LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
dlasy2.f
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1 *> \brief \b DLASY2 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 *
8 *> \htmlonly
9 *> Download DLASY2 + dependencies
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11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasy2.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasy2.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLASY2( 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 * DOUBLE PRECISION SCALE, XNORM
28 * ..
29 * .. Array Arguments ..
30 * DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
31 * $ X( LDX, * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> DLASY2 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 June 2016
170 *
171 *> \ingroup doubleSYauxiliary
172 *
173 * =====================================================================
174  SUBROUTINE dlasy2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
175  $ ldtr, b, ldb, scale, x, ldx, xnorm, info )
176 *
177 * -- LAPACK auxiliary routine (version 3.6.1) --
178 * -- LAPACK is a software package provided by Univ. of Tennessee, --
179 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
180 * June 2016
181 *
182 * .. Scalar Arguments ..
183  LOGICAL LTRANL, LTRANR
184  INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
185  DOUBLE PRECISION SCALE, XNORM
186 * ..
187 * .. Array Arguments ..
188  DOUBLE PRECISION B( ldb, * ), TL( ldtl, * ), TR( ldtr, * ),
189  $ x( ldx, * )
190 * ..
191 *
192 * =====================================================================
193 *
194 * .. Parameters ..
195  DOUBLE PRECISION ZERO, ONE
196  parameter ( zero = 0.0d+0, one = 1.0d+0 )
197  DOUBLE PRECISION TWO, HALF, EIGHT
198  parameter ( two = 2.0d+0, half = 0.5d+0, eight = 8.0d+0 )
199 * ..
200 * .. Local Scalars ..
201  LOGICAL BSWAP, XSWAP
202  INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
203  DOUBLE PRECISION 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  DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
211 * ..
212 * .. External Functions ..
213  INTEGER IDAMAX
214  DOUBLE PRECISION DLAMCH
215  EXTERNAL idamax, dlamch
216 * ..
217 * .. External Subroutines ..
218  EXTERNAL dcopy, dswap
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 = dlamch( 'P' )
243  smlnum = dlamch( '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 = idamax( 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 dcopy( 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 dswap( 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 dswap( 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 ) THEN
442  info = 1
443  t16( 4, 4 ) = smin
444  END IF
445  scale = one
446  IF( ( eight*smlnum )*abs( btmp( 1 ) ).GT.abs( t16( 1, 1 ) ) .OR.
447  $ ( eight*smlnum )*abs( btmp( 2 ) ).GT.abs( t16( 2, 2 ) ) .OR.
448  $ ( eight*smlnum )*abs( btmp( 3 ) ).GT.abs( t16( 3, 3 ) ) .OR.
449  $ ( eight*smlnum )*abs( btmp( 4 ) ).GT.abs( t16( 4, 4 ) ) ) THEN
450  scale = ( one / eight ) / max( abs( btmp( 1 ) ),
451  $ abs( btmp( 2 ) ), abs( btmp( 3 ) ), abs( btmp( 4 ) ) )
452  btmp( 1 ) = btmp( 1 )*scale
453  btmp( 2 ) = btmp( 2 )*scale
454  btmp( 3 ) = btmp( 3 )*scale
455  btmp( 4 ) = btmp( 4 )*scale
456  END IF
457  DO 120 i = 1, 4
458  k = 5 - i
459  temp = one / t16( k, k )
460  tmp( k ) = btmp( k )*temp
461  DO 110 j = k + 1, 4
462  tmp( k ) = tmp( k ) - ( temp*t16( k, j ) )*tmp( j )
463  110 CONTINUE
464  120 CONTINUE
465  DO 130 i = 1, 3
466  IF( jpiv( 4-i ).NE.4-i ) THEN
467  temp = tmp( 4-i )
468  tmp( 4-i ) = tmp( jpiv( 4-i ) )
469  tmp( jpiv( 4-i ) ) = temp
470  END IF
471  130 CONTINUE
472  x( 1, 1 ) = tmp( 1 )
473  x( 2, 1 ) = tmp( 2 )
474  x( 1, 2 ) = tmp( 3 )
475  x( 2, 2 ) = tmp( 4 )
476  xnorm = max( abs( tmp( 1 ) )+abs( tmp( 3 ) ),
477  $ abs( tmp( 2 ) )+abs( tmp( 4 ) ) )
478  RETURN
479 *
480 * End of DLASY2
481 *
482  END
dcopy
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:53
dlasy2
subroutine dlasy2(LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO)
DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
Definition: dlasy2.f:176
dswap
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:53