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[project @ 1996-07-19 01:29:05 by jwe] Initial revision
author jwe
date Fri, 19 Jul 1996 01:29:55 +0000
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children 15cddaacbc2d
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2328:b44c3b2a5fce 2329:30c606bec7a8
1 SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
2 $ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
3 *
4 * -- LAPACK auxiliary routine (version 2.0) --
5 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
6 * Courant Institute, Argonne National Lab, and Rice University
7 * October 31, 1992
8 *
9 * .. Scalar Arguments ..
10 LOGICAL LTRANL, LTRANR
11 INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
12 DOUBLE PRECISION SCALE, XNORM
13 * ..
14 * .. Array Arguments ..
15 DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
16 $ X( LDX, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
23 *
24 * op(TL)*X + ISGN*X*op(TR) = SCALE*B,
25 *
26 * where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
27 * -1. op(T) = T or T', where T' denotes the transpose of T.
28 *
29 * Arguments
30 * =========
31 *
32 * LTRANL (input) LOGICAL
33 * On entry, LTRANL specifies the op(TL):
34 * = .FALSE., op(TL) = TL,
35 * = .TRUE., op(TL) = TL'.
36 *
37 * LTRANR (input) LOGICAL
38 * On entry, LTRANR specifies the op(TR):
39 * = .FALSE., op(TR) = TR,
40 * = .TRUE., op(TR) = TR'.
41 *
42 * ISGN (input) INTEGER
43 * On entry, ISGN specifies the sign of the equation
44 * as described before. ISGN may only be 1 or -1.
45 *
46 * N1 (input) INTEGER
47 * On entry, N1 specifies the order of matrix TL.
48 * N1 may only be 0, 1 or 2.
49 *
50 * N2 (input) INTEGER
51 * On entry, N2 specifies the order of matrix TR.
52 * N2 may only be 0, 1 or 2.
53 *
54 * TL (input) DOUBLE PRECISION array, dimension (LDTL,2)
55 * On entry, TL contains an N1 by N1 matrix.
56 *
57 * LDTL (input) INTEGER
58 * The leading dimension of the matrix TL. LDTL >= max(1,N1).
59 *
60 * TR (input) DOUBLE PRECISION array, dimension (LDTR,2)
61 * On entry, TR contains an N2 by N2 matrix.
62 *
63 * LDTR (input) INTEGER
64 * The leading dimension of the matrix TR. LDTR >= max(1,N2).
65 *
66 * B (input) DOUBLE PRECISION array, dimension (LDB,2)
67 * On entry, the N1 by N2 matrix B contains the right-hand
68 * side of the equation.
69 *
70 * LDB (input) INTEGER
71 * The leading dimension of the matrix B. LDB >= max(1,N1).
72 *
73 * SCALE (output) DOUBLE PRECISION
74 * On exit, SCALE contains the scale factor. SCALE is chosen
75 * less than or equal to 1 to prevent the solution overflowing.
76 *
77 * X (output) DOUBLE PRECISION array, dimension (LDX,2)
78 * On exit, X contains the N1 by N2 solution.
79 *
80 * LDX (input) INTEGER
81 * The leading dimension of the matrix X. LDX >= max(1,N1).
82 *
83 * XNORM (output) DOUBLE PRECISION
84 * On exit, XNORM is the infinity-norm of the solution.
85 *
86 * INFO (output) INTEGER
87 * On exit, INFO is set to
88 * 0: successful exit.
89 * 1: TL and TR have too close eigenvalues, so TL or
90 * TR is perturbed to get a nonsingular equation.
91 * NOTE: In the interests of speed, this routine does not
92 * check the inputs for errors.
93 *
94 * =====================================================================
95 *
96 * .. Parameters ..
97 DOUBLE PRECISION ZERO, ONE
98 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
99 DOUBLE PRECISION TWO, HALF, EIGHT
100 PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 )
101 * ..
102 * .. Local Scalars ..
103 LOGICAL BSWAP, XSWAP
104 INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
105 DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
106 $ TEMP, U11, U12, U22, XMAX
107 * ..
108 * .. Local Arrays ..
109 LOGICAL BSWPIV( 4 ), XSWPIV( 4 )
110 INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
111 $ LOCU22( 4 )
112 DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
113 * ..
114 * .. External Functions ..
115 INTEGER IDAMAX
116 DOUBLE PRECISION DLAMCH
117 EXTERNAL IDAMAX, DLAMCH
118 * ..
119 * .. External Subroutines ..
120 EXTERNAL DCOPY, DSWAP
121 * ..
