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
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
|
SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
$ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
LOGICAL LTRANL, LTRANR
INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
DOUBLE PRECISION SCALE, XNORM
* ..
* .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
$ X( LDX, * )
* ..
*
* Purpose
* =======
*
* DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
*
* op(TL)*X + ISGN*X*op(TR) = SCALE*B,
*
* where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
* -1. op(T) = T or T', where T' denotes the transpose of T.
*
* Arguments
* =========
*
* LTRANL (input) LOGICAL
* On entry, LTRANL specifies the op(TL):
* = .FALSE., op(TL) = TL,
* = .TRUE., op(TL) = TL'.
*
* LTRANR (input) LOGICAL
* On entry, LTRANR specifies the op(TR):
* = .FALSE., op(TR) = TR,
* = .TRUE., op(TR) = TR'.
*
* ISGN (input) INTEGER
* On entry, ISGN specifies the sign of the equation
* as described before. ISGN may only be 1 or -1.
*
* N1 (input) INTEGER
* On entry, N1 specifies the order of matrix TL.
* N1 may only be 0, 1 or 2.
*
* N2 (input) INTEGER
* On entry, N2 specifies the order of matrix TR.
* N2 may only be 0, 1 or 2.
*
* TL (input) DOUBLE PRECISION array, dimension (LDTL,2)
* On entry, TL contains an N1 by N1 matrix.
*
* LDTL (input) INTEGER
* The leading dimension of the matrix TL. LDTL >= max(1,N1).
*
* TR (input) DOUBLE PRECISION array, dimension (LDTR,2)
* On entry, TR contains an N2 by N2 matrix.
*
* LDTR (input) INTEGER
* The leading dimension of the matrix TR. LDTR >= max(1,N2).
*
* B (input) DOUBLE PRECISION array, dimension (LDB,2)
* On entry, the N1 by N2 matrix B contains the right-hand
* side of the equation.
*
* LDB (input) INTEGER
* The leading dimension of the matrix B. LDB >= max(1,N1).
*
* SCALE (output) DOUBLE PRECISION
* On exit, SCALE contains the scale factor. SCALE is chosen
* less than or equal to 1 to prevent the solution overflowing.
*
* X (output) DOUBLE PRECISION array, dimension (LDX,2)
* On exit, X contains the N1 by N2 solution.
*
* LDX (input) INTEGER
* The leading dimension of the matrix X. LDX >= max(1,N1).
*
* XNORM (output) DOUBLE PRECISION
* On exit, XNORM is the infinity-norm of the solution.
*
* INFO (output) INTEGER
* On exit, INFO is set to
* 0: successful exit.
* 1: TL and TR have too close eigenvalues, so TL or
* TR is perturbed to get a nonsingular equation.
* NOTE: In the interests of speed, this routine does not
* check the inputs for errors.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TWO, HALF, EIGHT
PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL BSWAP, XSWAP
INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
$ TEMP, U11, U12, U22, XMAX
* ..
* .. Local Arrays ..
LOGICAL BSWPIV( 4 ), XSWPIV( 4 )
INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
$ LOCU22( 4 )
DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Data statements ..
DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
$ LOCU22 / 4, 3, 2, 1 /
DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
* ..
* .. Executable Statements ..
*
* Do not check the input parameters for errors
*
INFO = 0
*
* Quick return if possible
*
IF( N1.EQ.0 .OR. N2.EQ.0 )
$ RETURN
*
* Set constants to control overflow
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
SGN = ISGN
*
K = N1 + N1 + N2 - 2
GO TO ( 10, 20, 30, 50 )K
*
* 1 by 1: TL11*X + SGN*X*TR11 = B11
*
10 CONTINUE
TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
BET = ABS( TAU1 )
IF( BET.LE.SMLNUM ) THEN
TAU1 = SMLNUM
BET = SMLNUM
INFO = 1
END IF
*
SCALE = ONE
GAM = ABS( B( 1, 1 ) )
IF( SMLNUM*GAM.GT.BET )
$ SCALE = ONE / GAM
*
X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
XNORM = ABS( X( 1, 1 ) )
RETURN
*
* 1 by 2:
* TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12]
* [TR21 TR22]
*
20 CONTINUE
*
SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
$ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
$ SMLNUM )
TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
IF( LTRANR ) THEN
TMP( 2 ) = SGN*TR( 2, 1 )
TMP( 3 ) = SGN*TR( 1, 2 )
ELSE
TMP( 2 ) = SGN*TR( 1, 2 )
TMP( 3 ) = SGN*TR( 2, 1 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 1, 2 )
