Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.9/dist-packages/scipy/sparse/linalg/_isolve/

2# Distributed under the same license as SciPy.

4import warnings

5import numpy as np

6from numpy.linalg import LinAlgError

7from scipy.linalg import (get_blas_funcs, qr, solve, svd, qr_insert, lstsq)

8from scipy.sparse.linalg._isolve.utils import make_system

9from scipy._lib.deprecation import _deprecate_positional_args

12__all__ = ['gcrotmk']

15def _fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=(), outer_v=(),

16 prepend_outer_v=False):

17 """

18 FGMRES Arnoldi process, with optional projection or augmentation

20 Parameters

21 ----------

22 matvec : callable

23 Operation A*x

24 v0 : ndarray

25 Initial vector, normalized to nrm2(v0) == 1

26 m : int

27 Number of GMRES rounds

28 atol : float

29 Absolute tolerance for early exit

30 lpsolve : callable

31 Left preconditioner L

32 rpsolve : callable

33 Right preconditioner R

34 cs : list of (ndarray, ndarray)

35 Columns of matrices C and U in GCROT

36 outer_v : list of ndarrays

37 Augmentation vectors in LGMRES

38 prepend_outer_v : bool, optional

39 Whether augmentation vectors come before or after

40 Krylov iterates

42 Raises

43 ------

44 LinAlgError

45 If nans encountered

47 Returns

48 -------

49 Q, R : ndarray

50 QR decomposition of the upper Hessenberg H=QR

51 B : ndarray

52 Projections corresponding to matrix C

53 vs : list of ndarray

54 Columns of matrix V

55 zs : list of ndarray

56 Columns of matrix Z

57 y : ndarray

58 Solution to ||H y - e_1||_2 = min!

59 res : float

60 The final (preconditioned) residual norm

62 """

64 if lpsolve is None:

65 def lpsolve(x):

66 return x

67 if rpsolve is None:

68 def rpsolve(x):

69 return x

71 axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (v0,))

73 vs = [v0]

74 zs = []

75 y = None

76 res = np.nan

78 m = m + len(outer_v)

80 # Orthogonal projection coefficients

81 B = np.zeros((len(cs), m), dtype=v0.dtype)

83 # H is stored in QR factorized form

84 Q = np.ones((1, 1), dtype=v0.dtype)

85 R = np.zeros((1, 0), dtype=v0.dtype)

87 eps = np.finfo(v0.dtype).eps

89 breakdown = False

91 # FGMRES Arnoldi process

92 for j in range(m):

93 # L A Z = C B + V H

95 if prepend_outer_v and j < len(outer_v):

96 z, w = outer_v[j]

97 elif prepend_outer_v and j == len(outer_v):

98 z = rpsolve(v0)

99 w = None

100 elif not prepend_outer_v and j >= m - len(outer_v):

101 z, w = outer_v[j - (m - len(outer_v))]

102 else:

103 z = rpsolve(vs[-1])

104 w = None

105

106 if w is None:

107 w = lpsolve(matvec(z))

108 else:

109 # w is clobbered below

110 w = w.copy()

111

112 w_norm = nrm2(w)

113

114 # GCROT projection: L A -> (1 - C C^H) L A

115 # i.e. orthogonalize against C

116 for i, c in enumerate(cs):

117 alpha = dot(c, w)

118 B[i,j] = alpha

119 w = axpy(c, w, c.shape[0], -alpha) # w -= alpha*c

120

121 # Orthogonalize against V

122 hcur = np.zeros(j+2, dtype=Q.dtype)

123 for i, v in enumerate(vs):

124 alpha = dot(v, w)

125 hcur[i] = alpha

126 w = axpy(v, w, v.shape[0], -alpha) # w -= alpha*v

127 hcur[i+1] = nrm2(w)

128

129 with np.errstate(over='ignore', divide='ignore'):

130 # Careful with denormals

131 alpha = 1/hcur[-1]

132

133 if np.isfinite(alpha):

134 w = scal(alpha, w)

135

136 if not (hcur[-1] > eps * w_norm):

137 # w essentially in the span of previous vectors,

138 # or we have nans. Bail out after updating the QR

139 # solution.