122 * .. Intrinsic Functions ..
123 INTRINSIC ABS, MAX
124 * ..
125 * .. Data statements ..
126 DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
127 $ LOCU22 / 4, 3, 2, 1 /
128 DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
129 DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
130 * ..
131 * .. Executable Statements ..
132 *
133 * Do not check the input parameters for errors
134 *
135 INFO = 0
136 *
137 * Quick return if possible
138 *
139 IF( N1.EQ.0 .OR. N2.EQ.0 )
140 $ RETURN
141 *
142 * Set constants to control overflow
143 *
144 EPS = DLAMCH( 'P' )
145 SMLNUM = DLAMCH( 'S' ) / EPS
146 SGN = ISGN
147 *
148 K = N1 + N1 + N2 - 2
149 GO TO ( 10, 20, 30, 50 )K
150 *
151 * 1 by 1: TL11*X + SGN*X*TR11 = B11
152 *
153 10 CONTINUE
154 TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
155 BET = ABS( TAU1 )
156 IF( BET.LE.SMLNUM ) THEN
157 TAU1 = SMLNUM
158 BET = SMLNUM
159 INFO = 1
160 END IF
161 *
162 SCALE = ONE
163 GAM = ABS( B( 1, 1 ) )
164 IF( SMLNUM*GAM.GT.BET )
165 $ SCALE = ONE / GAM
166 *
167 X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
168 XNORM = ABS( X( 1, 1 ) )
169 RETURN
170 *
171 * 1 by 2:
172 * TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12]
173 * [TR21 TR22]
174 *
175 20 CONTINUE
176 *
177 SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
178 $ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
179 $ SMLNUM )
180 TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
181 TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
182 IF( LTRANR ) THEN
183 TMP( 2 ) = SGN*TR( 2, 1 )
184 TMP( 3 ) = SGN*TR( 1, 2 )
185 ELSE
186 TMP( 2 ) = SGN*TR( 1, 2 )
187 TMP( 3 ) = SGN*TR( 2, 1 )
188 END IF
189 BTMP( 1 ) = B( 1, 1 )
190 BTMP( 2 ) = B( 1, 2 )
191 GO TO 40
192 *
193 * 2 by 1:
194 * op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11]
195 * [TL21 TL22] [X21] [X21] [B21]
196 *
197 30 CONTINUE
198 SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
199 $ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
200 $ SMLNUM )
201 TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
202 TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
203 IF( LTRANL ) THEN
204 TMP( 2 ) = TL( 1, 2 )
205 TMP( 3 ) = TL( 2, 1 )
206 ELSE
207 TMP( 2 ) = TL( 2, 1 )
208 TMP( 3 ) = TL( 1, 2 )
209 END IF
210 BTMP( 1 ) = B( 1, 1 )
211 BTMP( 2 ) = B( 2, 1 )
212 40 CONTINUE
213 *
214 * Solve 2 by 2 system using complete pivoting.
215 * Set pivots less than SMIN to SMIN.
216 *
217 IPIV = IDAMAX( 4, TMP, 1 )
218 U11 = TMP( IPIV )
219 IF( ABS( U11 ).LE.SMIN ) THEN
220 INFO = 1
221 U11 = SMIN
222 END IF
223 U12 = TMP( LOCU12( IPIV ) )
224 L21 = TMP( LOCL21( IPIV ) ) / U11
225 U22 = TMP( LOCU22( IPIV ) ) - U12*L21
226 XSWAP = XSWPIV( IPIV )
227 BSWAP = BSWPIV( IPIV )
228 IF( ABS( U22 ).LE.SMIN ) THEN
229 INFO = 1
230 U22 = SMIN
231 END IF
232 IF( BSWAP ) THEN
233 TEMP = BTMP( 2 )
234 BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
235 BTMP( 1 ) = TEMP
236 ELSE
237 BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
238 END IF
239 SCALE = ONE
240 IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
241 $ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
242 SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
243 BTMP( 1 ) = BTMP( 1 )*SCALE
244 BTMP( 2 ) = BTMP( 2 )*SCALE
245 END IF
246 X2( 2 ) = BTMP( 2 ) / U22
247 X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
248 IF( XSWAP ) THEN
249 TEMP = X2( 2 )
250 X2( 2 ) = X2( 1 )
251 X2( 1 ) = TEMP
252 END IF
253 X( 1, 1 ) = X2( 1 )
254 IF( N1.EQ.1 ) THEN
255 X( 1, 2 ) = X2( 2 )
256 XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
257 ELSE
258 X( 2, 1 ) = X2( 2 )
259 XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
260 END IF
261 RETURN
262 *
263 * 2 by 2:
264 * op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
265 * [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
266 *
267 * Solve equivalent 4 by 4 system using complete pivoting.
