1*> \brief \b SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
22*                          LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
23*
24*       .. Scalar Arguments ..
25*       LOGICAL            LTRANL, LTRANR
26*       INTEGER            INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
27*       REAL               SCALE, XNORM
28*       ..
29*       .. Array Arguments ..
30*       REAL               B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
31*      $                   X( LDX, * )
32*       ..
33*
34*
35*> \par Purpose:
36*  =============
37*>
38*> \verbatim
39*>
40*> SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
41*>
42*>        op(TL)*X + ISGN*X*op(TR) = SCALE*B,
43*>
44*> where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
45*> -1.  op(T) = T or T**T, where T**T denotes the transpose of T.
46*> \endverbatim
47*
48*  Arguments:
49*  ==========
50*
51*> \param[in] LTRANL
52*> \verbatim
53*>          LTRANL is LOGICAL
54*>          On entry, LTRANL specifies the op(TL):
55*>             = .FALSE., op(TL) = TL,
56*>             = .TRUE., op(TL) = TL**T.
57*> \endverbatim
58*>
59*> \param[in] LTRANR
60*> \verbatim
61*>          LTRANR is LOGICAL
62*>          On entry, LTRANR specifies the op(TR):
63*>            = .FALSE., op(TR) = TR,
64*>            = .TRUE., op(TR) = TR**T.
65*> \endverbatim
66*>
67*> \param[in] ISGN
68*> \verbatim
69*>          ISGN is INTEGER
70*>          On entry, ISGN specifies the sign of the equation
71*>          as described before. ISGN may only be 1 or -1.
72*> \endverbatim
73*>
74*> \param[in] N1
75*> \verbatim
76*>          N1 is INTEGER
77*>          On entry, N1 specifies the order of matrix TL.
78*>          N1 may only be 0, 1 or 2.
79*> \endverbatim
80*>
81*> \param[in] N2
82*> \verbatim
83*>          N2 is INTEGER
84*>          On entry, N2 specifies the order of matrix TR.
85*>          N2 may only be 0, 1 or 2.
86*> \endverbatim
87*>
88*> \param[in] TL
89*> \verbatim
90*>          TL is REAL array, dimension (LDTL,2)
91*>          On entry, TL contains an N1 by N1 matrix.
92*> \endverbatim
93*>
94*> \param[in] LDTL
95*> \verbatim
96*>          LDTL is INTEGER
97*>          The leading dimension of the matrix TL. LDTL >= max(1,N1).
98*> \endverbatim
99*>
100*> \param[in] TR
101*> \verbatim
102*>          TR is REAL array, dimension (LDTR,2)
103*>          On entry, TR contains an N2 by N2 matrix.
104*> \endverbatim
105*>
106*> \param[in] LDTR
107*> \verbatim
108*>          LDTR is INTEGER
109*>          The leading dimension of the matrix TR. LDTR >= max(1,N2).
110*> \endverbatim
111*>
112*> \param[in] B
113*> \verbatim
114*>          B is REAL array, dimension (LDB,2)
115*>          On entry, the N1 by N2 matrix B contains the right-hand
116*>          side of the equation.
117*> \endverbatim
118*>
119*> \param[in] LDB
120*> \verbatim
121*>          LDB is INTEGER
122*>          The leading dimension of the matrix B. LDB >= max(1,N1).
123*> \endverbatim
124*>
125*> \param[out] SCALE
126*> \verbatim
127*>          SCALE is REAL
128*>          On exit, SCALE contains the scale factor. SCALE is chosen
129*>          less than or equal to 1 to prevent the solution overflowing.
130*> \endverbatim
131*>
132*> \param[out] X
133*> \verbatim
134*>          X is REAL array, dimension (LDX,2)
135*>          On exit, X contains the N1 by N2 solution.
136*> \endverbatim
137*>
138*> \param[in] LDX
139*> \verbatim
140*>          LDX is INTEGER
141*>          The leading dimension of the matrix X. LDX >= max(1,N1).
