-
-
Notifications
You must be signed in to change notification settings - Fork 5.1k
/
lobpcg.py
1081 lines (917 loc) · 39.9 KB
/
lobpcg.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
"""
Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
References
----------
.. [1] A. V. Knyazev (2001),
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method.
SIAM Journal on Scientific Computing 23, no. 2,
pp. 517-541. :doi:`10.1137/S1064827500366124`
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007),
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX)
in hypre and PETSc. :arxiv:`0705.2626`
.. [3] A. V. Knyazev's C and MATLAB implementations:
https://github.com/lobpcg/blopex
"""
import warnings
import numpy as np
from scipy.linalg import (inv, eigh, cho_factor, cho_solve,
cholesky, LinAlgError)
from scipy.sparse.linalg import LinearOperator
from scipy.sparse import issparse
__all__ = ["lobpcg"]
def _report_nonhermitian(M, name):
"""
Report if `M` is not a Hermitian matrix given its type.
"""
from scipy.linalg import norm
md = M - M.T.conj()
nmd = norm(md, 1)
tol = 10 * np.finfo(M.dtype).eps
tol = max(tol, tol * norm(M, 1))
if nmd > tol:
warnings.warn(
f"Matrix {name} of the type {M.dtype} is not Hermitian: "
f"condition: {nmd} < {tol} fails.",
UserWarning, stacklevel=4
)
def _as2d(ar):
"""
If the input array is 2D return it, if it is 1D, append a dimension,
making it a column vector.
"""
if ar.ndim == 2:
return ar
else: # Assume 1!
aux = np.array(ar, copy=False)
aux.shape = (ar.shape[0], 1)
return aux
def _makeMatMat(m):
if m is None:
return None
elif callable(m):
return lambda v: m(v)
else:
return lambda v: m @ v
def _matmul_inplace(x, y, verbosityLevel=0):
"""Perform 'np.matmul' in-place if possible.
If some sufficient conditions for inplace matmul are met, do so.
Otherwise try inplace update and fall back to overwrite if that fails.
"""
if x.flags["CARRAY"] and x.shape[1] == y.shape[1] and x.dtype == y.dtype:
# conditions where we can guarantee that inplace updates will work;
# i.e. x is not a view/slice, x & y have compatible dtypes, and the
# shape of the result of x @ y matches the shape of x.
np.matmul(x, y, out=x)
else:
# ideally, we'd have an exhaustive list of conditions above when
# inplace updates are possible; since we don't, we opportunistically
# try if it works, and fall back to overwriting if necessary
try:
np.matmul(x, y, out=x)
except Exception:
if verbosityLevel:
warnings.warn(
"Inplace update of x = x @ y failed, "
"x needs to be overwritten.",
UserWarning, stacklevel=3
)
x = x @ y
return x
def _applyConstraints(blockVectorV, factYBY, blockVectorBY, blockVectorY):
"""Changes blockVectorV in-place."""
YBV = blockVectorBY.T.conj() @ blockVectorV
tmp = cho_solve(factYBY, YBV)
blockVectorV -= blockVectorY @ tmp
def _b_orthonormalize(B, blockVectorV, blockVectorBV=None,
verbosityLevel=0):
"""in-place B-orthonormalize the given block vector using Cholesky."""
if blockVectorBV is None:
if B is None:
blockVectorBV = blockVectorV
else:
try:
blockVectorBV = B(blockVectorV)
except Exception as e:
if verbosityLevel:
warnings.warn(
f"Secondary MatMul call failed with error\n"
f"{e}\n",
UserWarning, stacklevel=3
)
return None, None, None
if blockVectorBV.shape != blockVectorV.shape:
raise ValueError(
f"The shape {blockVectorV.shape} "
f"of the orthogonalized matrix not preserved\n"
f"and changed to {blockVectorBV.shape} "
f"after multiplying by the secondary matrix.\n"
)
VBV = blockVectorV.T.conj() @ blockVectorBV
try:
