LAPACK  3.10.1 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 *
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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 *> \ingroup doubleSYauxiliary
170 *
171 * =====================================================================
172  SUBROUTINE dlasy2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
173  \$ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
174 *
175 * -- LAPACK auxiliary routine --
176 * -- LAPACK is a software package provided by Univ. of Tennessee, --
177 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
178 *
179 * .. Scalar Arguments ..
180  LOGICAL LTRANL, LTRANR
181  INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
182  DOUBLE PRECISION SCALE, XNORM
183 * ..
184 * .. Array Arguments ..
185  DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
186  \$ x( ldx, * )
187 * ..
188 *
189 * =====================================================================
190 *
191 * .. Parameters ..
192  DOUBLE PRECISION ZERO, ONE
193  parameter( zero = 0.0d+0, one = 1.0d+0 )
194  DOUBLE PRECISION TWO, HALF, EIGHT
195  parameter( two = 2.0d+0, half = 0.5d+0, eight = 8.0d+0 )
196 * ..
197 * .. Local Scalars ..
198  LOGICAL BSWAP, XSWAP
199  INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
200  DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
201  \$ temp, u11, u12, u22, xmax
202 * ..
203 * .. Local Arrays ..
204  LOGICAL BSWPIV( 4 ), XSWPIV( 4 )
205  INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
206  \$ locu22( 4 )
207  DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
208 * ..
209 * .. External Functions ..
210  INTEGER IDAMAX
211  DOUBLE PRECISION DLAMCH
212  EXTERNAL idamax, dlamch
213 * ..
214 * .. External Subroutines ..
215  EXTERNAL dcopy, dswap
216 * ..
217 * .. Intrinsic Functions ..
218  INTRINSIC abs, max
219 * ..
220 * .. Data statements ..
221  DATA locu12 / 3, 4, 1, 2 / , locl21 / 2, 1, 4, 3 / ,
222  \$ locu22 / 4, 3, 2, 1 /
223  DATA xswpiv / .false., .false., .true., .true. /
224  DATA bswpiv / .false., .true., .false., .true. /
225 * ..
226 * .. Executable Statements ..
227 *
228 * Do not check the input parameters for errors
229 *
230  info = 0
231 *
232 * Quick return if possible
233 *
234  IF( n1.EQ.0 .OR. n2.EQ.0 )
235  \$ RETURN
236 *
237 * Set constants to control overflow
238 *
239  eps = dlamch( 'P' )
240  smlnum = dlamch( 'S' ) / eps
241  sgn = isgn
242 *
243  k = n1 + n1 + n2 - 2
244  GO TO ( 10, 20, 30, 50 )k
245 *
246 * 1 by 1: TL11*X + SGN*X*TR11 = B11
247 *
248  10 CONTINUE
249  tau1 = tl( 1, 1 ) + sgn*tr( 1, 1 )
250  bet = abs( tau1 )
251  IF( bet.LE.smlnum ) THEN
252  tau1 = smlnum
253  bet = smlnum
254  info = 1
255  END IF
256 *
257  scale = one
258  gam = abs( b( 1, 1 ) )
259  IF( smlnum*gam.GT.bet )
260  \$ scale = one / gam
261 *
262  x( 1, 1 ) = ( b( 1, 1 )*scale ) / tau1
263  xnorm = abs( x( 1, 1 ) )
264  RETURN
265 *
266 * 1 by 2:
267 * TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12]
268 * [TR21 TR22]
269 *
270  20 CONTINUE
271 *
272  smin = max( eps*max( abs( tl( 1, 1 ) ), abs( tr( 1, 1 ) ),
273  \$ abs( tr( 1, 2 ) ), abs( tr( 2, 1 ) ), abs( tr( 2, 2 ) ) ),
274  \$ smlnum )
275  tmp( 1 ) = tl( 1, 1 ) + sgn*tr( 1, 1 )
276  tmp( 4 ) = tl( 1, 1 ) + sgn*tr( 2, 2 )
277  IF( ltranr ) THEN
278  tmp( 2 ) = sgn*tr( 2, 1 )
279  tmp( 3 ) = sgn*tr( 1, 2 )
280  ELSE
281  tmp( 2 ) = sgn*tr( 1, 2 )
282  tmp( 3 ) = sgn*tr( 2, 1 )
283  END IF
284  btmp( 1 ) = b( 1, 1 )
285  btmp( 2 ) = b( 1, 2 )
286  GO TO 40
287 *
288 * 2 by 1:
289 * op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11]
290 * [TL21 TL22] [X21] [X21] [B21]
291 *
292  30 CONTINUE
293  smin = max( eps*max( abs( tr( 1, 1 ) ), abs( tl( 1, 1 ) ),
294  \$ abs( tl( 1, 2 ) ), abs( tl( 2, 1 ) ), abs( tl( 2, 2 ) ) ),
295  \$ smlnum )
296  tmp( 1 ) = tl( 1, 1 ) + sgn*tr( 1, 1 )
297  tmp( 4 ) = tl( 2, 2 ) + sgn*tr( 1, 1 )
298  IF( ltranl ) THEN
299  tmp( 2 ) = tl( 1, 2 )
300  tmp( 3 ) = tl( 2, 1 )
301  ELSE
302  tmp( 2 ) = tl( 2, 1 )
303  tmp( 3 ) = tl( 1, 2 )
304  END IF
305  btmp( 1 ) = b( 1, 1 )
306  btmp( 2 ) = b( 2, 1 )
307  40 CONTINUE
308 *
309 * Solve 2 by 2 system using complete pivoting.
