Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - 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

SYNOPSIS

       SUBROUTINE CLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

           INTEGER        K, LDA, LDT, LDY, N, NB

           COMPLEX        A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

PURPOSE

       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.

ARGUMENTS

        N       (input) INTEGER
                The order of the matrix A.

        K       (input) INTEGER
                The offset for the reduction. Elements below the k-th
                subdiagonal in the first NB columns are reduced to zero.
                K < N.

        NB      (input) INTEGER
                The number of columns to be reduced.

        A       (input/output) 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.
                LDA     (input) INTEGER
                The leading dimension of the array A.  LDA >= max(1,N).

        TAU     (output) COMPLEX array, dimension (NB)
                The scalar factors of the elementary reflectors. See Further
                Details.

        T       (output) COMPLEX array, dimension (LDT,NB)
                The upper triangular matrix T.

        LDT     (input) INTEGER
                The leading dimension of the array T.  LDT >= NB.

        Y       (output) COMPLEX array, dimension (LDY,NB)
                The n-by-nb matrix Y.

        LDY     (input) INTEGER
                The leading dimension of the array Y. LDY >= N.

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   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
        ==========
        Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
        performance of reduction to Hessenberg form," ACM Transactions on
        Mathematical Software, 32(2):180-194, June 2006.

 LAPACK auxiliary routine (version 3.3.1)   April 2011                            CLAHR2(3lapack)