# VBV is a Cholesky factor from now on...
VBV = cholesky(VBV, overwrite_a=True)
VBV = inv(VBV, overwrite_a=True)
blockVectorV = _matmul_inplace(
blockVectorV, VBV,
verbosityLevel=verbosityLevel
)
if B is not None:
blockVectorBV = _matmul_inplace(
blockVectorBV, VBV,
verbosityLevel=verbosityLevel
)
return blockVectorV, blockVectorBV, VBV
except LinAlgError:
if verbosityLevel:
warnings.warn(
"Cholesky has failed.",
UserWarning, stacklevel=3
)
return None, None, None
def _get_indx(_lambda, num, largest):
"""Get `num` indices into `_lambda` depending on `largest` option."""
ii = np.argsort(_lambda)
if largest:
ii = ii[:-num - 1:-1]
else:
ii = ii[:num]
return ii
def _handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel):
if verbosityLevel:
_report_nonhermitian(gramA, "gramA")
_report_nonhermitian(gramB, "gramB")
def lobpcg(
A,
X,
B=None,
M=None,
Y=None,
tol=None,
maxiter=None,
largest=True,
verbosityLevel=0,
retLambdaHistory=False,
retResidualNormsHistory=False,
restartControl=20,
):
"""Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
LOBPCG is a preconditioned eigensolver for large real symmetric and complex
Hermitian definite generalized eigenproblems.
Parameters
----------
A : {sparse matrix, ndarray, LinearOperator, callable object}
The Hermitian linear operator of the problem, usually given by a
sparse matrix. Often called the "stiffness matrix".
X : ndarray, float32 or float64
Initial approximation to the ``k`` eigenvectors (non-sparse).
If `A` has ``shape=(n,n)`` then `X` must have ``shape=(n,k)``.
B : {sparse matrix, ndarray, LinearOperator, callable object}
Optional. By default ``B = None``, which is equivalent to identity.
The right hand side operator in a generalized eigenproblem if present.
Often called the "mass matrix". Must be Hermitian positive definite.
M : {sparse matrix, ndarray, LinearOperator, callable object}
Optional. By default ``M = None``, which is equivalent to identity.
Preconditioner aiming to accelerate convergence.
Y : ndarray, float32 or float64, default: None
An ``n-by-sizeY`` ndarray of constraints with ``sizeY < n``.
The iterations will be performed in the ``B``-orthogonal complement
of the column-space of `Y`. `Y` must be full rank if present.
tol : scalar, optional
The default is ``tol=n*sqrt(eps)``.
Solver tolerance for the stopping criterion.
maxiter : int, default: 20
Maximum number of iterations.
largest : bool, default: True
When True, solve for the largest eigenvalues, otherwise the smallest.
verbosityLevel : int, optional
By default ``verbosityLevel=0`` no output.
Controls the solver standard/screen output.
retLambdaHistory : bool, default: False
Whether to return iterative eigenvalue history.
retResidualNormsHistory : bool, default: False
Whether to return iterative history of residual norms.
restartControl : int, optional.
Iterations restart if the residuals jump ``2**restartControl`` times
compared to the smallest recorded in ``retResidualNormsHistory``.
The default is ``restartControl=20``, making the restarts rare for
backward compatibility.
Returns
-------
lambda : ndarray of the shape ``(k, )``.
Array of ``k`` approximate eigenvalues.
v : ndarray of the same shape as ``X.shape``.
An array of ``k`` approximate eigenvectors.
lambdaHistory : ndarray, optional.
The eigenvalue history, if `retLambdaHistory` is ``True``.
ResidualNormsHistory : ndarray, optional.
The history of residual norms, if `retResidualNormsHistory`
is ``True``.
Notes
-----
The iterative loop runs ``maxit=maxiter`` (20 if ``maxit=None``)
iterations at most and finishes earler if the tolerance is met.
Breaking backward compatibility with the previous version, LOBPCG
now returns the block of iterative vectors with the best accuracy rather
than the last one iterated, as a cure for possible divergence.
