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
290
291
292
293
294
295
296
297
298
299
300
301
|
SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
$ WR2, WI )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* March 31, 1993
*
* .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
* problem A - w B, with scaling as necessary to avoid over-/underflow.
*
* The scaling factor "s" results in a modified eigenvalue equation
*
* s A - w B
*
* where s is a non-negative scaling factor chosen so that w, w B,
* and s A do not overflow and, if possible, do not underflow, either.
*
* Arguments
* =========
*
* A (input) DOUBLE PRECISION array, dimension (LDA, 2)
* On entry, the 2 x 2 matrix A. It is assumed that its 1-norm
* is less than 1/SAFMIN. Entries less than
* sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= 2.
*
* B (input) DOUBLE PRECISION array, dimension (LDB, 2)
* On entry, the 2 x 2 upper triangular matrix B. It is
* assumed that the one-norm of B is less than 1/SAFMIN. The
* diagonals should be at least sqrt(SAFMIN) times the largest
* element of B (in absolute value); if a diagonal is smaller
* than that, then +/- sqrt(SAFMIN) will be used instead of
* that diagonal.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= 2.
*
* SAFMIN (input) DOUBLE PRECISION
* The smallest positive number s.t. 1/SAFMIN does not
* overflow. (This should always be DLAMCH('S') -- it is an
* argument in order to avoid having to call DLAMCH frequently.)
*
* SCALE1 (output) DOUBLE PRECISION
* A scaling factor used to avoid over-/underflow in the
* eigenvalue equation which defines the first eigenvalue. If
* the eigenvalues are complex, then the eigenvalues are
* ( WR1 +/- WI i ) / SCALE1 (which may lie outside the
* exponent range of the machine), SCALE1=SCALE2, and SCALE1
* will always be positive. If the eigenvalues are real, then
* the first (real) eigenvalue is WR1 / SCALE1 , but this may
* overflow or underflow, and in fact, SCALE1 may be zero or
* less than the underflow threshhold if the exact eigenvalue
* is sufficiently large.
*
* SCALE2 (output) DOUBLE PRECISION
* A scaling factor used to avoid over-/underflow in the
* eigenvalue equation which defines the second eigenvalue. If
* the eigenvalues are complex, then SCALE2=SCALE1. If the
* eigenvalues are real, then the second (real) eigenvalue is
* WR2 / SCALE2 , but this may overflow or underflow, and in
* fact, SCALE2 may be zero or less than the underflow
* threshhold if the exact eigenvalue is sufficiently large.
*
* WR1 (output) DOUBLE PRECISION
* If the eigenvalue is real, then WR1 is SCALE1 times the
* eigenvalue closest to the (2,2) element of A B**(-1). If the
* eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
* part of the eigenvalues.
*
* WR2 (output) DOUBLE PRECISION
* If the eigenvalue is real, then WR2 is SCALE2 times the
* other eigenvalue. If the eigenvalue is complex, then
* WR1=WR2 is SCALE1 times the real part of the eigenvalues.
*
* WI (output) DOUBLE PRECISION
* If the eigenvalue is real, then WI is zero. If the
* eigenvalue is complex, then WI is SCALE1 times the imaginary
* part of the eigenvalues. WI will always be non-negative.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = ONE / TWO )
DOUBLE PRECISION FUZZY1
PARAMETER ( FUZZY1 = ONE+1.0D-5 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
$ AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
$ BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
$ DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
$ SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
$ WSCALE, WSIZE, WSMALL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
RTMIN = SQRT( SAFMIN )
RTMAX = ONE / RTMIN
SAFMAX = ONE / SAFMIN
*
* Scale A
*
ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
$ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
ASCALE = ONE / ANORM
A11 = ASCALE*A( 1, 1 )
A21 = ASCALE*A( 2, 1 )
