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SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
* an orthogonal similarity transformation: Q' * A * Q = H .
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that A is already upper triangular in rows
* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
* set by a previous call to DGEBAL; otherwise they should be
* set to 1 and N respectively. See Further Details.
* 1 <= ILO <= IHI <= max(1,N).
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the n by n general matrix to be reduced.
* On exit, the upper triangle and the first subdiagonal of A
* are overwritten with the upper Hessenberg matrix H, and the
* elements below the first subdiagonal, with the array TAU,
* represent the orthogonal matrix Q as a product of elementary
* reflectors. See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* TAU (output) DOUBLE PRECISION array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) DOUBLE PRECISION array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of (ihi-ilo) elementary
* reflectors
*
* Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*
* 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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
* exit in A(i+2:ihi,i), and tau in TAU(i).
*
* The contents of A are illustrated by the following example, with
* n = 7, ilo = 2 and ihi = 6:
*
* on entry, on exit,
*
* ( a a a a a a a ) ( a a h h h h a )
* ( a a a a a a ) ( a h h h h a )
* ( a a a a a a ) ( h h h h h h )
* ( a a a a a a ) ( v2 h h h h h )
* ( a a a a a a ) ( v2 v3 h h h h )
* ( a a a a a a ) ( v2 v3 v4 h h h )
* ( 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 ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION AII
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEHD2', -INFO )
RETURN
END IF
*
DO 10 I = ILO, IHI - 1
*
* Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
*
CALL DLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAU( I ) )
AII = A( I+1, I )
A( I+1, I ) = ONE
*
* Apply H(i) to A(1:ihi,i+1:ihi) from the right
*
CALL DLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
$ A( 1, I+1 ), LDA, WORK )
*
* Apply H(i) to A(i+1:ihi,i+1:n) from the left
*
CALL DLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ),
$ A( I+1, I+1 ), LDA, WORK )
*
A( I+1, I ) = AII
10 CONTINUE
*
RETURN
*
* End of DGEHD2
*
END
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