If ``X.dtype == np.float32`` and user-provided operations/multiplications
by `A`, `B`, and `M` all preserve the ``np.float32`` data type,
all the calculations and the output are in ``np.float32``.
The size of the iteration history output equals to the number of the best
(limited by `maxit`) iterations plus 3: initial, final, and postprocessing.
If both `retLambdaHistory` and `retResidualNormsHistory` are ``True``,
the return tuple has the following format
``(lambda, V, lambda history, residual norms history)``.
In the following ``n`` denotes the matrix size and ``k`` the number
of required eigenvalues (smallest or largest).
The LOBPCG code internally solves eigenproblems of the size ``3k`` on every
iteration by calling the dense eigensolver `eigh`, so if ``k`` is not
small enough compared to ``n``, it makes no sense to call the LOBPCG code.
Moreover, if one calls the LOBPCG algorithm for ``5k > n``, it would likely
break internally, so the code calls the standard function `eigh` instead.
It is not that ``n`` should be large for the LOBPCG to work, but rather the
ratio ``n / k`` should be large. It you call LOBPCG with ``k=1``
and ``n=10``, it works though ``n`` is small. The method is intended
for extremely large ``n / k``.
The convergence speed depends basically on three factors:
1. Quality of the initial approximations `X` to the seeking eigenvectors.
Randomly distributed around the origin vectors work well if no better
choice is known.
2. Relative separation of the desired eigenvalues from the rest
of the eigenvalues. One can vary ``k`` to improve the separation.
3. Proper preconditioning to shrink the spectral spread.
For example, a rod vibration test problem (under tests
directory) is ill-conditioned for large ``n``, so convergence will be
slow, unless efficient preconditioning is used. For this specific
problem, a good simple preconditioner function would be a linear solve
for `A`, which is easy to code since `A` is tridiagonal.
References
----------
.. [1] A. V. Knyazev (2001),
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method.
SIAM Journal on Scientific Computing 23, no. 2,
pp. 517-541. :doi:`10.1137/S1064827500366124`
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov
(2007), Block Locally Optimal Preconditioned Eigenvalue Xolvers
(BLOPEX) in hypre and PETSc. :arxiv:`0705.2626`
.. [3] A. V. Knyazev's C and MATLAB implementations:
https://github.com/lobpcg/blopex
Examples
--------
Our first example is minimalistic - find the largest eigenvalue of
a diagonal matrix by solving the non-generalized eigenvalue problem
``A x = lambda x`` without constraints or preconditioning.
>>> import numpy as np
>>> from scipy.sparse import spdiags
>>> from scipy.sparse.linalg import LinearOperator, aslinearoperator
>>> from scipy.sparse.linalg import lobpcg
The square matrix size is
>>> n = 100
and its diagonal entries are 1, ..., 100 defined by
>>> vals = np.arange(1, n + 1).astype(np.int16)
The first mandatory input parameter in this test is
the sparse diagonal matrix `A`
of the eigenvalue problem ``A x = lambda x`` to solve.
>>> A = spdiags(vals, 0, n, n)
>>> A = A.astype(np.int16)
>>> A.toarray()
array([[ 1, 0, 0, ..., 0, 0, 0],
[ 0, 2, 0, ..., 0, 0, 0],
[ 0, 0, 3, ..., 0, 0, 0],
...,
[ 0, 0, 0, ..., 98, 0, 0],
[ 0, 0, 0, ..., 0, 99, 0],
[ 0, 0, 0, ..., 0, 0, 100]], dtype=int16)
The second mandatory input parameter `X` is a 2D array with the
row dimension determining the number of requested eigenvalues.
`X` is an initial guess for targeted eigenvectors.
`X` must have linearly independent columns.
If no initial approximations available, randomly oriented vectors
commonly work best, e.g., with components normally distributed
around zero or uniformly distributed on the interval [-1 1].
