# zlahrd.f (3) - Linux Manuals

zlahrd.f -

## SYNOPSIS

### Functions/Subroutines

subroutine zlahrd (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
ZLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

## Function/Subroutine Documentation

### subroutine zlahrd (integerN, integerK, integerNB, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( nb )TAU, complex*16, dimension( ldt, nb )T, integerLDT, complex*16, dimension( ldy, nb )Y, integerLDY)

ZLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Purpose:

``` ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1)
matrix A so that elements below the k-th subdiagonal are zero. The
reduction is performed by a unitary similarity transformation
Q**H * A * Q. The routine returns the matrices V and T which determine
Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.

This is an OBSOLETE auxiliary routine.
This routine will be 'deprecated' in a  future release.
Please use the new routine ZLAHR2 instead.
```

Parameters:

N

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

K

```          K is INTEGER
The offset for the reduction. Elements below the k-th
subdiagonal in the first NB columns are reduced to zero.
```

NB

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

A

```          A is COMPLEX*16 array, dimension (LDA,N-K+1)
On entry, the n-by-(n-k+1) general matrix A.
On exit, the elements on and above the k-th subdiagonal in
the first NB columns are overwritten with the corresponding
elements of the reduced matrix; the elements below the k-th
subdiagonal, with the array TAU, represent the matrix Q as a
product of elementary reflectors. The other columns of A are
unchanged. See Further Details.
```

LDA

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

TAU

```          TAU is COMPLEX*16 array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.
```

T

```          T is COMPLEX*16 array, dimension (LDT,NB)
The upper triangular matrix T.
```

LDT

```          LDT is INTEGER
The leading dimension of the array T.  LDT >= NB.
```

Y

```          Y is COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y.
```

LDY

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Further Details:

```  The matrix Q is represented as a product of nb 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+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
A(i+k+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the (n-k+1)-by-nb matrix
V which is needed, with T and Y, to apply the transformation to the
unreduced part of the matrix, using an update of the form:
A := (I - V*T*V**H) * (A - Y*V**H).

The contents of A on exit are illustrated by the following example
with n = 7, k = 3 and nb = 2:

( a   h   a   a   a )
( a   h   a   a   a )
( a   h   a   a   a )
( h   h   a   a   a )
( v1  h   a   a   a )
( v1  v2  a   a   a )
( v1  v2  a   a   a )

where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).
```

Definition at line 170 of file zlahrd.f.

## Author

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