.TH "hetrd_hb2st" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
hetrd_hb2st \- {he,sy}trd_hb2st: band to tridiagonal (2nd stage)
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBchetrd_hb2st\fP (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)"
.br
.RI "\fBCHETRD_HB2ST\fP reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T "
.ti -1c
.RI "subroutine \fBdsytrd_sb2st\fP (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)"
.br
.RI "\fBDSYTRD_SB2ST\fP reduces a real symmetric band matrix A to real symmetric tridiagonal form T "
.ti -1c
.RI "subroutine \fBssytrd_sb2st\fP (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)"
.br
.RI "\fBSSYTRD_SB2ST\fP reduces a real symmetric band matrix A to real symmetric tridiagonal form T "
.ti -1c
.RI "subroutine \fBzhetrd_hb2st\fP (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)"
.br
.RI "\fBZHETRD_HB2ST\fP reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine chetrd_hb2st (character stage1, character vect, character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) hous, integer lhous, complex, dimension( * ) work, integer lwork, integer info)"

.PP
\fBCHETRD_HB2ST\fP reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric
 tridiagonal form T by a unitary similarity transformation:
 Q**H * A * Q = T\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISTAGE1\fP 
.PP
.nf
          STAGE1 is CHARACTER*1
          = 'N':  'No': to mention that the stage 1 of the reduction
                  from dense to band using the chetrd_he2hb routine
                  was not called before this routine to reproduce AB\&.
                  In other term this routine is called as standalone\&.
          = 'Y':  'Yes': to mention that the stage 1 of the
                  reduction from dense to band using the chetrd_he2hb
                  routine has been called to produce AB (e\&.g\&., AB is
                  the output of chetrd_he2hb\&.
.fi
.PP
.br
\fIVECT\fP 
.PP
.nf
          VECT is CHARACTER*1
          = 'N':  No need for the Housholder representation,
                  and thus LHOUS is of size max(1, 4*N);
          = 'V':  the Householder representation is needed to
                  either generate or to apply Q later on,
                  then LHOUS is to be queried and computed\&.
                  (NOT AVAILABLE IN THIS RELEASE)\&.
.fi
.PP
.br
\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 matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KD >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is COMPLEX array, dimension (LDAB,N)
          On entry, 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)\&.
          On exit, the diagonal elements of AB are overwritten by the
          diagonal elements of the tridiagonal matrix T; if KD > 0, the
          elements on the first superdiagonal (if UPLO = 'U') or the
          first subdiagonal (if UPLO = 'L') are overwritten by the
          off-diagonal elements of T; the rest of AB is overwritten by
          values generated during the reduction\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'\&.
.fi
.PP
.br
\fIHOUS\fP 
.PP
.nf
          HOUS is COMPLEX array, dimension LHOUS, that
          store the Householder representation\&.
.fi
.PP
.br
\fILHOUS\fP 
.PP
.nf
          LHOUS is INTEGER
          The dimension of the array HOUS\&. LHOUS = MAX(1, dimension)
          If LWORK = -1, or LHOUS=-1,
          then a query is assumed; the routine
          only calculates the optimal size of the HOUS array, returns
          this value as the first entry of the HOUS array, and no error
          message related to LHOUS is issued by XERBLA\&.
          LHOUS = MAX(1, dimension) where
          dimension = 4*N if VECT='N'
          not available now if VECT='H'
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&. LWORK = MAX(1, dimension)
          If LWORK = -1, or LHOUS=-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   = (2KD+1)*N + KD*NTHREADS
          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\&.
.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

.SS "subroutine dsytrd_sb2st (character stage1, character vect, character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) hous, integer lhous, double precision, dimension( * ) work, integer lwork, integer info)"