Setting the initial approximations to dtype ``np.float32``
forces all iterative values to dtype ``np.float32`` speeding up
the run while still allowing accurate eigenvalue computations.
>>> k = 1
>>> rng = np.random.default_rng()
>>> X = rng.normal(size=(n, k))
>>> X = X.astype(np.float32)
>>> eigenvalues, _ = lobpcg(A, X, maxiter=60)
>>> eigenvalues
array([100.])
>>> eigenvalues.dtype
dtype('float32')
LOBPCG needs only access the matrix product with `A` rather
then the matrix itself. Since the matrix `A` is diagonal in
this example, one can write a function of the product
``A @ X`` using the diagonal values ``vals`` only, e.g., by
element-wise multiplication with broadcasting
>>> A_f = lambda X: vals[:, np.newaxis] * X
and use the handle ``A_f`` to this callable function as an input
>>> eigenvalues, _ = lobpcg(A_f, X, maxiter=60)
>>> eigenvalues
array([100.])
The next example illustrates computing 3 smallest eigenvalues of
the same matrix given by the function handle ``A_f`` with
constraints and preconditioning.
>>> k = 3
>>> X = rng.normal(size=(n, k))
Constraints - an optional input parameter is a 2D array comprising
of column vectors that the eigenvectors must be orthogonal to
>>> Y = np.eye(n, 3)
The preconditioner acts as the inverse of `A` in this example, but
in the reduced precision ``np.float32`` even though the initial `X`
and thus all iterates and the output are in full ``np.float64``.
>>> inv_vals = 1./vals
>>> inv_vals = inv_vals.astype(np.float32)
>>> M = lambda X: inv_vals[:, np.newaxis] * X
Let us now solve the eigenvalue problem for the matrix `A` first
without preconditioning requesting 80 iterations
>>> eigenvalues, _ = lobpcg(A_f, X, Y=Y, largest=False, maxiter=80)
>>> eigenvalues
array([4., 5., 6.])
>>> eigenvalues.dtype
dtype('float64')
With preconditioning we need only 20 iterations from the same `X`
>>> eigenvalues, _ = lobpcg(A_f, X, Y=Y, M=M, largest=False, maxiter=20)
>>> eigenvalues
array([4., 5., 6.])
Note that the vectors passed in `Y` are the eigenvectors of the 3
smallest eigenvalues. The results returned above are orthogonal to those.
Finally, the primary matrix `A` may be indefinite, e.g., after shifting
``vals`` by 50 from 1, ..., 100 to -49, ..., 50, we still can compute
the 3 smallest or largest eigenvalues.
>>> vals = vals - 50
>>> X = rng.normal(size=(n, k))
>>> eigenvalues, _ = lobpcg(A_f, X, largest=False, maxiter=99)
>>> eigenvalues
array([-49., -48., -47.])
>>> eigenvalues, _ = lobpcg(A_f, X, largest=True, maxiter=99)
>>> eigenvalues
array([50., 49., 48.])
"""
blockVectorX = X
bestblockVectorX = blockVectorX
blockVectorY = Y
residualTolerance = tol
if maxiter is None:
maxiter = 20
bestIterationNumber = maxiter
sizeY = 0
if blockVectorY is not None:
if len(blockVectorY.shape) != 2:
warnings.warn(
f"Expected rank-2 array for argument Y, instead got "
f"{len(blockVectorY.shape)}, "
f"so ignore it and use no constraints.",
UserWarning, stacklevel=2
)
blockVectorY = None
else:
sizeY = blockVectorY.shape[1]
