slahr2.f (3)  Linux Manuals
NAME
slahr2.f 
SYNOPSIS
Functions/Subroutines
subroutine slahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
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.
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 nBY(nk+1) matrix A so that elements below the kth 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.
KK is INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N.
NBNB is INTEGER The number of columns to be reduced.
AA is REAL array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth 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.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
TAUTAU is REAL array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.
TT is REAL array, dimension (LDT,NB) The upper triangular matrix T.
LDTLDT is INTEGER The leading dimension of the array T. LDT >= NB.
YY is REAL array, dimension (LDY,NB) The nbynb matrix Y.
LDYLDY 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+k1) = 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 (nk+1)bynb 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 LAPACK3.0's DLAHRD incorporating improvements proposed by QuintanaOrti and Van de Gejin. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK3.0's DLAHRD routine. (This subroutine is not backward compatible with LAPACK3.0's DLAHRD.)
References:

Gregorio QuintanaOrti and Robert van de Geijn, 'Improving the
performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180194, June 2006.
Definition at line 182 of file slahr2.f.
Author
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