# slahr2.f man page

slahr2.f —

## Synopsis

### Functions/Subroutines

subroutineslahr2(N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)SLAHR2reduces 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 slahr2 (integerN, integerK, integerNB, real, dimension( lda, * )A, integerLDA, real, dimension( nb )TAU, real, dimension( ldt, nb )T, integerLDT, real, dimension( ldy, nb )Y, integerLDY)

**SLAHR2** 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:**

```
SLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
matrix A so that elements below the k-th subdiagonal are zero. The
reduction is performed by an orthogonal similarity transformation
Q**T * A * Q. The routine returns the matrices V and T which determine
Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
This is an auxiliary routine called by SGEHRD.
```

**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 REAL 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 REAL array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.
```

*T*

```
T is REAL 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 REAL 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

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**T
where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
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 182 of file slahr2.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

slahr2(3) is an alias of slahr2.f(3).