# Block size.
if blockVectorX is None:
raise ValueError("The mandatory initial matrix X cannot be None")
if len(blockVectorX.shape) != 2:
raise ValueError("expected rank-2 array for argument X")
n, sizeX = blockVectorX.shape
# Data type of iterates, determined by X, must be inexact
if not np.issubdtype(blockVectorX.dtype, np.inexact):
warnings.warn(
f"Data type for argument X is {blockVectorX.dtype}, "
f"which is not inexact, so casted to np.float32.",
UserWarning, stacklevel=2
)
blockVectorX = np.asarray(blockVectorX, dtype=np.float32)
if retLambdaHistory:
lambdaHistory = np.zeros((maxiter + 3, sizeX),
dtype=blockVectorX.dtype)
if retResidualNormsHistory:
residualNormsHistory = np.zeros((maxiter + 3, sizeX),
dtype=blockVectorX.dtype)
if verbosityLevel:
aux = "Solving "
if B is None:
aux += "standard"
else:
aux += "generalized"
aux += " eigenvalue problem with"
if M is None:
aux += "out"
aux += " preconditioning\n\n"
aux += "matrix size %d\n" % n
aux += "block size %d\n\n" % sizeX
if blockVectorY is None:
aux += "No constraints\n\n"
else:
if sizeY > 1:
aux += "%d constraints\n\n" % sizeY
else:
aux += "%d constraint\n\n" % sizeY
print(aux)
if (n - sizeY) < (5 * sizeX):
warnings.warn(
f"The problem size {n} minus the constraints size {sizeY} "
f"is too small relative to the block size {sizeX}. "
f"Using a dense eigensolver instead of LOBPCG iterations."
f"No output of the history of the iterations.",
UserWarning, stacklevel=2
)
sizeX = min(sizeX, n)
if blockVectorY is not None:
raise NotImplementedError(
"The dense eigensolver does not support constraints."
)
# Define the closed range of indices of eigenvalues to return.
if largest:
eigvals = (n - sizeX, n - 1)
else:
eigvals = (0, sizeX - 1)
try:
if isinstance(A, LinearOperator):
A = A(np.eye(n, dtype=int))
elif callable(A):
A = A(np.eye(n, dtype=int))
if A.shape != (n, n):
raise ValueError(
f"The shape {A.shape} of the primary matrix\n"
f"defined by a callable object is wrong.\n"
)
elif issparse(A):
A = A.toarray()
else:
A = np.asarray(A)
except Exception as e:
raise Exception(
f"Primary MatMul call failed with error\n"
f"{e}\n")
if B is not None:
try:
if isinstance(B, LinearOperator):
B = B(np.eye(n, dtype=int))
elif callable(B):
B = B(np.eye(n, dtype=int))
if B.shape != (n, n):
raise ValueError(
f"The shape {B.shape} of the secondary matrix\n"
f"defined by a callable object is wrong.\n"
)
elif issparse(B):
B = B.toarray()
else:
B = np.asarray(B)
except Exception as e:
raise Exception(
f"Secondary MatMul call failed with error\n"
f"{e}\n")
try:
vals, vecs = eigh(A,
B,
subset_by_index=eigvals,
check_finite=False)
if largest:
# Reverse order to be compatible with eigs() in 'LM' mode.
vals = vals[::-1]
vecs = vecs[:, ::-1]
return vals, vecs
except Exception as e:
raise Exception(
f"Dense eigensolver failed with error\n"
f"{e}\n"
)
if (residualTolerance is None) or (residualTolerance <= 0.0):
residualTolerance = np.sqrt(np.finfo(blockVectorX.dtype).eps) * n
A = _makeMatMat(A)
B = _makeMatMat(B)
M = _makeMatMat(M)
# Apply constraints to X.
if blockVectorY is not None:
if B is not None:
blockVectorBY = B(blockVectorY)
if blockVectorBY.shape != blockVectorY.shape:
raise ValueError(
f"The shape {blockVectorY.shape} "
f"of the constraint not preserved\n"
f"and changed to {blockVectorBY.shape} "
f"after multiplying by the secondary matrix.\n"
)
else:
blockVectorBY = blockVectorY
# gramYBY is a dense array.
gramYBY = blockVectorY.T.conj() @ blockVectorBY
try:
