NAME

       zlahr2.f -

SYNOPSIS

   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.

Function/Subroutine Documentation

   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.

       Purpose:

            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.

       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.
                     K < N.

           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 >= 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   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.

Author

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