GO TO 40
*
* 2 by 1:
* op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11]
* [TL21 TL22] [X21] [X21] [B21]
*
30 CONTINUE
SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
$ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
$ SMLNUM )
TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
IF( LTRANL ) THEN
TMP( 2 ) = TL( 1, 2 )
TMP( 3 ) = TL( 2, 1 )
ELSE
TMP( 2 ) = TL( 2, 1 )
TMP( 3 ) = TL( 1, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 2, 1 )
40 CONTINUE
*
* Solve 2 by 2 system using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
IPIV = IDAMAX( 4, TMP, 1 )
U11 = TMP( IPIV )
IF( ABS( U11 ).LE.SMIN ) THEN
INFO = 1
U11 = SMIN
END IF
U12 = TMP( LOCU12( IPIV ) )
L21 = TMP( LOCL21( IPIV ) ) / U11
U22 = TMP( LOCU22( IPIV ) ) - U12*L21
XSWAP = XSWPIV( IPIV )
BSWAP = BSWPIV( IPIV )
IF( ABS( U22 ).LE.SMIN ) THEN
INFO = 1
U22 = SMIN
END IF
IF( BSWAP ) THEN
TEMP = BTMP( 2 )
BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
BTMP( 1 ) = TEMP
ELSE
BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
END IF
SCALE = ONE
IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
$ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
END IF
X2( 2 ) = BTMP( 2 ) / U22
X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
IF( XSWAP ) THEN
TEMP = X2( 2 )
X2( 2 ) = X2( 1 )
X2( 1 ) = TEMP
END IF
X( 1, 1 ) = X2( 1 )
IF( N1.EQ.1 ) THEN
X( 1, 2 ) = X2( 2 )
XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
ELSE
X( 2, 1 ) = X2( 2 )
XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
END IF
RETURN
*
* 2 by 2:
* op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
* [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
*
* Solve equivalent 4 by 4 system using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
50 CONTINUE
SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
$ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
$ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
SMIN = MAX( EPS*SMIN, SMLNUM )
BTMP( 1 ) = ZERO
CALL DCOPY( 16, BTMP, 0, T16, 1 )
T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
IF( LTRANL ) THEN
T16( 1, 2 ) = TL( 2, 1 )
T16( 2, 1 ) = TL( 1, 2 )
T16( 3, 4 ) = TL( 2, 1 )
T16( 4, 3 ) = TL( 1, 2 )
ELSE
T16( 1, 2 ) = TL( 1, 2 )
T16( 2, 1 ) = TL( 2, 1 )
T16( 3, 4 ) = TL( 1, 2 )
T16( 4, 3 ) = TL( 2, 1 )
END IF
IF( LTRANR ) THEN
T16( 1, 3 ) = SGN*TR( 1, 2 )
T16( 2, 4 ) = SGN*TR( 1, 2 )
T16( 3, 1 ) = SGN*TR( 2, 1 )
T16( 4, 2 ) = SGN*TR( 2, 1 )
ELSE
T16( 1, 3 ) = SGN*TR( 2, 1 )
T16( 2, 4 ) = SGN*TR( 2, 1 )
T16( 3, 1 ) = SGN*TR( 1, 2 )
T16( 4, 2 ) = SGN*TR( 1, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 2, 1 )
BTMP( 3 ) = B( 1, 2 )
BTMP( 4 ) = B( 2, 2 )
*
* Perform elimination
*
DO 100 I = 1, 3
XMAX = ZERO
DO 70 IP = I, 4
DO 60 JP = I, 4
IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
XMAX = ABS( T16( IP, JP ) )
IPSV = IP
JPSV = JP
END IF
60 CONTINUE
70 CONTINUE
IF( IPSV.NE.I ) THEN
CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
TEMP = BTMP( I )
BTMP( I ) = BTMP( IPSV )
BTMP( IPSV ) = TEMP
END IF
IF( JPSV.NE.I )
$ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
JPIV( I ) = JPSV
IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
INFO = 1
T16( I, I ) = SMIN
END IF
DO 90 J = I + 1, 4
T16( J, I ) = T16( J, I ) / T16( I, I )
BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
DO 80 K = I + 1, 4
T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
80 CONTINUE
90 CONTINUE
100 CONTINUE
IF( ABS( T16( 4, 4 ) ).LT.SMIN )
$ T16( 4, 4 ) = SMIN
SCALE = ONE
IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
$ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
BTMP( 3 ) = BTMP( 3 )*SCALE
BTMP( 4 ) = BTMP( 4 )*SCALE
END IF
DO 120 I = 1, 4
K = 5 - I
TEMP = ONE / T16( K, K )
TMP( K ) = BTMP( K )*TEMP
DO 110 J = K + 1, 4
TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
110 CONTINUE
120 CONTINUE
DO 130 I = 1, 3
IF( JPIV( 4-I ).NE.4-I ) THEN
TEMP = TMP( 4-I )
TMP( 4-I ) = TMP( JPIV( 4-I ) )
TMP( JPIV( 4-I ) ) = TEMP
END IF
130 CONTINUE
X( 1, 1 ) = TMP( 1 )
X( 2, 1 ) = TMP( 2 )
X( 1, 2 ) = TMP( 3 )
X( 2, 2 ) = TMP( 4 )
XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
$ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
RETURN
*
* End of DLASY2
*
END
|