# gramYBY is a Cholesky factor from now on...
gramYBY = cho_factor(gramYBY, overwrite_a=True)
except LinAlgError as e:
raise ValueError("Linearly dependent constraints") from e
_applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY)
##
# B-orthonormalize X.
blockVectorX, blockVectorBX, _ = _b_orthonormalize(
B, blockVectorX, verbosityLevel=verbosityLevel)
if blockVectorX is None:
raise ValueError("Linearly dependent initial approximations")
##
# Compute the initial Ritz vectors: solve the eigenproblem.
blockVectorAX = A(blockVectorX)
if blockVectorAX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the initial approximations not preserved\n"
f"and changed to {blockVectorAX.shape} "
f"after multiplying by the primary matrix.\n"
)
gramXAX = blockVectorX.T.conj() @ blockVectorAX
_lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
ii = _get_indx(_lambda, sizeX, largest)
_lambda = _lambda[ii]
if retLambdaHistory:
lambdaHistory[0, :] = _lambda
eigBlockVector = np.asarray(eigBlockVector[:, ii])
blockVectorX = _matmul_inplace(
blockVectorX, eigBlockVector,
verbosityLevel=verbosityLevel
)
blockVectorAX = _matmul_inplace(
blockVectorAX, eigBlockVector,
verbosityLevel=verbosityLevel
)
if B is not None:
blockVectorBX = _matmul_inplace(
blockVectorBX, eigBlockVector,
verbosityLevel=verbosityLevel
)
##
# Active index set.
activeMask = np.ones((sizeX,), dtype=bool)
##
# Main iteration loop.
blockVectorP = None # set during iteration
blockVectorAP = None
blockVectorBP = None
smallestResidualNorm = np.abs(np.finfo(blockVectorX.dtype).max)
iterationNumber = -1
restart = True
forcedRestart = False
explicitGramFlag = False
while iterationNumber < maxiter:
iterationNumber += 1
if B is not None:
aux = blockVectorBX * _lambda[np.newaxis, :]
else:
aux = blockVectorX * _lambda[np.newaxis, :]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
residualNorms = np.sqrt(np.abs(aux))
if retResidualNormsHistory:
residualNormsHistory[iterationNumber, :] = residualNorms
residualNorm = np.sum(np.abs(residualNorms)) / sizeX
if residualNorm < smallestResidualNorm:
smallestResidualNorm = residualNorm
bestIterationNumber = iterationNumber
bestblockVectorX = blockVectorX
elif residualNorm > 2**restartControl * smallestResidualNorm:
forcedRestart = True
blockVectorAX = A(blockVectorX)
if blockVectorAX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the restarted iterate not preserved\n"
f"and changed to {blockVectorAX.shape} "
f"after multiplying by the primary matrix.\n"
)
if B is not None:
blockVectorBX = B(blockVectorX)
if blockVectorBX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the restarted iterate not preserved\n"
f"and changed to {blockVectorBX.shape} "
f"after multiplying by the secondary matrix.\n"
)
ii = np.where(residualNorms > residualTolerance, True, False)
activeMask = activeMask & ii
currentBlockSize = activeMask.sum()
if verbosityLevel:
print(f"iteration {iterationNumber}")
print(f"current block size: {currentBlockSize}")
print(f"eigenvalue(s):\n{_lambda}")
print(f"residual norm(s):\n{residualNorms}")
if currentBlockSize == 0:
break
activeBlockVectorR = _as2d(blockVectorR[:, activeMask])
if iterationNumber > 0:
activeBlockVectorP = _as2d(blockVectorP[:, activeMask])
activeBlockVectorAP = _as2d(blockVectorAP[:, activeMask])
if B is not None:
activeBlockVectorBP = _as2d(blockVectorBP[:, activeMask])
if M is not None:
# Apply preconditioner T to the active residuals.
activeBlockVectorR = M(activeBlockVectorR)
##
# Apply constraints to the preconditioned residuals.
if blockVectorY is not None:
_applyConstraints(activeBlockVectorR,
gramYBY,
blockVectorBY,
blockVectorY)
