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+      SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
+*
+*  -- LAPACK auxiliary routine (version 3.1) --
+*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
+*     November 2006
+*
+*     .. Scalar Arguments ..
+      INTEGER            K, LDA, LDT, LDY, N, NB
+*     ..
+*     .. Array Arguments ..
+      DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
+     $                   Y( LDY, NB )
+*     ..
+*
+*  Purpose
+*  =======
+*
+*  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' * A * Q. The routine returns the matrices V and T which determine
+*  Q as a block reflector I - V*T*V', 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 DLAHR2 instead.
+*
+*  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.
+*
+*  NB      (input) INTEGER
+*          The number of columns to be reduced.
+*
+*  A       (input/output) 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.
+*
+*  LDA     (input) INTEGER
+*          The leading dimension of the array A.  LDA >= max(1,N).
+*
+*  TAU     (output) DOUBLE PRECISION array, dimension (NB)
+*          The scalar factors of the elementary reflectors. See Further
+*          Details.
+*
+*  T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
+*          The upper triangular matrix T.
+*
+*  LDT     (input) INTEGER
+*          The leading dimension of the array T.  LDT >= NB.
+*
+*  Y       (output) DOUBLE PRECISION 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'
+*
+*  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') * (A - Y*V').
+*
+*  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).
+*
+*  =====================================================================
+*
+*     .. Parameters ..
+      DOUBLE PRECISION   ZERO, ONE
+      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
+*     ..
+*     .. Local Scalars ..
+      INTEGER            I
+      DOUBLE PRECISION   EI
+*     ..
+*     .. External Subroutines ..
+      EXTERNAL           DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
+*     ..
+*     .. Intrinsic Functions ..
+      INTRINSIC          MIN
+*     ..
+*     .. Executable Statements ..
+*
+*     Quick return if possible
+*
+      IF( N.LE.1 )
+     $   RETURN
+*
+      DO 10 I = 1, NB
+         IF( I.GT.1 ) THEN
+*
+*           Update A(1:n,i)
+*
+*           Compute i-th column of A - Y * V'
+*
+            CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
+     $                  A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
+*
+*           Apply I - V * T' * V' to this column (call it b) from the
+*           left, using the last column of T as workspace
+*
+*           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
+*                    ( V2 )             ( b2 )
+*
+*           where V1 is unit lower triangular
+*
+*           w := V1' * b1
+*
+            CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
+            CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),
+     $                  LDA, T( 1, NB ), 1 )
+*
+*           w := w + V2'*b2
+*
+            CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
+     $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
+*
+*           w := T'*w
+*
+            CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
+     $                  T( 1, NB ), 1 )
+*
+*           b2 := b2 - V2*w
+*
+            CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
+     $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
+*
+*           b1 := b1 - V1*w
+*
+            CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
+     $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
+            CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
+*
+            A( K+I-1, I-1 ) = EI
+         END IF
+*
+*        Generate the elementary reflector H(i) to annihilate
+*        A(k+i+1:n,i)
+*
+         CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
+     $                TAU( I ) )
+         EI = A( K+I, I )
+         A( K+I, I ) = ONE
+*
+*        Compute  Y(1:n,i)
+*
+         CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
+     $               A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
+         CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA,
+     $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
+         CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
+     $               ONE, Y( 1, I ), 1 )
+         CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
+*
+*        Compute T(1:i,i)
+*
+         CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
+         CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
+     $               T( 1, I ), 1 )
+         T( I, I ) = TAU( I )
+*
+   10 CONTINUE
+      A( K+NB, NB ) = EI
+*
+      RETURN
+*
+*     End of DLAHRD
+*
+      END