forked from sloisel/numeric
-
Notifications
You must be signed in to change notification settings - Fork 0
/
documentation.html
1366 lines (1262 loc) · 46.5 KB
/
documentation.html
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
<!doctype html>
<html>
<head>
<link rel="SHORTCUT ICON" href="favicon.ico">
<link href='http://fonts.googleapis.com/css?family=Lato' rel='stylesheet' type='text/css'>
<link rel="stylesheet" type="text/css" href="resources/style.css">
<style>
table { font-size:14px; }
td { vertical-align:top; }
@media print
{
table { font-size:12px; }
td.navmain { font-size:26px; }
body { margin: 5mm 5mm 5mm 5mm; }
}
</style>
<title>Numeric Javascript: Documentation</title>
</head>
<body>
<!--#include file="resources/header.html" -->
<!--
This allows regression tests to run predictably:
<pre>
IN> numeric.seedrandom.seedrandom('1'); Math.random = numeric.seedrandom.random; Math.random();
OUT> 0.2694
</pre>
-->
<table cellspacing=5 style="border:5px solid black;">
<tr><td colspan=3 align="center" style="font-size:18px;">
<b>Reference card for the <tt>numeric</tt> module</b>
<tr valign="top"><td valign="top" width="33%">
<table>
<tr><td><b>Function</b><td><b>Description</b>
<tr><td colspan=2><hr>
<tr><td><tt>abs</tt><td>Absolute value
<tr><td><tt>acos</tt><td>Arc-cosine
<tr><td><tt>add</tt><td>Pointwise sum x+y
<tr><td><tt>addeq</tt><td>Pointwise sum x+=y
<tr><td><tt>all</tt><td>All the components of x are true
<tr><td><tt>and</tt><td>Pointwise x && y
<tr><td><tt>andeq</tt><td>Pointwise x &= y
<tr><td><tt>any</tt><td>One or more of the components of x are true
<tr><td><tt>asin</tt><td>Arc-sine
<tr><td><tt>atan</tt><td>Arc-tangeant
<tr><td><tt>atan2</tt><td>Arc-tangeant (two parameters)
<tr><td><tt>band</tt><td>Pointwise x & y
<tr><td><tt>bench</tt><td>Benchmarking routine
<tr><td><tt>bnot</tt><td>Binary negation ~x
<tr><td><tt>bor</tt><td>Binary or x|y
<tr><td><tt>bxor</tt><td>Binary xor x^y
<tr><td><tt>ccsDim</tt><td>Dimensions of sparse matrix
<tr><td><tt>ccsDot</tt><td>Sparse matrix-matrix product
<tr><td><tt>ccsFull</tt><td>Convert sparse to full
<tr><td><tt>ccsGather</tt><td>Gather entries of sparse matrix
<tr><td><tt>ccsGetBlock</tt><td>Get rows/columns of sparse matrix
<tr><td><tt>ccsLUP</tt><td>Compute LUP decomposition of sparse matrix
<tr><td><tt>ccsLUPSolve</tt><td>Solve Ax=b using LUP decomp
<tr><td><tt>ccsScatter</tt><td>Scatter entries of sparse matrix
<tr><td><tt>ccsSparse</tt><td>Convert from full to sparse
<tr><td><tt>ccsTSolve</tt><td>Solve upper/lower triangular system
<tr><td><tt>ccs<op></td><td>Supported ops include: add/div/mul/geq/etc...
<tr><td><tt>cLU</tt><td>Coordinate matrix LU decomposition
<tr><td><tt>cLUsolve</tt><td>Coordinate matrix LU solve
<tr><td><tt>cdelsq</tt><td>Coordinate matrix Laplacian
<tr><td><tt>cdotMV</tt><td>Coordinate matrix-vector product
<tr><td><tt>ceil</tt><td>Pointwise Math.ceil(x)
<tr><td><tt>cgrid</tt><td>Coordinate grid for cdelsq
<tr><td><tt>clone</tt><td>Deep copy of Array
<tr><td><tt>cos</tt><td>Pointwise Math.cos(x)
<tr><td><tt>det</tt><td>Determinant
<tr><td><tt>diag</tt><td>Create diagonal matrix
<tr><td><tt>dim</tt><td>Get Array dimensions
<tr><td><tt>div</tt><td>Pointwise x/y
<tr><td><tt>diveq</tt><td>Pointwise x/=y
<tr><td><tt>dopri</tt><td>Numerical integration of ODE using Dormand-Prince RK method. Returns an object Dopri.