##
# B-orthogonalize the preconditioned residuals to X.
if B is not None:
activeBlockVectorR = activeBlockVectorR - (
blockVectorX @
(blockVectorBX.T.conj() @ activeBlockVectorR)
)
else:
activeBlockVectorR = activeBlockVectorR - (
blockVectorX @
(blockVectorX.T.conj() @ activeBlockVectorR)
)
##
# B-orthonormalize the preconditioned residuals.
aux = _b_orthonormalize(
B, activeBlockVectorR, verbosityLevel=verbosityLevel)
activeBlockVectorR, activeBlockVectorBR, _ = aux
if activeBlockVectorR is None:
warnings.warn(
f"Failed at iteration {iterationNumber} with accuracies "
f"{residualNorms}\n not reaching the requested "
f"tolerance {residualTolerance}.",
UserWarning, stacklevel=2
)
break
activeBlockVectorAR = A(activeBlockVectorR)
if iterationNumber > 0:
if B is not None:
aux = _b_orthonormalize(
B, activeBlockVectorP, activeBlockVectorBP,
verbosityLevel=verbosityLevel
)
activeBlockVectorP, activeBlockVectorBP, invR = aux
else:
aux = _b_orthonormalize(B, activeBlockVectorP,
verbosityLevel=verbosityLevel)
activeBlockVectorP, _, invR = aux
# Function _b_orthonormalize returns None if Cholesky fails
if activeBlockVectorP is not None:
activeBlockVectorAP = _matmul_inplace(
activeBlockVectorAP, invR,
verbosityLevel=verbosityLevel
)
restart = forcedRestart
else:
restart = True
##
# Perform the Rayleigh Ritz Procedure:
# Compute symmetric Gram matrices:
if activeBlockVectorAR.dtype == "float32":
myeps = 1
else:
myeps = np.sqrt(np.finfo(activeBlockVectorR.dtype).eps)
if residualNorms.max() > myeps and not explicitGramFlag:
explicitGramFlag = False
else:
# Once explicitGramFlag, forever explicitGramFlag.
explicitGramFlag = True
# Shared memory assingments to simplify the code
if B is None:
blockVectorBX = blockVectorX
activeBlockVectorBR = activeBlockVectorR
if not restart:
activeBlockVectorBP = activeBlockVectorP
# Common submatrices:
gramXAR = np.dot(blockVectorX.T.conj(), activeBlockVectorAR)
gramRAR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR)
gramDtype = activeBlockVectorAR.dtype
if explicitGramFlag:
gramRAR = (gramRAR + gramRAR.T.conj()) / 2
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
gramXAX = (gramXAX + gramXAX.T.conj()) / 2
gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX)
gramRBR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBR)
gramXBR = np.dot(blockVectorX.T.conj(), activeBlockVectorBR)
else:
gramXAX = np.diag(_lambda).astype(gramDtype)
gramXBX = np.eye(sizeX, dtype=gramDtype)
gramRBR = np.eye(currentBlockSize, dtype=gramDtype)
gramXBR = np.zeros((sizeX, currentBlockSize), dtype=gramDtype)
if not restart:
gramXAP = np.dot(blockVectorX.T.conj(), activeBlockVectorAP)
gramRAP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP)
gramPAP = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP)
gramXBP = np.dot(blockVectorX.T.conj(), activeBlockVectorBP)
gramRBP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP)
if explicitGramFlag:
gramPAP = (gramPAP + gramPAP.T.conj()) / 2
gramPBP = np.dot(activeBlockVectorP.T.conj(),
activeBlockVectorBP)
else:
gramPBP = np.eye(currentBlockSize, dtype=gramDtype)
gramA = np.block(
[
[gramXAX, gramXAR, gramXAP],
[gramXAR.T.conj(), gramRAR, gramRAP],
[gramXAP.T.conj(), gramRAP.T.conj(), gramPAP],
]
)
gramB = np.block(
[
[gramXBX, gramXBR, gramXBP],
[gramXBR.T.conj(), gramRBR, gramRBP],
[gramXBP.T.conj(), gramRBP.T.conj(), gramPBP],
]
)
_handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel)
try:
_lambda, eigBlockVector = eigh(gramA,
gramB,
check_finite=False)
except LinAlgError as e:
