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zlahrd.f(3) LAPACK zlahrd.f(3)

zlahrd.f -


subroutine zlahrd (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
 
ZLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

ZLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
Purpose:
 ZLAHRD 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 a 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 OBSOLETE auxiliary routine. This routine will be 'deprecated' in a future release. Please use the new routine ZLAHR2 instead.
Parameters:
N
          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.
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 >= max(1,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+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 h a a a ) ( a h a a a ) ( a h 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).
Definition at line 170 of file zlahrd.f.

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