140 breakdown = True

141

142 vs.append(w)

143 zs.append(z)

144

145 # Arnoldi LSQ problem

146

147 # Add new column to H=Q@R, padding other columns with zeros

148 Q2 = np.zeros((j+2, j+2), dtype=Q.dtype, order='F')

149 Q2[:j+1,:j+1] = Q

150 Q2[j+1,j+1] = 1

151

152 R2 = np.zeros((j+2, j), dtype=R.dtype, order='F')

153 R2[:j+1,:] = R

154

155 Q, R = qr_insert(Q2, R2, hcur, j, which='col',

156 overwrite_qru=True, check_finite=False)

157

158 # Transformed least squares problem

159 # || Q R y - inner_res_0 * e_1 ||_2 = min!

160 # Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0]

161

162 # Residual is immediately known

163 res = abs(Q[0,-1])

164

165 # Check for termination

166 if res < atol or breakdown:

167 break

168

169 if not np.isfinite(R[j,j]):

170 # nans encountered, bail out

171 raise LinAlgError()

172

173 # -- Get the LSQ problem solution

174

175 # The problem is triangular, but the condition number may be

176 # bad (or in case of breakdown the last diagonal entry may be

177 # zero), so use lstsq instead of trtrs.

178 y, _, _, _, = lstsq(R[:j+1,:j+1], Q[0,:j+1].conj())

179

180 B = B[:,:j+1]

181

182 return Q, R, B, vs, zs, y, res

183

184

185@_deprecate_positional_args(version="1.14.0")

186def gcrotmk(A, b, x0=None, *, tol=1e-5, maxiter=1000, M=None, callback=None,

187 m=20, k=None, CU=None, discard_C=False, truncate='oldest',

188 atol=None):

189 """

190 Solve a matrix equation using flexible GCROT(m,k) algorithm.

191

192 Parameters

193 ----------

194 A : {sparse matrix, ndarray, LinearOperator}

195 The real or complex N-by-N matrix of the linear system.

196 Alternatively, ``A`` can be a linear operator which can

197 produce ``Ax`` using, e.g.,

198 ``scipy.sparse.linalg.LinearOperator``.

199 b : ndarray

200 Right hand side of the linear system. Has shape (N,) or (N,1).

201 x0 : ndarray

202 Starting guess for the solution.

203 tol, atol : float, optional

204 Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``.

205 The default for ``atol`` is `tol`.

206

207 .. warning::

208

209 The default value for `atol` will be changed in a future release.

210 For future compatibility, specify `atol` explicitly.

211 maxiter : int, optional

212 Maximum number of iterations. Iteration will stop after maxiter

213 steps even if the specified tolerance has not been achieved.

214 M : {sparse matrix, ndarray, LinearOperator}, optional

215 Preconditioner for A. The preconditioner should approximate the

216 inverse of A. gcrotmk is a 'flexible' algorithm and the preconditioner

217 can vary from iteration to iteration. Effective preconditioning

218 dramatically improves the rate of convergence, which implies that

219 fewer iterations are needed to reach a given error tolerance.

220 callback : function, optional

221 User-supplied function to call after each iteration. It is called

222 as callback(xk), where xk is the current solution vector.

223 m : int, optional

224 Number of inner FGMRES iterations per each outer iteration.

225 Default: 20

226 k : int, optional

227 Number of vectors to carry between inner FGMRES iterations.

228 According to [2]_, good values are around m.

229 Default: m

230 CU : list of tuples, optional

231 List of tuples ``(c, u)`` which contain the columns of the matrices

232 C and U in the GCROT(m,k) algorithm. For details, see [2]_.

233 The list given and vectors contained in it are modified in-place.

234 If not given, start from empty matrices. The ``c`` elements in the

235 tuples can be ``None``, in which case the vectors are recomputed

236 via ``c = A u`` on start and orthogonalized as described in [3]_.

237 discard_C : bool, optional

238 Discard the C-vectors at the end. Useful if recycling Krylov subspaces

239 for different linear systems.

240 truncate : {'oldest', 'smallest'}, optional

241 Truncation scheme to use. Drop: oldest vectors, or vectors with

242 smallest singular values using the scheme discussed in [1,2].

