1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
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
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
|
*> \brief \b SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLANV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slanv2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slanv2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slanv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
* .. Scalar Arguments ..
* REAL A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
*> matrix in standard form:
*>
*> [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ]
*> [ C D ] [ SN CS ] [ CC DD ] [-SN CS ]
*>
*> where either
*> 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
*> 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
*> conjugate eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in,out] A
*> \verbatim
*> A is REAL
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is REAL
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is REAL
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is REAL
*> On entry, the elements of the input matrix.
*> On exit, they are overwritten by the elements of the
*> standardised Schur form.
*> \endverbatim
*>
*> \param[out] RT1R
*> \verbatim
*> RT1R is REAL
*> \endverbatim
*>
*> \param[out] RT1I
*> \verbatim
*> RT1I is REAL
*> \endverbatim
*>
*> \param[out] RT2R
*> \verbatim
*> RT2R is REAL
*> \endverbatim
*>
*> \param[out] RT2I
*> \verbatim
*> RT2I is REAL
*> The real and imaginary parts of the eigenvalues. If the
*> eigenvalues are a complex conjugate pair, RT1I > 0.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is REAL
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is REAL
*> Parameters of the rotation matrix.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup realOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Modified by V. Sima, Research Institute for Informatics, Bucharest,
*> Romania, to reduce the risk of cancellation errors,
*> when computing real eigenvalues, and to ensure, if possible, that
*> abs(RT1R) >= abs(RT2R).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
REAL A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0E+0, HALF = 0.5E+0, ONE = 1.0E+0 )
REAL MULTPL
PARAMETER ( MULTPL = 4.0E+0 )
* ..
* .. Local Scalars ..
REAL AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
$ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z
* ..
* .. External Functions ..
REAL SLAMCH, SLAPY2
EXTERNAL SLAMCH, SLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
EPS = SLAMCH( 'P' )
IF( C.EQ.ZERO ) THEN
CS = ONE
SN = ZERO
GO TO 10
*
ELSE IF( B.EQ.ZERO ) THEN
*
* Swap rows and columns
*
CS = ZERO
SN = ONE
TEMP = D
D = A
A = TEMP
B = -C
C = ZERO
GO TO 10
ELSE IF( (A-D).EQ.ZERO .AND. SIGN( ONE, B ).NE.
$ SIGN( ONE, C ) ) THEN
CS = ONE
SN = ZERO
GO TO 10
ELSE
*
TEMP = A - D
P = HALF*TEMP
BCMAX = MAX( ABS( B ), ABS( C ) )
BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
SCALE = MAX( ABS( P ), BCMAX )
Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
*
* If Z is of the order of the machine accuracy, postpone the
* decision on the nature of eigenvalues
*
IF( Z.GE.MULTPL*EPS ) THEN
*
* Real eigenvalues. Compute A and D.
*
Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
A = D + Z
D = D - ( BCMAX / Z )*BCMIS
*
* Compute B and the rotation matrix
*
TAU = SLAPY2( C, Z )
CS = Z / TAU
SN = C / TAU
B = B - C
C = ZERO
ELSE
*
* Complex eigenvalues, or real (almost) equal eigenvalues.
* Make diagonal elements equal.
*
SIGMA = B + C
TAU = SLAPY2( SIGMA, TEMP )
CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
*
* Compute [ AA BB ] = [ A B ] [ CS -SN ]
* [ CC DD ] [ C D ] [ SN CS ]
*
AA = A*CS + B*SN
BB = -A*SN + B*CS
CC = C*CS + D*SN
DD = -C*SN + D*CS
*
* Compute [ A B ] = [ CS SN ] [ AA BB ]
* [ C D ] [-SN CS ] [ CC DD ]
*
A = AA*CS + CC*SN
B = BB*CS + DD*SN
C = -AA*SN + CC*CS
D = -BB*SN + DD*CS
*
TEMP = HALF*( A+D )
A = TEMP
D = TEMP
*
IF( C.NE.ZERO ) THEN
IF( B.NE.ZERO ) THEN
IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
*
* Real eigenvalues: reduce to upper triangular form
*
SAB = SQRT( ABS( B ) )
SAC = SQRT( ABS( C ) )
P = SIGN( SAB*SAC, C )
TAU = ONE / SQRT( ABS( B+C ) )
A = TEMP + P
D = TEMP - P
B = B - C
C = ZERO
CS1 = SAB*TAU
SN1 = SAC*TAU
TEMP = CS*CS1 - SN*SN1
SN = CS*SN1 + SN*CS1
CS = TEMP
END IF
ELSE
B = -C
C = ZERO
TEMP = CS
CS = -SN
SN = TEMP
END IF
END IF
END IF
*
END IF
*
10 CONTINUE
*
* Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
*
RT1R = A
RT2R = D
IF( C.EQ.ZERO ) THEN
RT1I = ZERO
RT2I = ZERO
ELSE
RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
RT2I = -RT1I
END IF
RETURN
*
* End of SLANV2
*
END
|
