1*> \brief \b SLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaqr1.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
22*
23*       .. Scalar Arguments ..
24*       REAL               SI1, SI2, SR1, SR2
25*       INTEGER            LDH, N
26*       ..
27*       .. Array Arguments ..
28*       REAL               H( LDH, * ), V( * )
29*       ..
30*
31*
32*> \par Purpose:
33*  =============
34*>
35*> \verbatim
36*>
37*>      Given a 2-by-2 or 3-by-3 matrix H, SLAQR1 sets v to a
38*>      scalar multiple of the first column of the product
39*>
40*>      (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
41*>
42*>      scaling to avoid overflows and most underflows. It
43*>      is assumed that either
44*>
45*>              1) sr1 = sr2 and si1 = -si2
46*>          or
47*>              2) si1 = si2 = 0.
48*>
49*>      This is useful for starting double implicit shift bulges
50*>      in the QR algorithm.
51*> \endverbatim
52*
53*  Arguments:
54*  ==========
55*
56*> \param[in] N
57*> \verbatim
58*>          N is INTEGER
59*>              Order of the matrix H. N must be either 2 or 3.
60*> \endverbatim
61*>
62*> \param[in] H
63*> \verbatim
64*>          H is REAL array, dimension (LDH,N)
65*>              The 2-by-2 or 3-by-3 matrix H in (*).
66*> \endverbatim
67*>
68*> \param[in] LDH
69*> \verbatim
70*>          LDH is INTEGER
71*>              The leading dimension of H as declared in
72*>              the calling procedure.  LDH >= N
73*> \endverbatim
74*>
75*> \param[in] SR1
76*> \verbatim
77*>          SR1 is REAL
78*> \endverbatim
79*>
80*> \param[in] SI1
81*> \verbatim
82*>          SI1 is REAL
83*> \endverbatim
84*>
85*> \param[in] SR2
86*> \verbatim
87*>          SR2 is REAL
88*> \endverbatim
89*>
90*> \param[in] SI2
91*> \verbatim
92*>          SI2 is REAL
93*>              The shifts in (*).
94*> \endverbatim
95*>
96*> \param[out] V
97*> \verbatim
98*>          V is REAL array, dimension (N)
99*>              A scalar multiple of the first column of the
100*>              matrix K in (*).
101*> \endverbatim
102*
103*  Authors:
104*  ========
105*
106*> \author Univ. of Tennessee
107*> \author Univ. of California Berkeley
108*> \author Univ. of Colorado Denver
109*> \author NAG Ltd.
110*
111*> \ingroup realOTHERauxiliary
112*
113*> \par Contributors:
114*  ==================
115*>
116*>       Karen Braman and Ralph Byers, Department of Mathematics,
117*>       University of Kansas, USA
118*>
119*  =====================================================================
120      SUBROUTINE SLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
121*
122*  -- LAPACK auxiliary routine --
123*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
124*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
125*
126*     .. Scalar Arguments ..
127      REAL               SI1, SI2, SR1, SR2
128      INTEGER            LDH, N
129*     ..
130*     .. Array Arguments ..
131      REAL               H( LDH, * ), V( * )
132*     ..
133*
134*  ================================================================
135*
136*     .. Parameters ..
137      REAL               ZERO
138      PARAMETER          ( ZERO = 0.0e0 )
139*     ..
140*     .. Local Scalars ..
141      REAL               H21S, H31S, S
142*     ..
143*     .. Intrinsic Functions ..
144      INTRINSIC          ABS
145*     ..
146*     .. Executable Statements ..
147*
148*     Quick return if possible
149*
150      IF( N.NE.2 .AND. N.NE.3 ) THEN
151         RETURN
152      END IF
153*
154      IF( N.EQ.2 ) THEN
155         S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) )
156         IF( S.EQ.ZERO ) THEN
157            V( 1 ) = ZERO
158            V( 2 ) = ZERO
159         ELSE
160            H21S = H( 2, 1 ) / S
161            V( 1 ) = H21S*H( 1, 2 ) + ( H( 1, 1 )-SR1 )*
162     $               ( ( H( 1, 1 )-SR2 ) / S ) - SI1*( SI2 / S )
163            V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 )
164         END IF
165      ELSE
166         S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) ) +
167     $       ABS( H( 3, 1 ) )
168         IF( S.EQ.ZERO ) THEN
169            V( 1 ) = ZERO
170            V( 2 ) = ZERO
171            V( 3 ) = ZERO
172         ELSE
173            H21S = H( 2, 1 ) / S
174            H31S = H( 3, 1 ) / S
175            V( 1 ) = ( H( 1, 1 )-SR1 )*( ( H( 1, 1 )-SR2 ) / S ) -
176     $               SI1*( SI2 / S ) + H( 1, 2 )*H21S + H( 1, 3 )*H31S
177            V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 ) +
178     $               H( 2, 3 )*H31S
179            V( 3 ) = H31S*( H( 1, 1 )+H( 3, 3 )-SR1-SR2 ) +
180     $               H21S*H( 3, 2 )
181         END IF
182      END IF
183      END
184