142*> \endverbatim
143*>
144*> \param[out] XNORM
145*> \verbatim
146*>          XNORM is REAL
147*>          On exit, XNORM is the infinity-norm of the solution.
148*> \endverbatim
149*>
150*> \param[out] INFO
151*> \verbatim
152*>          INFO is INTEGER
153*>          On exit, INFO is set to
154*>             0: successful exit.
155*>             1: TL and TR have too close eigenvalues, so TL or
156*>                TR is perturbed to get a nonsingular equation.
157*>          NOTE: In the interests of speed, this routine does not
158*>                check the inputs for errors.
159*> \endverbatim
160*
161*  Authors:
162*  ========
163*
164*> \author Univ. of Tennessee
165*> \author Univ. of California Berkeley
166*> \author Univ. of Colorado Denver
167*> \author NAG Ltd.
168*
169*> \date September 2012
170*
171*> \ingroup realSYauxiliary
172*
173*  =====================================================================
174      SUBROUTINE SLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
175     $                   LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
176*
177*  -- LAPACK auxiliary routine (version 3.4.2) --
178*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
179*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
180*     September 2012
181*
182*     .. Scalar Arguments ..
183      LOGICAL            LTRANL, LTRANR
184      INTEGER            INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
185      REAL               SCALE, XNORM
186*     ..
187*     .. Array Arguments ..
188      REAL               B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
189     $                   X( LDX, * )
190*     ..
191*
192* =====================================================================
193*
194*     .. Parameters ..
195      REAL               ZERO, ONE
196      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
197      REAL               TWO, HALF, EIGHT
198      PARAMETER          ( TWO = 2.0E+0, HALF = 0.5E+0, EIGHT = 8.0E+0 )
199*     ..
200*     .. Local Scalars ..
201      LOGICAL            BSWAP, XSWAP
202      INTEGER            I, IP, IPIV, IPSV, J, JP, JPSV, K
203      REAL               BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
204     $                   TEMP, U11, U12, U22, XMAX
205*     ..
206*     .. Local Arrays ..
207      LOGICAL            BSWPIV( 4 ), XSWPIV( 4 )
208      INTEGER            JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
209     $                   LOCU22( 4 )
210      REAL               BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
211*     ..
212*     .. External Functions ..
213      INTEGER            ISAMAX
214      REAL               SLAMCH
215      EXTERNAL           ISAMAX, SLAMCH
216*     ..
217*     .. External Subroutines ..
218      EXTERNAL           SCOPY, SSWAP
219*     ..
220*     .. Intrinsic Functions ..
221      INTRINSIC          ABS, MAX
222*     ..
223*     .. Data statements ..
224      DATA               LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
225     $                   LOCU22 / 4, 3, 2, 1 /
226      DATA               XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
227      DATA               BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
228*     ..
229*     .. Executable Statements ..