<tr><td><tt>Dopri.at</tt><td>Evaluate the ODE solution at a point
</table>
<td valign="top" width="33%">
<table>
<tr><td><b>Function</b><td><b>Description</b>
<tr><td colspan=2><hr>
<tr><td><tt>dot</tt><td>Matrix-Matrix, Matrix-Vector and Vector-Matrix product
<tr><td><tt>eig</tt><td>Eigenvalues and eigenvectors
<tr><td><tt>epsilon</tt><td>2.220446049250313e-16
<tr><td><tt>eq</tt><td>Pointwise comparison x === y
<tr><td><tt>exp</tt><td>Pointwise Math.exp(x)
<tr><td><tt>floor</tt><td>Poinwise Math.floor(x)
<tr><td><tt>geq</tt><td>Pointwise x>=y
<tr><td><tt>getBlock</tt><td>Extract a block from a matrix
<tr><td><tt>getDiag</tt><td>Get the diagonal of a matrix
<tr><td><tt>gt</tt><td>Pointwise x>y
<tr><td><tt>identity</tt><td>Identity matrix
<tr><td><tt>imageURL</tt><td>Encode a matrix as an image URL
<tr><td><tt>inv</tt><td>Matrix inverse
<tr><td><tt>isFinite</tt><td>Pointwise isFinite(x)
<tr><td><tt>isNaN</tt><td>Pointwise isNaN(x)
<tr><td><tt>largeArray</tt><td>Don't prettyPrint Arrays larger than this
<tr><td><tt>leq</tt><td>Pointwise x<=y
<tr><td><tt>linspace</tt><td>Generate evenly spaced values
<tr><td><tt>log</tt><td>Pointwise Math.log(x)
<tr><td><tt>lshift</tt><td>Pointwise x<<y
<tr><td><tt>lshifteq</tt><td>Pointwise x<<=y
<tr><td><tt>lt</tt><td>Pointwise x<y
<tr><td><tt>LU</tt><td>Dense LU decomposition
<tr><td><tt>LUsolve</tt><td>Dense LU solve
<tr><td><tt>mapreduce</tt><td>Make a pointwise map-reduce function
<tr><td><tt>mod</tt><td>Pointwise x%y
<tr><td><tt>modeq</tt><td>Pointwise x%=y
<tr><td><tt>mul</tt><td>Pointwise x*y
<tr><td><tt>neg</tt><td>Pointwise -x
<tr><td><tt>neq</tt><td>Pointwise x!==y
<tr><td><tt>norm2</tt><td>Square root of the sum of the square of the entries of x
<tr><td><tt>norm2Squared</tt><td>Sum of squares of entries of x
<tr><td><tt>norminf</tt><td>Largest modulus entry of x
<tr><td><tt>not</tt><td>Pointwise logical negation !x
<tr><td><tt>or</tt><td>Pointwise logical or x||y
<tr><td><tt>oreq</tt><td>Pointwise x|=y
<tr><td><tt>parseCSV</tt><td>Parse a CSV file into an Array
<tr><td><tt>parseDate</tt><td>Pointwise parseDate(x)
<tr><td><tt>parseFloat</tt><td>Pointwise parseFloat(x)
<tr><td><tt>pointwise</tt><td>Create a pointwise function
<tr><td><tt>pow</tt><td>Pointwise Math.pow(x)
<tr><td><tt>precision</tt><td>Number of digits to prettyPrint
<tr><td><tt>prettyPrint</tt><td>Pretty-prints x
<tr><td><tt>random</tt><td>Create an Array of random numbers
<tr><td><tt>rep</tt><td>Create an Array by duplicating values
</table>
<td valign="top" width="33%">
<table>
