numeric-linalg

001
002
003
004
005
006
007
008
009
010
011
012
013
014
015
016
017
018
019
020
021
022
023
024
025
026
027
028
029
030
031
032
033
034
035
036
037
038
039
040
041
042
043
044
045
046
047
048
049
050
051
052
053
054
055
056
057
058
059
060
061
062
063
064
065
066
067
068
069
070
071
072
073
074
075
076
077
078
079
080
081
082
083
084
085
086
087
088
089
090
091
092
093
094
095
096
097
098
099
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183

*> \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.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLAQR1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaqr1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaqr1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaqr1.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
*       .. Scalar Arguments ..
*       REAL               SI1, SI2, SR1, SR2
*       INTEGER            LDH, N
*       ..
*       .. Array Arguments ..
*       REAL               H( LDH, * ), V( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>      Given a 2-by-2 or 3-by-3 matrix H, SLAQR1 sets v to a
*>      scalar multiple of the first column of the product
*>
*>      (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
*>
*>      scaling to avoid overflows and most underflows. It
*>      is assumed that either
*>
*>              1) sr1 = sr2 and si1 = -si2
*>          or
*>              2) si1 = si2 = 0.
*>
*>      This is useful for starting double implicit shift bulges
*>      in the QR algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>              Order of the matrix H. N must be either 2 or 3.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*>          H is REAL array, dimension (LDH,N)
*>              The 2-by-2 or 3-by-3 matrix H in (*).
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>              The leading dimension of H as declared in
*>              the calling procedure.  LDH >= N
*> \endverbatim
*>
*> \param[in] SR1
*> \verbatim
*>          SR1 is REAL
*> \endverbatim
*>
*> \param[in] SI1
*> \verbatim
*>          SI1 is REAL
*> \endverbatim
*>
*> \param[in] SR2
*> \verbatim
*>          SR2 is REAL
*> \endverbatim
*>
*> \param[in] SI2
*> \verbatim
*>          SI2 is REAL
*>              The shifts in (*).
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is REAL array, dimension (N)
*>              A scalar multiple of the first column of the
*>              matrix K in (*).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr1
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*>
*  =====================================================================
      SUBROUTINE SLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      REAL               SI1, SI2, SR1, SR2
      INTEGER            LDH, N
*     ..
*     .. Array Arguments ..
      REAL               H( LDH, * ), V( * )
*     ..
*
*  ================================================================
*
*     .. Parameters ..
      REAL               ZERO
      PARAMETER          ( ZERO = 0.0e0 )
*     ..
*     .. Local Scalars ..
      REAL               H21S, H31S, S
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.NE.2 .AND. N.NE.3 ) THEN
         RETURN
      END IF
*
      IF( N.EQ.2 ) THEN
         S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) )
         IF( S.EQ.ZERO ) THEN
            V( 1 ) = ZERO
            V( 2 ) = ZERO
         ELSE
            H21S = H( 2, 1 ) / S
            V( 1 ) = H21S*H( 1, 2 ) + ( H( 1, 1 )-SR1 )*
     $               ( ( H( 1, 1 )-SR2 ) / S ) - SI1*( SI2 / S )
            V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 )
         END IF
      ELSE
         S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) ) +
     $       ABS( H( 3, 1 ) )
         IF( S.EQ.ZERO ) THEN
            V( 1 ) = ZERO
            V( 2 ) = ZERO
            V( 3 ) = ZERO
         ELSE
            H21S = H( 2, 1 ) / S
            H31S = H( 3, 1 ) / S
            V( 1 ) = ( H( 1, 1 )-SR1 )*( ( H( 1, 1 )-SR2 ) / S ) -
     $               SI1*( SI2 / S ) + H( 1, 2 )*H21S + H( 1, 3 )*H31S
            V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 ) +
     $               H( 2, 3 )*H31S
            V( 3 ) = H31S*( H( 1, 1 )+H( 3, 3 )-SR1-SR2 ) +
     $               H21S*H( 3, 2 )
         END IF
      END IF
      END