1*> \brief \b DLAQR1 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/dlaqr1.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
22*
23*       .. Scalar Arguments ..
24*       DOUBLE PRECISION   SI1, SI2, SR1, SR2
25*       INTEGER            LDH, N
26*       ..
27*       .. Array Arguments ..
28*       DOUBLE PRECISION   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, DLAQR1 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 DOUBLE PRECISION array of 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.GE.N
73*> \endverbatim
74*>
75*> \param[in] SR1
76*> \verbatim
77*>          SR1 is DOUBLE PRECISION
78*> \endverbatim
79*>
80*> \param[in] SI1
81*> \verbatim
82*>          SI1 is DOUBLE PRECISION
83*> \endverbatim
84*>
85*> \param[in] SR2
86*> \verbatim
87*>          SR2 is DOUBLE PRECISION
88*> \endverbatim
89*>
90*> \param[in] SI2
91*> \verbatim
92*>          SI2 is DOUBLE PRECISION
93*>              The shifts in (*).
94*> \endverbatim
95*>
96*> \param[out] V
97*> \verbatim
98*>          V is DOUBLE PRECISION array of 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*> \date September 2012
112*
113*> \ingroup doubleOTHERauxiliary
114*
115*> \par Contributors:
116*  ==================
117*>
118*>       Karen Braman and Ralph Byers, Department of Mathematics,
119*>       University of Kansas, USA
120*>
121*  =====================================================================
122      SUBROUTINE DLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
123*
124*  -- LAPACK auxiliary routine (version 3.4.2) --
125*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
126*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
127*     September 2012
128*
129*     .. Scalar Arguments ..
130      DOUBLE PRECISION   SI1, SI2, SR1, SR2
131      INTEGER            LDH, N
132*     ..
133*     .. Array Arguments ..
134      DOUBLE PRECISION   H( LDH, * ), V( * )
135*     ..
136*
137*  ================================================================
138*
139*     .. Parameters ..
140      DOUBLE PRECISION   ZERO
141      PARAMETER          ( ZERO = 0.0d0 )
142*     ..
143*     .. Local Scalars ..
144      DOUBLE PRECISION   H21S, H31S, S
145*     ..
146*     .. Intrinsic Functions ..
147      INTRINSIC          ABS
148*     ..
149*     .. Executable Statements ..
150      IF( N.EQ.2 ) THEN
151         S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) )
152         IF( S.EQ.ZERO ) THEN
153            V( 1 ) = ZERO
154            V( 2 ) = ZERO
155         ELSE
156            H21S = H( 2, 1 ) / S
157            V( 1 ) = H21S*H( 1, 2 ) + ( H( 1, 1 )-SR1 )*
158     $               ( ( H( 1, 1 )-SR2 ) / S ) - SI1*( SI2 / S )
159            V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 )
160         END IF
161      ELSE
162         S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) ) +
163     $       ABS( H( 3, 1 ) )
164         IF( S.EQ.ZERO ) THEN
165            V( 1 ) = ZERO
166            V( 2 ) = ZERO
167            V( 3 ) = ZERO
168         ELSE
169            H21S = H( 2, 1 ) / S
170            H31S = H( 3, 1 ) / S
171            V( 1 ) = ( H( 1, 1 )-SR1 )*( ( H( 1, 1 )-SR2 ) / S ) -
172     $               SI1*( SI2 / S ) + H( 1, 2 )*H21S + H( 1, 3 )*H31S
173            V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 ) +
174     $               H( 2, 3 )*H31S
175            V( 3 ) = H31S*( H( 1, 1 )+H( 3, 3 )-SR1-SR2 ) +
176     $               H21S*H( 3, 2 )
177         END IF
178      END IF
179      END
180