.PP
\fBDSYTRD_SB2ST\fP reduces a real symmetric band matrix A to real symmetric tridiagonal form T  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric
 tridiagonal form T by a orthogonal similarity transformation:
 Q**T * A * Q = T\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISTAGE1\fP 
.PP
.nf
          STAGE1 is CHARACTER*1
          = 'N':  'No': to mention that the stage 1 of the reduction
                  from dense to band using the dsytrd_sy2sb routine
                  was not called before this routine to reproduce AB\&.
                  In other term this routine is called as standalone\&.
          = 'Y':  'Yes': to mention that the stage 1 of the
                  reduction from dense to band using the dsytrd_sy2sb
                  routine has been called to produce AB (e\&.g\&., AB is
                  the output of dsytrd_sy2sb\&.
.fi
.PP
.br
\fIVECT\fP 
.PP
.nf
          VECT is CHARACTER*1
          = 'N':  No need for the Housholder representation,
                  and thus LHOUS is of size max(1, 4*N);
          = 'V':  the Householder representation is needed to
                  either generate or to apply Q later on,
                  then LHOUS is to be queried and computed\&.
                  (NOT AVAILABLE IN THIS RELEASE)\&.
.fi
.PP
.br
\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 matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KD >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, 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)\&.
          On exit, the diagonal elements of AB are overwritten by the
          diagonal elements of the tridiagonal matrix T; if KD > 0, the
          elements on the first superdiagonal (if UPLO = 'U') or the
          first subdiagonal (if UPLO = 'L') are overwritten by the
          off-diagonal elements of T; the rest of AB is overwritten by
          values generated during the reduction\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'\&.
.fi
.PP
.br
\fIHOUS\fP 
.PP
.nf
          HOUS is DOUBLE PRECISION array, dimension LHOUS, that
          store the Householder representation\&.
.fi
.PP
.br
\fILHOUS\fP 
.PP
.nf
          LHOUS is INTEGER
          The dimension of the array HOUS\&. LHOUS = MAX(1, dimension)
          If LWORK = -1, or LHOUS=-1,
          then a query is assumed; the routine
          only calculates the optimal size of the HOUS array, returns
          this value as the first entry of the HOUS array, and no error
          message related to LHOUS is issued by XERBLA\&.
          LHOUS = MAX(1, dimension) where
          dimension = 4*N if VECT='N'
          not available now if VECT='H'
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&. LWORK = MAX(1, dimension)
          If LWORK = -1, or LHOUS=-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   = (2KD+1)*N + KD*NTHREADS
          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\&.
.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

.SS "subroutine ssytrd_sb2st (character stage1, character vect, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) hous, integer lhous, real, dimension( * ) work, integer lwork, integer info)"

.PP
\fBSSYTRD_SB2ST\fP reduces a real symmetric band matrix A to real symmetric tridiagonal form T  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric
 tridiagonal form T by a orthogonal similarity transformation:
 Q**T * A * Q = T\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISTAGE1\fP 
.PP
.nf
          STAGE1 is CHARACTER*1
          = 'N':  'No': to mention that the stage 1 of the reduction
                  from dense to band using the ssytrd_sy2sb routine
                  was not called before this routine to reproduce AB\&.
                  In other term this routine is called as standalone\&.
          = 'Y':  'Yes': to mention that the stage 1 of the
                  reduction from dense to band using the ssytrd_sy2sb
                  routine has been called to produce AB (e\&.g\&., AB is
                  the output of ssytrd_sy2sb\&.
.fi
.PP
.br
\fIVECT\fP 
.PP
.nf
          VECT is CHARACTER*1
          = 'N':  No need for the Housholder representation,
                  and thus LHOUS is of size max(1, 4*N);
          = 'V':  the Householder representation is needed to
                  either generate or to apply Q later on,
                  then LHOUS is to be queried and computed\&.
                  (NOT AVAILABLE IN THIS RELEASE)\&.
.fi
.PP
.br
\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 matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KD >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is REAL array, dimension (LDAB,N)
          On entry, 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)\&.
          On exit, the diagonal elements of AB are overwritten by the
          diagonal elements of the tridiagonal matrix T; if KD > 0, the
          elements on the first superdiagonal (if UPLO = 'U') or the
          first subdiagonal (if UPLO = 'L') are overwritten by the
          off-diagonal elements of T; the rest of AB is overwritten by
          values generated during the reduction\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'\&.
.fi
.PP
.br
\fIHOUS\fP 
.PP
.nf
          HOUS is REAL array, dimension LHOUS, that
          store the Householder representation\&.
.fi
.PP
.br
\fILHOUS\fP 
.PP
.nf
          LHOUS is INTEGER
          The dimension of the array HOUS\&. LHOUS = MAX(1, dimension)
          If LWORK = -1, or LHOUS=-1,
          then a query is assumed; the routine
          only calculates the optimal size of the HOUS array, returns
          this value as the first entry of the HOUS array, and no error
          message related to LHOUS is issued by XERBLA\&.
          LHOUS = MAX(1, dimension) where
          dimension = 4*N if VECT='N'
          not available now if VECT='H'
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&. LWORK = MAX(1, dimension)
          If LWORK = -1, or LHOUS=-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   = (2KD+1)*N + KD*NTHREADS
          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\&.
.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