230*
231*     Do not check the input parameters for errors
232*
233      INFO = 0
234*
235*     Quick return if possible
236*
237      IF( N1.EQ.0 .OR. N2.EQ.0 )
238     $   RETURN
239*
240*     Set constants to control overflow
241*
242      EPS = SLAMCH( 'P' )
243      SMLNUM = SLAMCH( 'S' ) / EPS
244      SGN = ISGN
245*
246      K = N1 + N1 + N2 - 2
247      GO TO ( 10, 20, 30, 50 )K
248*
249*     1 by 1: TL11*X + SGN*X*TR11 = B11
250*
251   10 CONTINUE
252      TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
253      BET = ABS( TAU1 )
254      IF( BET.LE.SMLNUM ) THEN
255         TAU1 = SMLNUM
256         BET = SMLNUM
257         INFO = 1
258      END IF
259*
260      SCALE = ONE
261      GAM = ABS( B( 1, 1 ) )
262      IF( SMLNUM*GAM.GT.BET )
263     $   SCALE = ONE / GAM
264*
265      X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
266      XNORM = ABS( X( 1, 1 ) )
267      RETURN
268*
269*     1 by 2:
270*     TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12]  = [B11 B12]
271*                                       [TR21 TR22]
272*
273   20 CONTINUE
274*
275      SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
276     $       ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
277     $       SMLNUM )
278      TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
279      TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
280      IF( LTRANR ) THEN
281         TMP( 2 ) = SGN*TR( 2, 1 )
282         TMP( 3 ) = SGN*TR( 1, 2 )
283      ELSE
284         TMP( 2 ) = SGN*TR( 1, 2 )
285         TMP( 3 ) = SGN*TR( 2, 1 )
286      END IF
287      BTMP( 1 ) = B( 1, 1 )
288      BTMP( 2 ) = B( 1, 2 )
289      GO TO 40
290*
291*     2 by 1:
292*          op[TL11 TL12]*[X11] + ISGN* [X11]*TR11  = [B11]
293*            [TL21 TL22] [X21]         [X21]         [B21]
294*
295   30 CONTINUE
296      SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
297     $       ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
298     $       SMLNUM )
299      TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
300      TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
301      IF( LTRANL ) THEN
302         TMP( 2 ) = TL( 1, 2 )
303         TMP( 3 ) = TL( 2, 1 )
304      ELSE
305         TMP( 2 ) = TL( 2, 1 )
306         TMP( 3 ) = TL( 1, 2 )
307      END IF
308      BTMP( 1 ) = B( 1, 1 )
309      BTMP( 2 ) = B( 2, 1 )
310   40 CONTINUE
311*
312*     Solve 2 by 2 system using complete pivoting.
313*     Set pivots less than SMIN to SMIN.
314*
315      IPIV = ISAMAX( 4, TMP, 1 )
316      U11 = TMP( IPIV )
317      IF( ABS( U11 ).LE.SMIN ) THEN
318         INFO = 1
319         U11 = SMIN
320      END IF
321      U12 = TMP( LOCU12( IPIV ) )
322      L21 = TMP( LOCL21( IPIV ) ) / U11
323      U22 = TMP( LOCU22( IPIV ) ) - U12*L21
324      XSWAP = XSWPIV( IPIV )
325      BSWAP = BSWPIV( IPIV )
326      IF( ABS( U22 ).LE.SMIN ) THEN
327         INFO = 1
328         U22 = SMIN
329      END IF
330      IF( BSWAP ) THEN
331         TEMP = BTMP( 2 )
332         BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
333         BTMP( 1 ) = TEMP
334      ELSE
335         BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
336      END IF
337      SCALE = ONE
338      IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
339     $    ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
340         SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
341         BTMP( 1 ) = BTMP( 1 )*SCALE
342         BTMP( 2 ) = BTMP( 2 )*SCALE
343      END IF
344      X2( 2 ) = BTMP( 2 ) / U22
345      X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
346      IF( XSWAP ) THEN
347         TEMP = X2( 2 )
348         X2( 2 ) = X2( 1 )
349         X2( 1 ) = TEMP
350      END IF
351      X( 1, 1 ) = X2( 1 )
352      IF( N1.EQ.1 ) THEN
353         X( 1, 2 ) = X2( 2 )
354         XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
355      ELSE
356         X( 2, 1 ) = X2( 2 )
357         XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
358      END IF
359      RETURN
360*
361*     2 by 2:
362*     op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
363*       [TL21 TL22] [X21 X22]        [X21 X22]   [TR21 TR22]   [B21 B22]
364*
365*     Solve equivalent 4 by 4 system using complete pivoting.
366*     Set pivots less than SMIN to SMIN.
