.TH "hetrd_he2hb" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
hetrd_he2hb \- {he,sy}trd_he2hb: full to band (1st stage)
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBchetrd_he2hb\fP (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)"
.br
.RI "\fBCHETRD_HE2HB\fP "
.ti -1c
.RI "subroutine \fBdsytrd_sy2sb\fP (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)"
.br
.RI "\fBDSYTRD_SY2SB\fP "
.ti -1c
.RI "subroutine \fBssytrd_sy2sb\fP (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)"
.br
.RI "\fBSSYTRD_SY2SB\fP "
.ti -1c
.RI "subroutine \fBzhetrd_he2hb\fP (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)"
.br
.RI "\fBZHETRD_HE2HB\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine chetrd_he2hb (character uplo, integer n, integer kd, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)"

.PP
\fBCHETRD_HE2HB\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian
 band-diagonal form AB by a unitary similarity transformation:
 Q**H * A * Q = AB\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIKD\fP 
.PP
.nf
          KD is INTEGER
          The number of superdiagonals of the reduced matrix if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KD >= 0\&.
          The reduced matrix is stored in the array AB\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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 diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the unitary
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the unitary matrix Q as a product
          of elementary reflectors\&. See Further Details\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is COMPLEX array, dimension (LDAB,N)
          On exit, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first KD+1 rows of the array\&.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd)\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          TAU is COMPLEX array, dimension (N-KD)
          The scalar factors of the elementary reflectors (see Further
          Details)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension (LWORK)
          On exit, if INFO = 0, or if LWORK=-1, 
          WORK(1) returns the size of LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK which should be calculated
          by a workspace query\&. LWORK = MAX(1, LWORK_QUERY)
          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\&.
          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
          where FACTOPTNB is the blocking used by the QR or LQ
          algorithm, usually FACTOPTNB=128 is a good choice otherwise
          putting LWORK=-1 will provide the size of WORK\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  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 
.fi
.PP
.RE
.PP
.PP
.nf
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(k)**H \&. \&. \&. H(2)**H H(1)**H, where k = n-kd\&.

  Each H(i) has the form

     H(i) = I - tau * v * v**H

  where tau is a complex scalar, and v is a complex vector with
  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
  A(i,i+kd+1:n), and tau in TAU(i)\&.

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) \&. \&. \&. H(k), where k = n-kd\&.

  Each H(i) has the form

     H(i) = I - tau * v * v**H

  where tau is a complex scalar, and v is a complex vector with
  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
  A(i+kd+2:n,i), and tau in TAU(i)\&.

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
    (                            ab    )              (  v1     v2     v3     ab/v4 ab )

  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i)\&..fi
.PP
 
.SS "subroutine dsytrd_sy2sb (character uplo, integer n, integer kd, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)"

.PP
\fBDSYTRD_SY2SB\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric
 band-diagonal form AB by a orthogonal similarity transformation:
 Q**T * A * Q = AB\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIKD\fP 
.PP
.nf
          KD is INTEGER
          The number of superdiagonals of the reduced matrix if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KD >= 0\&.
          The reduced matrix is stored in the array AB\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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 diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors\&. See Further Details\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On exit, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array\&.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd)\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          TAU is DOUBLE PRECISION array, dimension (N-KD)
          The scalar factors of the elementary reflectors (see Further
          Details)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, or if LWORK=-1, 
          WORK(1) returns the size of LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK which should be calculated
          by a workspace query\&. LWORK = MAX(1, LWORK_QUERY)
          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\&.
          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
          where FACTOPTNB is the blocking used by the QR or LQ
          algorithm, usually FACTOPTNB=128 is a good choice otherwise
          putting LWORK=-1 will provide the size of WORK\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  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 
.fi
.PP
.RE
.PP
.PP
.nf
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(k)**T \&. \&. \&. H(2)**T H(1)**T, where k = n-kd\&.

  Each H(i) has the form

     H(i) = I - tau * v * v**T

  where tau is a real scalar, and v is a real vector with
  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
  A(i,i+kd+1:n), and tau in TAU(i)\&.

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) \&. \&. \&. H(k), where k = n-kd\&.

