1*> \brief \b SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
22*
23*       .. Scalar Arguments ..
24*       INTEGER            IHI, ILO, INFO, LDA, N
25*       ..
26*       .. Array Arguments ..
27*       REAL               A( LDA, * ), TAU( * ), WORK( * )
28*       ..
29*
30*
31*> \par Purpose:
32*  =============
33*>
34*> \verbatim
35*>
36*> SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
37*> an orthogonal similarity transformation:  Q**T * A * Q = H .
38*> \endverbatim
39*
40*  Arguments:
41*  ==========
42*
43*> \param[in] N
44*> \verbatim
45*>          N is INTEGER
46*>          The order of the matrix A.  N >= 0.
47*> \endverbatim
48*>
49*> \param[in] ILO
50*> \verbatim
51*>          ILO is INTEGER
52*> \endverbatim
53*>
54*> \param[in] IHI
55*> \verbatim
56*>          IHI is INTEGER
57*>
58*>          It is assumed that A is already upper triangular in rows
59*>          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
60*>          set by a previous call to SGEBAL; otherwise they should be
61*>          set to 1 and N respectively. See Further Details.
62*>          1 <= ILO <= IHI <= max(1,N).
63*> \endverbatim
64*>
65*> \param[in,out] A
66*> \verbatim
67*>          A is REAL array, dimension (LDA,N)
68*>          On entry, the n by n general matrix to be reduced.
69*>          On exit, the upper triangle and the first subdiagonal of A
70*>          are overwritten with the upper Hessenberg matrix H, and the
71*>          elements below the first subdiagonal, with the array TAU,
72*>          represent the orthogonal matrix Q as a product of elementary
73*>          reflectors. See Further Details.
74*> \endverbatim
75*>
76*> \param[in] LDA
77*> \verbatim
78*>          LDA is INTEGER
79*>          The leading dimension of the array A.  LDA >= max(1,N).
80*> \endverbatim
81*>
82*> \param[out] TAU
83*> \verbatim
84*>          TAU is REAL array, dimension (N-1)
85*>          The scalar factors of the elementary reflectors (see Further
86*>          Details).
87*> \endverbatim
88*>
89*> \param[out] WORK
90*> \verbatim
91*>          WORK is REAL array, dimension (N)
92*> \endverbatim
93*>
94*> \param[out] INFO
95*> \verbatim
96*>          INFO is INTEGER
97*>          = 0:  successful exit.
98*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
99*> \endverbatim
100*
101*  Authors:
102*  ========
103*
104*> \author Univ. of Tennessee
105*> \author Univ. of California Berkeley
106*> \author Univ. of Colorado Denver
107*> \author NAG Ltd.
108*
109*> \date September 2012
110*
111*> \ingroup realGEcomputational
112*
113*> \par Further Details:
114*  =====================
115*>
116*> \verbatim
117*>
118*>  The matrix Q is represented as a product of (ihi-ilo) elementary
119*>  reflectors
120*>
121*>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
122*>
123*>  Each H(i) has the form
124*>
125*>     H(i) = I - tau * v * v**T
126*>
127*>  where tau is a real scalar, and v is a real vector with
128*>  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
129*>  exit in A(i+2:ihi,i), and tau in TAU(i).
130*>
131*>  The contents of A are illustrated by the following example, with
132*>  n = 7, ilo = 2 and ihi = 6:
133*>
134*>  on entry,                        on exit,
135*>
136*>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
137*>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
138*>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
139*>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
140*>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
141*>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
142*>  (                         a )    (                          a )
143*>
144*>  where a denotes an element of the original matrix A, h denotes a
145*>  modified element of the upper Hessenberg matrix H, and vi denotes an
146*>  element of the vector defining H(i).
147*> \endverbatim
148*>
149*  =====================================================================
150      SUBROUTINE SGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
151*
152*  -- LAPACK computational routine (version 3.4.2) --
153*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
154*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
155*     September 2012
156*
157*     .. Scalar Arguments ..
158      INTEGER            IHI, ILO, INFO, LDA, N
159*     ..
160*     .. Array Arguments ..
161      REAL               A( LDA, * ), TAU( * ), WORK( * )
162*     ..
163*
164*  =====================================================================
165*
166*     .. Parameters ..
167      REAL               ONE
168      PARAMETER          ( ONE = 1.0E+0 )
169*     ..
170*     .. Local Scalars ..
171      INTEGER            I
172      REAL               AII
173*     ..
174*     .. External Subroutines ..
175      EXTERNAL           SLARF, SLARFG, XERBLA
176*     ..
177*     .. Intrinsic Functions ..
178      INTRINSIC          MAX, MIN
179*     ..
180*     .. Executable Statements ..
181*
182*     Test the input parameters
183*
184      INFO = 0
185      IF( N.LT.0 ) THEN
186         INFO = -1
187      ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
188         INFO = -2
189      ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
190         INFO = -3
191      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
192         INFO = -5
193      END IF
194      IF( INFO.NE.0 ) THEN
195         CALL XERBLA( 'SGEHD2', -INFO )
196         RETURN
197      END IF
198*
199      DO 10 I = ILO, IHI - 1
200*
201*        Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
202*
203         CALL SLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
204     $                TAU( I ) )
205         AII = A( I+1, I )
206         A( I+1, I ) = ONE
207*
208*        Apply H(i) to A(1:ihi,i+1:ihi) from the right
209*
210         CALL SLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
211     $               A( 1, I+1 ), LDA, WORK )
212*
213*        Apply H(i) to A(i+1:ihi,i+1:n) from the left
214*
215         CALL SLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ),
216     $               A( I+1, I+1 ), LDA, WORK )
217*
218         A( I+1, I ) = AII
219   10 CONTINUE
220*
221      RETURN
222*
223*     End of SGEHD2
224*
225      END
226c $Id$
227