<tr><td><b>Function</b><td><b>Description</b>
<tr><td colspan=2><hr>
<tr><td><tt>round</tt><td>Pointwise Math.round(x)
<tr><td><tt>rrshift</tt><td>Pointwise x>>>y
<tr><td><tt>rrshifteq</tt><td>Pointwise x>>>=y
<tr><td><tt>rshift</tt><td>Pointwise x>>y
<tr><td><tt>rshifteq</tt><td>Pointwise x>>=y
<tr><td><tt>same</tt><td>x and y are entrywise identical
<tr><td><tt>seedrandom</tt><td>The seedrandom module
<tr><td><tt>setBlock</tt><td>Set a block of a matrix
<tr><td><tt>sin</tt><td>Pointwise Math.sin(x)
<tr><td><tt>solve</tt><td>Solve Ax=b
<tr><td><tt>solveLP</tt><td>Solve a linear programming problem
<tr><td><tt>solveQP</tt><td>Solve a quadratic programming problem
<tr><td><tt>spline</tt><td>Create a Spline object
<tr><td><tt>Spline.at</tt><td>Evaluate the Spline at a point
<tr><td><tt>Spline.diff</tt><td>Differentiate the Spline
<tr><td><tt>Spline.roots</tt><td>Find all the roots of the Spline
<tr><td><tt>sqrt</tt><td>Pointwise Math.sqrt(x)
<tr><td><tt>sub</tt><td>Pointwise x-y
<tr><td><tt>subeq</tt><td>Pointwise x-=y
<tr><td><tt>sum</tt><td>Sum all the entries of x
<tr><td><tt>svd</tt><td>Singular value decomposition
<tr><td><tt>t</tt><td>Create a tensor type T (may be complex-valued)
<tr><td><tt>T.<numericfun></tt><td>Supported <numericfun> are: abs, add, cos, diag, div, dot, exp, getBlock, getDiag, inv, log, mul, neg, norm2, setBlock, sin, sub, transpose
<tr><td><tt>T.conj</tt><td>Pointwise complex conjugate
<tr><td><tt>T.fft</tt><td>Fast Fourier transform
<tr><td><tt>T.get</tt><td>Read an entry
<tr><td><tt>T.getRow</tt><td>Get a row
<tr><td><tt>T.getRows</tt><td>Get a range of rows
<tr><td><tt>T.ifft</tt><td>Inverse FFT
<tr><td><tt>T.reciprocal</tt><td>Pointwise 1/z
<tr><td><tt>T.set</tt><td>Set an entry
<tr><td><tt>T.setRow</tt><td>Set a row
<tr><td><tt>T.setRows</tt><td>Set a range of rows
<tr><td><tt>T.transjugate</tt><td>The conjugate-transpose of a matrix
<tr><td><tt>tan</tt><td>Pointwise Math.tan(x)
<tr><td><tt>tensor</tt><td>Tensor product ret[i][j] = x[i]*y[j]
<tr><td><tt>toCSV</tt><td>Make a CSV file
<tr><td><tt>transpose</tt><td>Matrix transpose
<tr><td><tt>uncmin</tt><td>Unconstrained optimization
<tr><td><tt>version</tt><td>Version string for the numeric library
<tr><td><tt>xor</tt><td>Pointwise x^y
<tr><td><tt>xoreq</tt><td>Pointwise x^=y
</table></table>
<br>
<h1>Numerical analysis in Javascript</h1>
<a href="http://www.numericjs.com/">Numeric Javascript</a> is
library that provides many useful functions for numerical
calculations, particularly for linear algebra (vectors and matrices).