243 See [2]_ for detailed comparison.

244 Default: 'oldest'

245

246 Returns

247 -------

248 x : ndarray

249 The solution found.

250 info : int

251 Provides convergence information:

252

253 * 0 : successful exit

254 * >0 : convergence to tolerance not achieved, number of iterations

255

256 Examples

257 --------

258 >>> import numpy as np

259 >>> from scipy.sparse import csc_matrix

260 >>> from scipy.sparse.linalg import gcrotmk

261 >>> R = np.random.randn(5, 5)

262 >>> A = csc_matrix(R)

263 >>> b = np.random.randn(5)

264 >>> x, exit_code = gcrotmk(A, b, atol=1e-5)

265 >>> print(exit_code)

266 0

267 >>> np.allclose(A.dot(x), b)

268 True

269

270 References

271 ----------

272 .. [1] E. de Sturler, ''Truncation strategies for optimal Krylov subspace

273 methods'', SIAM J. Numer. Anal. 36, 864 (1999).

274 .. [2] J.E. Hicken and D.W. Zingg, ''A simplified and flexible variant

275 of GCROT for solving nonsymmetric linear systems'',

276 SIAM J. Sci. Comput. 32, 172 (2010).

277 .. [3] M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti,

278 ''Recycling Krylov subspaces for sequences of linear systems'',

279 SIAM J. Sci. Comput. 28, 1651 (2006).

280

281 """

282 A,M,x,b,postprocess = make_system(A,M,x0,b)

283

284 if not np.isfinite(b).all():

285 raise ValueError("RHS must contain only finite numbers")

286

287 if truncate not in ('oldest', 'smallest'):

288 raise ValueError(f"Invalid value for 'truncate': {truncate!r}")

289

290 if atol is None:

291 warnings.warn("scipy.sparse.linalg.gcrotmk called without specifying `atol`. "

292 "The default value will change in the future. To preserve "

293 "current behavior, set ``atol=tol``.",

294 category=DeprecationWarning, stacklevel=2)

295 atol = tol

296

297 matvec = A.matvec

298 psolve = M.matvec

299

300 if CU is None:

301 CU = []

302

303 if k is None:

304 k = m

305

306 axpy, dot, scal = None, None, None

307

308 if x0 is None:

309 r = b.copy()

310 else:

311 r = b - matvec(x)

312

313 axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (x, r))

314

315 b_norm = nrm2(b)

316 if b_norm == 0:

317 x = b

318 return (postprocess(x), 0)

319

320 if discard_C:

321 CU[:] = [(None, u) for c, u in CU]

322

323 # Reorthogonalize old vectors

324 if CU:

325 # Sort already existing vectors to the front

326 CU.sort(key=lambda cu: cu[0] is not None)

327

328 # Fill-in missing ones

329 C = np.empty((A.shape[0], len(CU)), dtype=r.dtype, order='F')

330 us = []

331 j = 0

332 while CU:

333 # More memory-efficient: throw away old vectors as we go

334 c, u = CU.pop(0)

335 if c is None:

336 c = matvec(u)

337 C[:,j] = c

338 j += 1

339 us.append(u)

340

341 # Orthogonalize

342 Q, R, P = qr(C, overwrite_a=True, mode='economic', pivoting=True)

343 del C

344

345 # C := Q

346 cs = list(Q.T)

347

348 # U := U P R^-1, back-substitution

349 new_us = []

350 for j in range(len(cs)):

351 u = us[P[j]]

352 for i in range(j):

353 u = axpy(us[P[i]], u, u.shape[0], -R[i,j])

354 if abs(R[j,j]) < 1e-12 * abs(R[0,0]):

355 # discard rest of the vectors

356 break

357 u = scal(1.0/R[j,j], u)

358 new_us.append(u)

359

360 # Form the new CU lists

361 CU[:] = list(zip(cs, new_us))[::-1]

362

363 if CU:

364 axpy, dot = get_blas_funcs(['axpy', 'dot'], (r,))

365

366 # Solve first the projection operation with respect to the CU

367 # vectors. This corresponds to modifying the initial guess to

368 # be

369 #

370 # x' = x + U y

371 # y = argmin_y || b - A (x + U y) ||^2

372 #

373 # The solution is y = C^H (b - A x)

374 for c, u in CU:

375 yc = dot(c, r)

376 x = axpy(u, x, x.shape[0], yc)

377 r = axpy(c, r, r.shape[0], -yc)

378

379 # GCROT main iteration

380 for j_outer in range(maxiter):

381 # -- callback

382 if callback is not None:

383 callback(x)

384

385 beta = nrm2(r)

386

387 # -- check stopping condition

388 beta_tol = max(atol, tol * b_norm)

