N', LHOUS2 = max(1, 4*n);
          if VECT='V', option not yet available.

WORK

          WORK is COMPLEX*16 array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. 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, and no error
          message related to LWORK is issued by XERBLA.
          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.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.
  http://doi.acm.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.
  http://doi.acm.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.
  http://hpc.sagepub.com/content/28/2/196

Definition at line 222 of file zhetrd_2stage.f.

Author

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