367*
368   50 CONTINUE
369      SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
370     $       ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
371      SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
372     $       ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
373      SMIN = MAX( EPS*SMIN, SMLNUM )
374      BTMP( 1 ) = ZERO
375      CALL SCOPY( 16, BTMP, 0, T16, 1 )
376      T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
377      T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
378      T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
379      T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
380      IF( LTRANL ) THEN
381         T16( 1, 2 ) = TL( 2, 1 )
382         T16( 2, 1 ) = TL( 1, 2 )
383         T16( 3, 4 ) = TL( 2, 1 )
384         T16( 4, 3 ) = TL( 1, 2 )
385      ELSE
386         T16( 1, 2 ) = TL( 1, 2 )
387         T16( 2, 1 ) = TL( 2, 1 )
388         T16( 3, 4 ) = TL( 1, 2 )
389         T16( 4, 3 ) = TL( 2, 1 )
390      END IF
391      IF( LTRANR ) THEN
392         T16( 1, 3 ) = SGN*TR( 1, 2 )
393         T16( 2, 4 ) = SGN*TR( 1, 2 )
394         T16( 3, 1 ) = SGN*TR( 2, 1 )
395         T16( 4, 2 ) = SGN*TR( 2, 1 )
396      ELSE
397         T16( 1, 3 ) = SGN*TR( 2, 1 )
398         T16( 2, 4 ) = SGN*TR( 2, 1 )
399         T16( 3, 1 ) = SGN*TR( 1, 2 )
400         T16( 4, 2 ) = SGN*TR( 1, 2 )
401      END IF
402      BTMP( 1 ) = B( 1, 1 )
403      BTMP( 2 ) = B( 2, 1 )
404      BTMP( 3 ) = B( 1, 2 )
405      BTMP( 4 ) = B( 2, 2 )
406*
407*     Perform elimination
408*
409      DO 100 I = 1, 3
410         XMAX = ZERO
411         DO 70 IP = I, 4
412            DO 60 JP = I, 4
413               IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
414                  XMAX = ABS( T16( IP, JP ) )
415                  IPSV = IP
416                  JPSV = JP
417               END IF
418   60       CONTINUE
419   70    CONTINUE
420         IF( IPSV.NE.I ) THEN
421            CALL SSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
422            TEMP = BTMP( I )
423            BTMP( I ) = BTMP( IPSV )
424            BTMP( IPSV ) = TEMP
425         END IF
426         IF( JPSV.NE.I )
427     $      CALL SSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
428         JPIV( I ) = JPSV
429         IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
430            INFO = 1
431            T16( I, I ) = SMIN
432         END IF
433         DO 90 J = I + 1, 4
434            T16( J, I ) = T16( J, I ) / T16( I, I )
435            BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
436            DO 80 K = I + 1, 4
437               T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
438   80       CONTINUE
439   90    CONTINUE
440  100 CONTINUE
441      IF( ABS( T16( 4, 4 ) ).LT.SMIN )
442     $   T16( 4, 4 ) = SMIN
443      SCALE = ONE
444      IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
445     $    ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
446     $    ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
447     $    ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
448         SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
449     $           ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
450         BTMP( 1 ) = BTMP( 1 )*SCALE
451         BTMP( 2 ) = BTMP( 2 )*SCALE
452         BTMP( 3 ) = BTMP( 3 )*SCALE
453         BTMP( 4 ) = BTMP( 4 )*SCALE
454      END IF
455      DO 120 I = 1, 4
456         K = 5 - I
457         TEMP = ONE / T16( K, K )
458         TMP( K ) = BTMP( K )*TEMP
459         DO 110 J = K + 1, 4
460            TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
461  110    CONTINUE
462  120 CONTINUE
463      DO 130 I = 1, 3
464         IF( JPIV( 4-I ).NE.4-I ) THEN
465            TEMP = TMP( 4-I )
466            TMP( 4-I ) = TMP( JPIV( 4-I ) )
467            TMP( JPIV( 4-I ) ) = TEMP
468         END IF
469  130 CONTINUE
470      X( 1, 1 ) = TMP( 1 )
471      X( 2, 1 ) = TMP( 2 )
472      X( 1, 2 ) = TMP( 3 )
473      X( 2, 2 ) = TMP( 4 )
474      XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
475     $        ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
476      RETURN
477*
478*     End of SLASY2
479*
480      END
481c $Id$
482