LAPACK
3.6.1
LAPACK: Linear Algebra PACKage
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subroutine dlahrd | ( | integer | N, |
integer | K, | ||
integer | NB, | ||
double precision, dimension( lda, * ) | A, | ||
integer | LDA, | ||
double precision, dimension( nb ) | TAU, | ||
double precision, dimension( ldt, nb ) | T, | ||
integer | LDT, | ||
double precision, dimension( ldy, nb ) | Y, | ||
integer | LDY | ||
) |
DLAHRD 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.
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This routine is deprecated and has been replaced by routine DLAHR2. DLAHRD 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.
[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. |
[in] | NB | NB is INTEGER The number of columns to be reduced. |
[in,out] | 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. |
[in] | LDA | LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). |
[out] | TAU | TAU is DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. |
[out] | T | T is DOUBLE PRECISION 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 DOUBLE PRECISION 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**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 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 169 of file dlahrd.f.