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LAPACK
3.4.2
LAPACK: Linear Algebra PACKage
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Go to the source code of this file.
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. | |
| subroutine zlahr2 | ( | integer | N, |
| integer | K, | ||
| integer | NB, | ||
| complex*16, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| complex*16, dimension( nb ) | TAU, | ||
| complex*16, dimension( ldt, nb ) | T, | ||
| integer | LDT, | ||
| complex*16, dimension( ldy, nb ) | Y, | ||
| integer | 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.
Download ZLAHR2 + dependencies [TGZ] [ZIP] [TXT]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.
| [in] | N | N is INTEGER
The order of the matrix A. |
| [in] | 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. |
| [in] | NB | NB is INTEGER
The number of columns to be reduced. |
| [in,out] | 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. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | TAU | TAU is COMPLEX*16 array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details. |
| [out] | T | T is COMPLEX*16 array, dimension (LDT,NB)
The upper triangular matrix T. |
| [in] | LDT | LDT is INTEGER
The leading dimension of the array T. LDT >= NB. |
| [out] | Y | Y is COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y. |
| [in] | LDY | LDY is INTEGER
The leading dimension of the array Y. LDY >= N. |
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.) Definition at line 182 of file zlahr2.f.
1.8.1.1