You can create vectors and matrices and multiply them:
<pre>
IN> A = [[1,2,3],[4,5,6]];
OUT> [[1,2,3],
[4,5,6]]
IN> x = [7,8,9]
OUT> [7,8,9]
IN> numeric.dot(A,x);
OUT> [50,122]
</pre>
The example shown above can be executed in the
<a href="http://www.numericjs.com/workshop.php">Javascript Workshop</a> or at any
Javascript prompt. The Workshop provides plotting capabilities:<br>
<img src="resources/workshop.png"><br>
The function <tt>workshop.plot()</tt> is essentially the <a href="http://code.google.com/p/flot/">flot</a>
plotting command.<br><br>
The <tt>numeric</tt> library provides functions that implement most of the usual Javascript
operators for vectors and matrices:
<pre>
IN> x = [7,8,9];
y = [10,1,2];
numeric['+'](x,y)
OUT> [17,9,11]
IN> numeric['>'](x,y)
OUT> [false,true,true]
</pre>
These operators can also be called with plain Javascript function names:
<pre>
IN> numeric.add([7,8,9],[10,1,2])
OUT> [17,9,11]
</pre>
You can also use these operators with three or more parameters:
<pre>
IN> numeric.add([1,2],[3,4],[5,6],[7,8])
OUT> [16,20]
</pre>
The function <tt>numeric.inv()</tt> can be used to compute the inverse of an invertible matrix:
<pre>
IN> A = [[1,2,3],[4,5,6],[7,1,9]]
OUT> [[1,2,3],
[4,5,6],
[7,1,9]]
IN> Ainv = numeric.inv(A);
OUT> [[-0.9286,0.3571,0.07143],
[-0.1429,0.2857,-0.1429],
[0.7381,-0.3095,0.07143]]
</pre>
The function <tt>numeric.prettyPrint()</tt> is used to print most of the examples in this documentation.
It formats objects, arrays and numbers so that they can be understood easily. All output is automatically
formatted using <tt>numeric.prettyPrint()</tt> when in the
<a href="http://www.numericjs.com/workshop.php">Workshop</a>. In order to present the information clearly and
succintly, the function <tt>numeric.prettyPrint()</tt> lays out matrices so that all the numbers align.
Furthermore, numbers are given approximately using the <tt>numeric.precision</tt> variable:
<pre>
IN> numeric.precision = 10; x = 3.141592653589793
OUT> 3.141592654
IN> numeric.precision = 4; x
OUT> 3.142
</pre>
The default precision is 4 digits. In addition to printing approximate numbers,
the function <tt>numeric.prettyPrint()</tt> will replace large arrays with the string <tt>...Large Array...</tt>:
<pre>
IN> numeric.identity(100)
OUT> ...Large Array...
</pre>
By default, this happens with the Array's length is more than 50. This can be controlled by setting the
variable <tt>numeric.largeArray</tt> to an appropriate value:
<pre>
IN> numeric.largeArray = 2; A = numeric.identity(4)
OUT> ...Large Array...
IN> numeric.largeArray = 50; A
OUT> [[1,0,0,0],
[0,1,0,0],
[0,0,1,0],
[0,0,0,1]]
</pre>
In particular, if you want to print all Arrays regardless of size, set <tt>numeric.largeArray = Infinity</tt>.
<br><br>
<h1>Math Object functions</h1>
The <tt>Math</tt> object functions have also been adapted to work on Arrays as follows:
<pre>
IN> numeric.exp([1,2]);
OUT> [2.718,7.389]
IN> numeric.exp([[1,2],[3,4]])
OUT> [[2.718, 7.389],
[20.09, 54.6]]
IN> numeric.abs([-2,3])
OUT> [2,3]
IN> numeric.acos([0.1,0.2])
OUT> [1.471,1.369]
IN> numeric.asin([0.1,0.2])
OUT> [0.1002,0.2014]
IN> numeric.atan([1,2])
OUT> [0.7854,1.107]
IN> numeric.atan2([1,2],[3,4])
OUT> [0.3218,0.4636]
IN> numeric.ceil([-2.2,3.3])
OUT> [-2,4]
IN> numeric.floor([-2.2,3.3])
OUT> [-3,3]
IN> numeric.log([1,2])
OUT> [0,0.6931]
IN> numeric.pow([2,3],[0.25,7.1])
OUT> [1.189,2441]
IN> numeric.round([-2.2,3.3])
OUT> [-2,3]
IN> numeric.sin([1,2])
OUT> [0.8415,0.9093]
IN> numeric.sqrt([1,2])
OUT> [1,1.414]
IN> numeric.tan([1,2])
OUT> [1.557,-2.185]
</pre>
<h1>Utility functions</h1>
The function <tt>numeric.dim()</tt> allows you to compute the dimensions of an Array.
