SLAHRD(3)      LAPACK routine of NEC Numeric Library Collection      SLAHRD(3)



NAME
       SLAHRD

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



PURPOSE
            SLAHRD 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.

            This is an OBSOLETE auxiliary routine.
            This routine will be 'deprecated' in a  future release.
            Please use the new routine SLAHR2 instead.




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

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

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

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

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

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

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

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

           LDY       (input)
                     LDY is 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**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).



LAPACK routine                  31 October 2017                      SLAHRD(3)