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LAPACK
3.7.0
LAPACK: Linear Algebra PACKage
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subroutine clahr2 | ( | integer | N, |
integer | K, | ||
integer | NB, | ||
complex, dimension( lda, * ) | A, | ||
integer | LDA, | ||
complex, dimension( nb ) | TAU, | ||
complex, dimension( ldt, nb ) | T, | ||
integer | LDT, | ||
complex, dimension( ldy, nb ) | Y, | ||
integer | LDY | ||
) |
CLAHR2 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.
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CLAHR2 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 CGEHRD.
[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 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 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. |
[out] | T | T is COMPLEX 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 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 183 of file clahr2.f.