---
name: "slahr2"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/slahr2/3lapack"
---

# slahr2(3lapack)

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

## NAME

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

## SYNOPSIS

- SUBROUTINE SLAHR2(

  N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

- INTEGER

  K, LDA, LDT, LDY, N, NB

- REAL

  A( LDA, \* ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

## 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' \* A \* Q. The routine returns the matrices V and T which determine Q as a block reflector I - V\*T\*V', and also the matrix Y = A \* V \* T. This is an auxiliary routine called by SGEHRD.\

## ARGUMENTS

- N (input) INTEGER

  The order of the matrix A.

- K (input) INTEGER

  The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. K \< N.

- NB (input) INTEGER

  The number of columns to be reduced.

- A (input/output) 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     (input) INTEGER The leading dimension of the array A.  LDA \>= max(1,N).

- TAU (output) REAL array, dimension (NB)

  The scalar factors of the elementary reflectors. See Further Details.

- T (output) REAL array, dimension (LDT,NB)

  The upper triangular matrix T.

- LDT (input) INTEGER

  The leading dimension of the array T.  LDT \>= NB.

- Y (output) REAL array, dimension (LDY,NB)

  The n-by-nb matrix Y.

- LDY (input) INTEGER

  The leading dimension of the array Y. LDY \>= N.

## 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'\
 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') \* (A - Y\*V').\
 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 file is a slight modification of LAPACK-3.0's SLAHRD 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 routine. This function is not backward compatible with LAPACK3.0.\