# dsytrd_sy2sb.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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dsytrd_sy2sb.f

## SYNOPSIS

### Functions/Subroutines

subroutine dsytrd_sy2sb (UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO)
DSYTRD_SY2SB

## Function/Subroutine Documentation

### 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)

DSYTRD_SY2SB

Purpose:

``` 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.
```

Parameters:

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.
```

KD

```          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.
```

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 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.
```

LDA

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

AB

```          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).
```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.
```

TAU

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

WORK

```          WORK is DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, or if LWORK=-1,
WORK(1) returns the size of LWORK.
```

LWORK

```          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.
```

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
```

```  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

Definition at line 245 of file dsytrd_sy2sb.f.
```

## Author

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