# raise ValueError("eigh failed in lobpcg iterations") from e
if verbosityLevel:
warnings.warn(
f"eigh failed at iteration {iterationNumber} \n"
f"with error {e} causing a restart.\n",
UserWarning, stacklevel=2
)
# try again after dropping the direction vectors P from RR
restart = True
if restart:
gramA = np.block([[gramXAX, gramXAR], [gramXAR.T.conj(), gramRAR]])
gramB = np.block([[gramXBX, gramXBR], [gramXBR.T.conj(), gramRBR]])
_handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel)
try:
_lambda, eigBlockVector = eigh(gramA,
gramB,
check_finite=False)
except LinAlgError as e:
# raise ValueError("eigh failed in lobpcg iterations") from e
warnings.warn(
f"eigh failed at iteration {iterationNumber} with error\n"
f"{e}\n",
UserWarning, stacklevel=2
)
break
ii = _get_indx(_lambda, sizeX, largest)
_lambda = _lambda[ii]
eigBlockVector = eigBlockVector[:, ii]
if retLambdaHistory:
lambdaHistory[iterationNumber + 1, :] = _lambda
# Compute Ritz vectors.
if B is not None:
if not restart:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:
sizeX + currentBlockSize]
eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
pp += np.dot(activeBlockVectorP, eigBlockVectorP)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
app += np.dot(activeBlockVectorAP, eigBlockVectorP)
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
bpp += np.dot(activeBlockVectorBP, eigBlockVectorP)
else:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp
blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp
else:
if not restart:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:
sizeX + currentBlockSize]
eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
pp += np.dot(activeBlockVectorP, eigBlockVectorP)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
app += np.dot(activeBlockVectorAP, eigBlockVectorP)
else:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
blockVectorP, blockVectorAP = pp, app
if B is not None:
aux = blockVectorBX * _lambda[np.newaxis, :]
else:
aux = blockVectorX * _lambda[np.newaxis, :]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
residualNorms = np.sqrt(np.abs(aux))
# Use old lambda in case of early loop exit.
if retLambdaHistory:
lambdaHistory[iterationNumber + 1, :] = _lambda
if retResidualNormsHistory:
residualNormsHistory[iterationNumber + 1, :] = residualNorms
residualNorm = np.sum(np.abs(residualNorms)) / sizeX
if residualNorm < smallestResidualNorm:
smallestResidualNorm = residualNorm
bestIterationNumber = iterationNumber + 1
bestblockVectorX = blockVectorX
if np.max(np.abs(residualNorms)) > residualTolerance:
warnings.warn(
f"Exited at iteration {iterationNumber} with accuracies \n"
f"{residualNorms}\n"
f"not reaching the requested tolerance {residualTolerance}.\n"
f"Use iteration {bestIterationNumber} instead with accuracy \n"
f"{smallestResidualNorm}.\n",
UserWarning, stacklevel=2
)
if verbosityLevel:
print(f"Final iterative eigenvalue(s):\n{_lambda}")
print(f"Final iterative residual norm(s):\n{residualNorms}")
blockVectorX = bestblockVectorX
# Making eigenvectors "exactly" satisfy the blockVectorY constrains
if blockVectorY is not None:
_applyConstraints(blockVectorX,
gramYBY,
blockVectorBY,
blockVectorY)
# Making eigenvectors "exactly" othonormalized by final "exact" RR
blockVectorAX = A(blockVectorX)
if blockVectorAX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the postprocessing iterate not preserved\n"
f"and changed to {blockVectorAX.shape} "
f"after multiplying by the primary matrix.\n"
)
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
blockVectorBX = blockVectorX