<pre>
IN> numeric.dim([1,2])
OUT> [2]
IN> numeric.dim([[1,2,3],[4,5,6]])
OUT> [2,3]
</pre>
You can perform a deep comparison of Arrays using <tt>numeric.same()</tt>:
<pre>
IN> numeric.same([1,2],[1,2])
OUT> true
IN> numeric.same([1,2],[1,2,3])
OUT> false
IN> numeric.same([1,2],[[1],[2]])
OUT> false
IN> numeric.same([[1,2],[3,4]],[[1,2],[3,4]])
OUT> true
IN> numeric.same([[1,2],[3,4]],[[1,2],[3,5]])
OUT> false
IN> numeric.same([[1,2],[2,4]],[[1,2],[3,4]])
OUT> false
</pre>
You can create a multidimensional Array from a given value using <tt>numeric.rep()</tt>
<pre>
IN> numeric.rep([3],5)
OUT> [5,5,5]
IN> numeric.rep([2,3],0)
OUT> [[0,0,0],
[0,0,0]]
</pre>
You can loop over Arrays as you normally would. However, in order to quickly generate optimized
loops, the <tt>numeric</tt> library provides a few efficient loop-generation mechanisms. For example, the
<tt>numeric.mapreduce()</tt> function can be used to make a function that computes the sum of all the
entries of an Array.
<pre>
IN> sum = numeric.mapreduce('accum += xi','0'); sum([1,2,3])
OUT> 6
IN> sum([[1,2,3],[4,5,6]])
OUT> 21
</pre>
The functions <tt>numeric.any()</tt> and <tt>numeric.all()</tt> allow you to check whether any or all entries
of an Array are boolean true values.
<pre>
IN> numeric.any([false,true])
OUT> true
IN> numeric.any([[0,0,3.14],[0,false,0]])
OUT> true
IN> numeric.any([0,0,false])
OUT> false
IN> numeric.all([false,true])
OUT> false
IN> numeric.all([[1,4,3.14],["no",true,-1]])
OUT> true
IN> numeric.all([0,0,false])
OUT> false
</pre>
You can create a diagonal matrix using <tt>numeric.diag()</tt>
<pre>
IN> numeric.diag([1,2,3])
OUT> [[1,0,0],
[0,2,0],
[0,0,3]]
</pre>
The function <tt>numeric.identity()</tt> returns the identity matrix.