A12 = ASCALE*A( 1, 2 )
A22 = ASCALE*A( 2, 2 )
*
* Perturb B if necessary to insure non-singularity
*
B11 = B( 1, 1 )
B12 = B( 1, 2 )
B22 = B( 2, 2 )
BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
IF( ABS( B11 ).LT.BMIN )
$ B11 = SIGN( BMIN, B11 )
IF( ABS( B22 ).LT.BMIN )
$ B22 = SIGN( BMIN, B22 )
*
* Scale B
*
BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
BSCALE = ONE / BSIZE
B11 = B11*BSCALE
B12 = B12*BSCALE
B22 = B22*BSCALE
*
* Compute larger eigenvalue by method described by C. van Loan
*
* ( AS is A shifted by -SHIFT*B )
*
BINV11 = ONE / B11
BINV22 = ONE / B22
S1 = A11*BINV11
S2 = A22*BINV22
IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
AS12 = A12 - S1*B12
AS22 = A22 - S1*B22
SS = A21*( BINV11*BINV22 )
ABI22 = AS22*BINV22 - SS*B12
PP = HALF*ABI22
SHIFT = S1
ELSE
AS12 = A12 - S2*B12
AS11 = A11 - S2*B11
SS = A21*( BINV11*BINV22 )
ABI22 = -SS*B12
PP = HALF*( AS11*BINV11+ABI22 )
SHIFT = S2
END IF
QQ = SS*AS12
IF( ABS( PP*RTMIN ).GE.ONE ) THEN
DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
R = SQRT( ABS( DISCR ) )*RTMAX
ELSE
IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
R = SQRT( ABS( DISCR ) )*RTMIN
ELSE
DISCR = PP**2 + QQ
R = SQRT( ABS( DISCR ) )
END IF
END IF
*
* Note: the test of R in the following IF is to cover the case when
* DISCR is small and negative and is flushed to zero during
* the calculation of R. On machines which have a consistent
* flush-to-zero threshhold and handle numbers above that
* threshhold correctly, it would not be necessary.
*
IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
SUM = PP + SIGN( R, PP )
DIFF = PP - SIGN( R, PP )
WBIG = SHIFT + SUM
*
* Compute smaller eigenvalue
*
WSMALL = SHIFT + DIFF
IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
WSMALL = WDET / WBIG
END IF
*
* Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
* for WR1.
*
IF( PP.GT.ABI22 ) THEN
WR1 = MIN( WBIG, WSMALL )
WR2 = MAX( WBIG, WSMALL )
ELSE
WR1 = MAX( WBIG, WSMALL )
WR2 = MIN( WBIG, WSMALL )
END IF
WI = ZERO
ELSE
*
* Complex eigenvalues
*
WR1 = SHIFT + PP
WR2 = WR1
WI = R
END IF
*
* Further scaling to avoid underflow and overflow in computing
* SCALE1 and overflow in computing w*B.
*
* This scale factor (WSCALE) is bounded from above using C1 and C2,
* and from below using C3 and C4.
* C1 implements the condition s A must never overflow.
* C2 implements the condition w B must never overflow.
* C3, with C2,
* implement the condition that s A - w B must never overflow.
* C4 implements the condition s should not underflow.
* C5 implements the condition max(s,|w|) should be at least 2.
*
C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
C2 = SAFMIN*MAX( ONE, BNORM )
C3 = BSIZE*SAFMIN
IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
ELSE
C4 = ONE
END IF
IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
C5 = MIN( ONE, ASCALE*BSIZE )
ELSE
C5 = ONE
END IF
*
* Scale first eigenvalue
*
WABS = ABS( WR1 ) + ABS( WI )
WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
$ MIN( C4, HALF*MAX( WABS, C5 ) ) )
IF( WSIZE.NE.ONE ) THEN
WSCALE = ONE / WSIZE
IF( WSIZE.GT.ONE ) THEN
SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
$ MIN( ASCALE, BSIZE )
ELSE
SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
$ MAX( ASCALE, BSIZE )
END IF
WR1 = WR1*WSCALE
IF( WI.NE.ZERO ) THEN
WI = WI*WSCALE
WR2 = WR1
SCALE2 = SCALE1
END IF
ELSE
SCALE1 = ASCALE*BSIZE
SCALE2 = SCALE1
END IF
*
* Scale second eigenvalue (if real)
*
IF( WI.EQ.ZERO ) THEN
WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
$ MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
IF( WSIZE.NE.ONE ) THEN
WSCALE = ONE / WSIZE
IF( WSIZE.GT.ONE ) THEN
SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
$ MIN( ASCALE, BSIZE )
ELSE
SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
$ MAX( ASCALE, BSIZE )
END IF
WR2 = WR2*WSCALE
ELSE
SCALE2 = ASCALE*BSIZE
END IF
END IF
*
* End of DLAG2
*
RETURN
END
|
