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 3.0) --
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
383