zlahr2 (3)  Linux Man Pages
NAME
zlahr2.f 
SYNOPSIS
Functions/Subroutines
subroutine zlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
ZLAHR2 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 zlahr2 (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)
ZLAHR2 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:

ZLAHR2 reduces the first NB columns of A complex general nBY(nk+1) matrix A so that elements below the kth 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 ZGEHRD.
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 COMPLEX*16 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 COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.
TT is COMPLEX*16 array, dimension (LDT,NB) The upper triangular matrix T.
LDTLDT is INTEGER The leading dimension of the array T. LDT >= NB.
YY is COMPLEX*16 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**H where tau is a complex scalar, and v is a complex 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**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 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 zlahr2.f.
Author
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