# chetrd_2stage.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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## NAME

chetrd_2stage.f

## SYNOPSIS

### Functions/Subroutines

subroutine chetrd_2stage (VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
CHETRD_2STAGE

## Function/Subroutine 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, that
store the Householder representation of the stage2
band to tridiagonal.
```

LHOUS2

```          LHOUS2 is INTEGER
The dimension of the array HOUS2. LHOUS2 = MAX(1, dimension)
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.
LHOUS2 = MAX(1, dimension) where
dimension = 4*N if VECT='N'
not available now if VECT='H'
```

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)
+ (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
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

NAG Ltd.

Date:

November 2017

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).
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 227 of file chetrd_2stage.f.

## Author

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