310 * Set pivots less than SMIN to SMIN.
311 *
312  ipiv = idamax( 4, tmp, 1 )
313  u11 = tmp( ipiv )
314  IF( abs( u11 ).LE.smin ) THEN
315  info = 1
316  u11 = smin
317  END IF
318  u12 = tmp( locu12( ipiv ) )
319  l21 = tmp( locl21( ipiv ) ) / u11
320  u22 = tmp( locu22( ipiv ) ) - u12*l21
321  xswap = xswpiv( ipiv )
322  bswap = bswpiv( ipiv )
323  IF( abs( u22 ).LE.smin ) THEN
324  info = 1
325  u22 = smin
326  END IF
327  IF( bswap ) THEN
328  temp = btmp( 2 )
329  btmp( 2 ) = btmp( 1 ) - l21*temp
330  btmp( 1 ) = temp
331  ELSE
332  btmp( 2 ) = btmp( 2 ) - l21*btmp( 1 )
333  END IF
334  scale = one
335  IF( ( two*smlnum )*abs( btmp( 2 ) ).GT.abs( u22 ) .OR.
336  \$ ( two*smlnum )*abs( btmp( 1 ) ).GT.abs( u11 ) ) THEN
337  scale = half / max( abs( btmp( 1 ) ), abs( btmp( 2 ) ) )
338  btmp( 1 ) = btmp( 1 )*scale
339  btmp( 2 ) = btmp( 2 )*scale
340  END IF
341  x2( 2 ) = btmp( 2 ) / u22
342  x2( 1 ) = btmp( 1 ) / u11 - ( u12 / u11 )*x2( 2 )
343  IF( xswap ) THEN
344  temp = x2( 2 )
345  x2( 2 ) = x2( 1 )
346  x2( 1 ) = temp
347  END IF
348  x( 1, 1 ) = x2( 1 )
349  IF( n1.EQ.1 ) THEN
350  x( 1, 2 ) = x2( 2 )
351  xnorm = abs( x( 1, 1 ) ) + abs( x( 1, 2 ) )
352  ELSE
353  x( 2, 1 ) = x2( 2 )
354  xnorm = max( abs( x( 1, 1 ) ), abs( x( 2, 1 ) ) )
355  END IF
356  RETURN
357 *
358 * 2 by 2:
359 * op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
360 * [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
361 *
362 * Solve equivalent 4 by 4 system using complete pivoting.
363 * Set pivots less than SMIN to SMIN.
364 *
365  50 CONTINUE
366  smin = max( abs( tr( 1, 1 ) ), abs( tr( 1, 2 ) ),
367  \$ abs( tr( 2, 1 ) ), abs( tr( 2, 2 ) ) )
368  smin = max( smin, abs( tl( 1, 1 ) ), abs( tl( 1, 2 ) ),
369  \$ abs( tl( 2, 1 ) ), abs( tl( 2, 2 ) ) )
370  smin = max( eps*smin, smlnum )
371  btmp( 1 ) = zero
372  CALL dcopy( 16, btmp, 0, t16, 1 )
373  t16( 1, 1 ) = tl( 1, 1 ) + sgn*tr( 1, 1 )
374  t16( 2, 2 ) = tl( 2, 2 ) + sgn*tr( 1, 1 )
375  t16( 3, 3 ) = tl( 1, 1 ) + sgn*tr( 2, 2 )
376  t16( 4, 4 ) = tl( 2, 2 ) + sgn*tr( 2, 2 )
377  IF( ltranl ) THEN
378  t16( 1, 2 ) = tl( 2, 1 )
379  t16( 2, 1 ) = tl( 1, 2 )
380  t16( 3, 4 ) = tl( 2, 1 )
381  t16( 4, 3 ) = tl( 1, 2 )
382  ELSE
383  t16( 1, 2 ) = tl( 1, 2 )
384  t16( 2, 1 ) = tl( 2, 1 )
385  t16( 3, 4 ) = tl( 1, 2 )
386  t16( 4, 3 ) = tl( 2, 1 )
387  END IF
388  IF( ltranr ) THEN
389  t16( 1, 3 ) = sgn*tr( 1, 2 )
390  t16( 2, 4 ) = sgn*tr( 1, 2 )
391  t16( 3, 1 ) = sgn*tr( 2, 1 )
392  t16( 4, 2 ) = sgn*tr( 2, 1 )
393  ELSE
