# clahr2.f man page

clahr2.f

## Synopsis

### Functions/Subroutines

subroutine clahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

## Function/Subroutine Documentation

### subroutine clahr2 (integer N, integer K, integer NB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( nb ) TAU, complex, dimension( ldt, nb ) T, integer LDT, complex, dimension( ldy, nb ) Y, integer LDY)

CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Purpose:

``` CLAHR2 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 an 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 auxiliary routine called by CGEHRD.```
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.
K < N.```

NB

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

A

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

T

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

LDY

```          LDY is INTEGER
The leading dimension of the array Y. LDY >= N.```
Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

December 2016

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   a   a   a   a )
( a   a   a   a   a )
( a   a   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).

This subroutine is a slight modification of LAPACK-3.0's DLAHRD
incorporating improvements proposed by Quintana-Orti and Van de
Gejin. Note that the entries of A(1:K,2:NB) differ from those
returned by the original LAPACK-3.0's DLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)```
References:

Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

Definition at line 183 of file clahr2.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page clahr2(3) is an alias of clahr2.f(3).

Tue Nov 14 2017 Version 3.8.0 LAPACK