268 * Set pivots less than SMIN to SMIN.
269 *
270 50 CONTINUE
271 SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
272 $ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
273 SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
274 $ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
275 SMIN = MAX( EPS*SMIN, SMLNUM )
276 BTMP( 1 ) = ZERO
277 CALL DCOPY( 16, BTMP, 0, T16, 1 )
278 T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
279 T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
280 T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
281 T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
282 IF( LTRANL ) THEN
283 T16( 1, 2 ) = TL( 2, 1 )
284 T16( 2, 1 ) = TL( 1, 2 )
285 T16( 3, 4 ) = TL( 2, 1 )
286 T16( 4, 3 ) = TL( 1, 2 )
287 ELSE
288 T16( 1, 2 ) = TL( 1, 2 )
289 T16( 2, 1 ) = TL( 2, 1 )
290 T16( 3, 4 ) = TL( 1, 2 )
291 T16( 4, 3 ) = TL( 2, 1 )
292 END IF
293 IF( LTRANR ) THEN
294 T16( 1, 3 ) = SGN*TR( 1, 2 )
295 T16( 2, 4 ) = SGN*TR( 1, 2 )
296 T16( 3, 1 ) = SGN*TR( 2, 1 )
297 T16( 4, 2 ) = SGN*TR( 2, 1 )
298 ELSE
299 T16( 1, 3 ) = SGN*TR( 2, 1 )
300 T16( 2, 4 ) = SGN*TR( 2, 1 )
301 T16( 3, 1 ) = SGN*TR( 1, 2 )
302 T16( 4, 2 ) = SGN*TR( 1, 2 )
303 END IF
304 BTMP( 1 ) = B( 1, 1 )
305 BTMP( 2 ) = B( 2, 1 )
306 BTMP( 3 ) = B( 1, 2 )
307 BTMP( 4 ) = B( 2, 2 )
308 *
309 * Perform elimination
310 *
311 DO 100 I = 1, 3
312 XMAX = ZERO
313 DO 70 IP = I, 4
314 DO 60 JP = I, 4
315 IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
316 XMAX = ABS( T16( IP, JP ) )
317 IPSV = IP
318 JPSV = JP
319 END IF
320 60 CONTINUE
321 70 CONTINUE
322 IF( IPSV.NE.I ) THEN
323 CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
324 TEMP = BTMP( I )
325 BTMP( I ) = BTMP( IPSV )
326 BTMP( IPSV ) = TEMP
327 END IF
328 IF( JPSV.NE.I )
329 $ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
330 JPIV( I ) = JPSV
331 IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
332 INFO = 1
333 T16( I, I ) = SMIN
334 END IF
335 DO 90 J = I + 1, 4
336 T16( J, I ) = T16( J, I ) / T16( I, I )
337 BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
338 DO 80 K = I + 1, 4
339 T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
340 80 CONTINUE
341 90 CONTINUE
342 100 CONTINUE
343 IF( ABS( T16( 4, 4 ) ).LT.SMIN )
344 $ T16( 4, 4 ) = SMIN
345 SCALE = ONE
346 IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
347 $ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
348 $ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
349 $ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
350 SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
351 $ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
352 BTMP( 1 ) = BTMP( 1 )*SCALE
353 BTMP( 2 ) = BTMP( 2 )*SCALE
354 BTMP( 3 ) = BTMP( 3 )*SCALE
355 BTMP( 4 ) = BTMP( 4 )*SCALE
356 END IF
357 DO 120 I = 1, 4
358 K = 5 - I
359 TEMP = ONE / T16( K, K )
360 TMP( K ) = BTMP( K )*TEMP
361 DO 110 J = K + 1, 4
362 TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
363 110 CONTINUE
364 120 CONTINUE
365 DO 130 I = 1, 3
366 IF( JPIV( 4-I ).NE.4-I ) THEN
367 TEMP = TMP( 4-I )
368 TMP( 4-I ) = TMP( JPIV( 4-I ) )
369 TMP( JPIV( 4-I ) ) = TEMP
370 END IF
371 130 CONTINUE
372 X( 1, 1 ) = TMP( 1 )
373 X( 2, 1 ) = TMP( 2 )
374 X( 1, 2 ) = TMP( 3 )
375 X( 2, 2 ) = TMP( 4 )
376 XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
377 $ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
378 RETURN
379 *
380 * End of DLASY2
381 *
382 END