394  t16( 1, 3 ) = sgn*tr( 2, 1 )
395  t16( 2, 4 ) = sgn*tr( 2, 1 )
396  t16( 3, 1 ) = sgn*tr( 1, 2 )
397  t16( 4, 2 ) = sgn*tr( 1, 2 )
398  END IF
399  btmp( 1 ) = b( 1, 1 )
400  btmp( 2 ) = b( 2, 1 )
401  btmp( 3 ) = b( 1, 2 )
402  btmp( 4 ) = b( 2, 2 )
403 *
404 * Perform elimination
405 *
406  DO 100 i = 1, 3
407  xmax = zero
408  DO 70 ip = i, 4
409  DO 60 jp = i, 4
410  IF( abs( t16( ip, jp ) ).GE.xmax ) THEN
411  xmax = abs( t16( ip, jp ) )
412  ipsv = ip
413  jpsv = jp
414  END IF
415  60 CONTINUE
416  70 CONTINUE
417  IF( ipsv.NE.i ) THEN
418  CALL dswap( 4, t16( ipsv, 1 ), 4, t16( i, 1 ), 4 )
419  temp = btmp( i )
420  btmp( i ) = btmp( ipsv )
421  btmp( ipsv ) = temp
422  END IF
423  IF( jpsv.NE.i )
424  \$ CALL dswap( 4, t16( 1, jpsv ), 1, t16( 1, i ), 1 )
425  jpiv( i ) = jpsv
426  IF( abs( t16( i, i ) ).LT.smin ) THEN
427  info = 1
428  t16( i, i ) = smin
429  END IF
430  DO 90 j = i + 1, 4
431  t16( j, i ) = t16( j, i ) / t16( i, i )
432  btmp( j ) = btmp( j ) - t16( j, i )*btmp( i )
433  DO 80 k = i + 1, 4
434  t16( j, k ) = t16( j, k ) - t16( j, i )*t16( i, k )
435  80 CONTINUE
436  90 CONTINUE
437  100 CONTINUE
438  IF( abs( t16( 4, 4 ) ).LT.smin ) THEN
439  info = 1
440  t16( 4, 4 ) = smin
441  END IF
442  scale = one
443  IF( ( eight*smlnum )*abs( btmp( 1 ) ).GT.abs( t16( 1, 1 ) ) .OR.
444  \$ ( eight*smlnum )*abs( btmp( 2 ) ).GT.abs( t16( 2, 2 ) ) .OR.
445  \$ ( eight*smlnum )*abs( btmp( 3 ) ).GT.abs( t16( 3, 3 ) ) .OR.
446  \$ ( eight*smlnum )*abs( btmp( 4 ) ).GT.abs( t16( 4, 4 ) ) ) THEN
447  scale = ( one / eight ) / max( abs( btmp( 1 ) ),
448  \$ abs( btmp( 2 ) ), abs( btmp( 3 ) ), abs( btmp( 4 ) ) )
449  btmp( 1 ) = btmp( 1 )*scale
450  btmp( 2 ) = btmp( 2 )*scale
451  btmp( 3 ) = btmp( 3 )*scale
452  btmp( 4 ) = btmp( 4 )*scale
453  END IF
454  DO 120 i = 1, 4
455  k = 5 - i
456  temp = one / t16( k, k )
457  tmp( k ) = btmp( k )*temp
458  DO 110 j = k + 1, 4
459  tmp( k ) = tmp( k ) - ( temp*t16( k, j ) )*tmp( j )
460  110 CONTINUE
461  120 CONTINUE
462  DO 130 i = 1, 3
463  IF( jpiv( 4-i ).NE.4-i ) THEN
464  temp = tmp( 4-i )
465  tmp( 4-i ) = tmp( jpiv( 4-i ) )
466  tmp( jpiv( 4-i ) ) = temp
467  END IF
468  130 CONTINUE
469  x( 1, 1 ) = tmp( 1 )
470  x( 2, 1 ) = tmp( 2 )
471  x( 1, 2 ) = tmp( 3 )
472  x( 2, 2 ) = tmp( 4 )
473  xnorm = max( abs( tmp( 1 ) )+abs( tmp( 3 ) ),
474  \$ abs( tmp( 2 ) )+abs( tmp( 4 ) ) )
475  RETURN
476 *
477 * End of DLASY2
478 *
479  END
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
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:174