389

390 if beta <= beta_tol and (j_outer > 0 or CU):

391 # recompute residual to avoid rounding error

392 r = b - matvec(x)

393 beta = nrm2(r)

394

395 if beta <= beta_tol:

396 j_outer = -1

397 break

398

399 ml = m + max(k - len(CU), 0)

400

401 cs = [c for c, u in CU]

402

403 try:

404 Q, R, B, vs, zs, y, pres = _fgmres(matvec,

405 r/beta,

406 ml,

407 rpsolve=psolve,

408 atol=max(atol, tol*b_norm)/beta,

409 cs=cs)

410 y *= beta

411 except LinAlgError:

412 # Floating point over/underflow, non-finite result from

413 # matmul etc. -- report failure.

414 break

415

416 #

417 # At this point,

418 #

419 # [A U, A Z] = [C, V] G; G = [ I B ]

420 # [ 0 H ]

421 #

422 # where [C, V] has orthonormal columns, and r = beta v_0. Moreover,

423 #

424 # || b - A (x + Z y + U q) ||_2 = || r - C B y - V H y - C q ||_2 = min!

425 #

426 # from which y = argmin_y || beta e_1 - H y ||_2, and q = -B y

427 #

428

429 #

430 # GCROT(m,k) update

431 #

432

433 # Define new outer vectors

434

435 # ux := (Z - U B) y

436 ux = zs[0]*y[0]

437 for z, yc in zip(zs[1:], y[1:]):

438 ux = axpy(z, ux, ux.shape[0], yc) # ux += z*yc

439 by = B.dot(y)

440 for cu, byc in zip(CU, by):

441 c, u = cu

442 ux = axpy(u, ux, ux.shape[0], -byc) # ux -= u*byc

443

444 # cx := V H y

445 hy = Q.dot(R.dot(y))

446 cx = vs[0] * hy[0]

447 for v, hyc in zip(vs[1:], hy[1:]):

448 cx = axpy(v, cx, cx.shape[0], hyc) # cx += v*hyc

449

450 # Normalize cx, maintaining cx = A ux

451 # This new cx is orthogonal to the previous C, by construction

452 try:

453 alpha = 1/nrm2(cx)

454 if not np.isfinite(alpha):

455 raise FloatingPointError()

456 except (FloatingPointError, ZeroDivisionError):

457 # Cannot update, so skip it

458 continue

459

460 cx = scal(alpha, cx)

461 ux = scal(alpha, ux)

462

463 # Update residual and solution

464 gamma = dot(cx, r)

465 r = axpy(cx, r, r.shape[0], -gamma) # r -= gamma*cx

466 x = axpy(ux, x, x.shape[0], gamma) # x += gamma*ux

467

468 # Truncate CU

469 if truncate == 'oldest':

470 while len(CU) >= k and CU:

471 del CU[0]

472 elif truncate == 'smallest':

473 if len(CU) >= k and CU:

474 # cf. [1,2]

475 D = solve(R[:-1,:].T, B.T).T

476 W, sigma, V = svd(D)

477

478 # C := C W[:,:k-1], U := U W[:,:k-1]

479 new_CU = []

480 for j, w in enumerate(W[:,:k-1].T):

481 c, u = CU[0]

482 c = c * w[0]

483 u = u * w[0]

484 for cup, wp in zip(CU[1:], w[1:]):

485 cp, up = cup

486 c = axpy(cp, c, c.shape[0], wp)

487 u = axpy(up, u, u.shape[0], wp)

488

489 # Reorthogonalize at the same time; not necessary

490 # in exact arithmetic, but floating point error

491 # tends to accumulate here

492 for cp, up in new_CU:

493 alpha = dot(cp, c)

494 c = axpy(cp, c, c.shape[0], -alpha)

495 u = axpy(up, u, u.shape[0], -alpha)

496 alpha = nrm2(c)

497 c = scal(1.0/alpha, c)

498 u = scal(1.0/alpha, u)

499

500 new_CU.append((c, u))

501 CU[:] = new_CU

502

503 # Add new vector to CU

504 CU.append((cx, ux))

505

506 # Include the solution vector to the span

507 CU.append((None, x.copy()))

508 if discard_C:

509 CU[:] = [(None, uz) for cz, uz in CU]

510

511 return postprocess(x), j_outer + 1

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