subroutine dlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
DLAHR2 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.
DLAHR2 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:
DLAHR2 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 DGEHRD.
Parameters:
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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.
T
T is DOUBLE PRECISION 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 DOUBLE PRECISION 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 California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
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:
Definition at line 183 of file dlahr2.f.
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