.SS "subroutine zhetrd_hb2st (character stage1, character vect, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) hous, integer lhous, complex*16, dimension( * ) work, integer lwork, integer info)"

.PP
\fBZHETRD_HB2ST\fP reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric
 tridiagonal form T by a unitary similarity transformation:
 Q**H * A * Q = T\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISTAGE1\fP 
.PP
.nf
          STAGE1 is CHARACTER*1
          = 'N':  'No': to mention that the stage 1 of the reduction
                  from dense to band using the zhetrd_he2hb routine
                  was not called before this routine to reproduce AB\&.
                  In other term this routine is called as standalone\&.
          = 'Y':  'Yes': to mention that the stage 1 of the
                  reduction from dense to band using the zhetrd_he2hb
                  routine has been called to produce AB (e\&.g\&., AB is
                  the output of zhetrd_he2hb\&.
.fi
.PP
.br
\fIVECT\fP 
.PP
.nf
          VECT is CHARACTER*1
          = 'N':  No need for the Housholder representation,
                  and thus LHOUS is of size max(1, 4*N);
          = 'V':  the Householder representation is needed to
                  either generate or to apply Q later on,
                  then LHOUS is to be queried and computed\&.
                  (NOT AVAILABLE IN THIS RELEASE)\&.
.fi
.PP
.br
\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 matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KD >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is COMPLEX*16 array, dimension (LDAB,N)
          On entry, 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)\&.
          On exit, the diagonal elements of AB are overwritten by the
          diagonal elements of the tridiagonal matrix T; if KD > 0, the
          elements on the first superdiagonal (if UPLO = 'U') or the
          first subdiagonal (if UPLO = 'L') are overwritten by the
          off-diagonal elements of T; the rest of AB is overwritten by
          values generated during the reduction\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'\&.
.fi
.PP
.br
\fIHOUS\fP 
.PP
.nf
          HOUS is COMPLEX*16 array, dimension LHOUS, that
          store the Householder representation\&.
.fi
.PP
.br
\fILHOUS\fP 
.PP
.nf
          LHOUS is INTEGER
          The dimension of the array HOUS\&. LHOUS = MAX(1, dimension)
          If LWORK = -1, or LHOUS=-1,
          then a query is assumed; the routine
          only calculates the optimal size of the HOUS array, returns
          this value as the first entry of the HOUS array, and no error
          message related to LHOUS is issued by XERBLA\&.
          LHOUS = MAX(1, dimension) where
          dimension = 4*N if VECT='N'
          not available now if VECT='H'
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&. LWORK = MAX(1, dimension)
          If LWORK = -1, or LHOUS=-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   = (2KD+1)*N + KD*NTHREADS
          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\&.
.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

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.