subroutine zlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
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 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 ZGEHRD.
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*16 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*16 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.T
T is COMPLEX*16 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*16 array, dimension (LDY,NB) The n-by-nb matrix Y.LDY
LDY is INTEGER The leading dimension of the array Y. LDY >= N.
Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:
September 2012Further 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.)
Gregorio Quintana-Orti and Robert van de Geijn, 'Improving theDefinition at line 182 of file zlahr2.f.
performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.