hetrd_2stage(3)

NAME
hetrd_2stage - {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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