# clatrd.f - Man Page

## Synopsis

### Functions/Subroutines

subroutine clatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

## Function/Subroutine Documentation

### subroutine clatrd (character UPLO, integer N, integer NB, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) E, complex, dimension( * ) TAU, complex, dimension( ldw, * ) W, integer LDW)

CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Purpose:

``` CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
Hermitian tridiagonal form by a unitary similarity
transformation Q**H * A * Q, and returns the matrices V and W which are
needed to apply the transformation to the unreduced part of A.

If UPLO = 'U', CLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by CHETRD.```
Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular```

N

```          N is INTEGER
The order of the matrix A.```

NB

```          NB is INTEGER
The number of rows and columns to be reduced.```

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 last NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements above the diagonal
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors;
if UPLO = 'L', the first NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements below the diagonal
with the array TAU, represent the  unitary 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).```

E

```          E is REAL array, dimension (N-1)
If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
elements of the last NB columns of the reduced matrix;
if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
the first NB columns of the reduced matrix.```

TAU

```          TAU is COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors, stored in
TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
See Further Details.```

W

```          W is COMPLEX array, dimension (LDW,NB)
The n-by-nb matrix W required to update the unreduced part
of A.```

LDW

```          LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).```
Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

December 2016

Further Details:

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

Q = H(n) H(n-1) . . . H(n-nb+1).

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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
and tau in TAU(i-1).

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

Q = H(1) H(2) . . . H(nb).

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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V
which is needed, with W, to apply the transformation to the unreduced
part of the matrix, using a Hermitian rank-2k update of the form:
A := A - V*W**H - W*V**H.

The contents of A on exit are illustrated by the following examples
with n = 5 and nb = 2:

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

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes
an element of the original matrix that is unchanged, and vi denotes
an element of the vector defining H(i).```

Definition at line 201 of file clatrd.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page clatrd(3) is an alias of clatrd.f(3).

Tue Nov 14 2017 Version 3.8.0 LAPACK