  Each H(i) has the form

     H(i) = I - tau * v * v**T

  where tau is a real scalar, and v is a real vector with
  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
  A(i+kd+2:n,i), and tau in TAU(i)\&.

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
    (                            ab    )              (  v1     v2     v3     ab/v4 ab )

  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i)\&..fi
.PP
 
.SS "subroutine ssytrd_sy2sb (character uplo, integer n, integer kd, real, dimension( lda, * ) a, integer lda, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)"

.PP
\fBSSYTRD_SY2SB\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric
 band-diagonal form AB by a orthogonal similarity transformation:
 Q**T * A * Q = AB\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIKD\fP 
.PP
.nf
          KD is INTEGER
          The number of superdiagonals of the reduced matrix if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KD >= 0\&.
          The reduced matrix is stored in the array AB\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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 diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors\&. See Further Details\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is REAL array, dimension (LDAB,N)
          On exit, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array\&.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd)\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          TAU is REAL array, dimension (N-KD)
          The scalar factors of the elementary reflectors (see Further
          Details)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (LWORK)
          On exit, if INFO = 0, or if LWORK=-1, 
          WORK(1) returns the size of LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK which should be calculated
          by a workspace query\&. LWORK = MAX(1, LWORK_QUERY)
          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\&.
          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
          where FACTOPTNB is the blocking used by the QR or LQ
          algorithm, usually FACTOPTNB=128 is a good choice otherwise
          putting LWORK=-1 will provide the size of WORK\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  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 
.fi
.PP
.RE
.PP
.PP
.nf
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(k)**T \&. \&. \&. H(2)**T H(1)**T, where k = n-kd\&.

  Each H(i) has the form

     H(i) = I - tau * v * v**T

  where tau is a real scalar, and v is a real vector with
  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
  A(i,i+kd+1:n), and tau in TAU(i)\&.

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) \&. \&. \&. H(k), where k = n-kd\&.

  Each H(i) has the form

     H(i) = I - tau * v * v**T

  where tau is a real scalar, and v is a real vector with
  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
  A(i+kd+2:n,i), and tau in TAU(i)\&.

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
    (                            ab    )              (  v1     v2     v3     ab/v4 ab )

  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i)\&..fi
.PP
 
.SS "subroutine zhetrd_he2hb (character uplo, integer n, integer kd, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)"

.PP
\fBZHETRD_HE2HB\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian
 band-diagonal form AB by a unitary similarity transformation:
 Q**H * A * Q = AB\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIKD\fP 
.PP
.nf
          KD is INTEGER
          The number of superdiagonals of the reduced matrix if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KD >= 0\&.
          The reduced matrix is stored in the array AB\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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 diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the unitary
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the unitary matrix Q as a product
          of elementary reflectors\&. See Further Details\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is COMPLEX*16 array, dimension (LDAB,N)
          On exit, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first KD+1 rows of the array\&.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd)\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          TAU is COMPLEX*16 array, dimension (N-KD)
          The scalar factors of the elementary reflectors (see Further
          Details)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (LWORK)
          On exit, if INFO = 0, or if LWORK=-1, 
          WORK(1) returns the size of LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK which should be calculated
          by a workspace query\&. LWORK = MAX(1, LWORK_QUERY)
          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\&.
          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
          where FACTOPTNB is the blocking used by the QR or LQ
          algorithm, usually FACTOPTNB=128 is a good choice otherwise
          putting LWORK=-1 will provide the size of WORK\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  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 
.fi
.PP
.RE
.PP
.PP
.nf
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(k)**H \&. \&. \&. H(2)**H H(1)**H, where k = n-kd\&.

  Each H(i) has the form

     H(i) = I - tau * v * v**H

  where tau is a complex scalar, and v is a complex vector with
  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
  A(i,i+kd+1:n), and tau in TAU(i)\&.

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) \&. \&. \&. H(k), where k = n-kd\&.

  Each H(i) has the form

     H(i) = I - tau * v * v**H

  where tau is a complex scalar, and v is a complex vector with
  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
  A(i+kd+2:n,i), and tau in TAU(i)\&.

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
    (                            ab    )              (  v1     v2     v3     ab/v4 ab )

  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i)\&..fi
.PP
 
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.