<pre>
IN> numeric.identity(3)
OUT> [[1,0,0],
[0,1,0],
[0,0,1]]
</pre>
Random Arrays can also be created:
<pre >
IN> numeric.random([2,3])
OUT> [[0.05303,0.1537,0.7280],
[0.3839,0.08818,0.6316]]
</pre>
You can generate a vector of evenly spaced values:
<pre>
IN> numeric.linspace(1,5);
OUT> [1,2,3,4,5]
IN> numeric.linspace(1,3,5);
OUT> [1,1.5,2,2.5,3]
</pre>
<!--
<pre>
IN> numeric.blockMatrix([[[[1,2],[3,4]],[[5,6],[7,8]]],
[[[11,12],[13,14]],[[15,16],[17,18]]]])
OUT> [[ 1, 2, 5, 6],
[ 3, 4, 7, 8],
[11,12,15,16],
[13,14,17,18]]
</pre>
-->
<h1>Arithmetic operations</h1>
The standard arithmetic operations have been vectorized:
<pre>
IN> numeric.addVV([1,2],[3,4])
OUT> [4,6]
IN> numeric.addVS([1,2],3)
OUT> [4,5]
</pre>
There are also polymorphic functions:
<pre>
IN> numeric.add(1,[2,3])
OUT> [3,4]
IN> numeric.add([1,2,3],[4,5,6])
OUT> [5,7,9]
</pre>
The other arithmetic operations are available:
<pre>
IN> numeric.sub([1,2],[3,4])
OUT> [-2,-2]
IN> numeric.mul([1,2],[3,4])
OUT> [3,8]
IN> numeric.div([1,2],[3,4])
OUT> [0.3333,0.5]
</pre>
The in-place operators (such as +=) are also available:
<pre>
IN> v = [1,2,3,4]; numeric.addeq(v,3); v
OUT> [4,5,6,7]
IN> numeric.subeq([1,2,3],[5,3,1])
OUT> [-4,-1,2]
</pre>
Unary operators:
<pre>
IN> numeric.neg([1,-2,3])
OUT> [-1,2,-3]
IN> numeric.isFinite([10,NaN,Infinity])
OUT> [true,false,false]
IN> numeric.isNaN([10,NaN,Infinity])
OUT> [false,true,false]
</pre>
<!--
<pre>
IN> n = 41; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 42; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 43; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 44; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 45; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 46; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
</pre>
-->
<h1>Linear algebra</h1>
Matrix products are implemented in the functions
<tt>numeric.dotVV()</tt>
<tt>numeric.dotVM()</tt>
<tt>numeric.dotMV()</tt>
<tt>numeric.dotMM()</tt>:
<pre>
IN> numeric.dotVV([1,2],[3,4])
OUT> 11
IN> numeric.dotVM([1,2],[[3,4],[5,6]])
OUT> [13,16]
IN> numeric.dotMV([[1,2],[3,4]],[5,6])
OUT> [17,39]
IN> numeric.dotMMbig([[1,2],[3,4]],[[5,6],[7,8]])
OUT> [[19,22],
[43,50]]
IN> numeric.dotMMsmall([[1,2],[3,4]],[[5,6],[7,8]])
OUT> [[19,22],
[43,50]]
IN> numeric.dot([1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9])
OUT> 285
</pre>
The function <tt>numeric.dot()</tt> is "polymorphic" and selects the appropriate Matrix product:
<pre>
IN> numeric.dot([1,2,3],[4,5,6])
OUT> 32
IN> numeric.dot([[1,2,3],[4,5,6]],[7,8,9])
OUT> [50,122]
</pre>
Solving the linear problem Ax=b (<a href="https://github.com/yuzeh">Dan Huang</a>):
<pre>
IN> numeric.solve([[1,2],[3,4]],[17,39])
OUT> [5,6]
</pre>
The algorithm is based on the LU decomposition:
<pre>
IN> LU = numeric.LU([[1,2],[3,4]])
OUT> {LU:[[3 ,4 ],
[0.3333,0.6667]],
P:[1,1]}
IN> numeric.LUsolve(LU,[17,39])
OUT> [5,6]
</pre>
<!--
Stress testing
<pre>
IN> ns = [5,6,10,16,25,40,41];
for(j=0;j<ns.length;j++) {
n = ns[j];
for(k=0;k<10;++k) {
A = numeric.random([n,n]);
x = numeric.random([n]);
b = numeric.dot(A,x);
y = numeric.solve(A,b);
if(!(numeric.norminf(numeric.sub(x,y))<1e-10)) throw new Error(
"numeric.solve Stress Test:"+numeric.prettyPrint({j:j,k:k,A:A,x:x,b:b,y:y}));
}
}
"numeric.solve Stress Test OK";
OUT> "numeric.solve Stress Test OK"
</pre>
-->
The determinant:
<pre>
IN> numeric.det([[1,2],[3,4]]);
OUT> -2
IN> numeric.det([[6,8,4,2,8,5],[3,5,2,4,9,2],[7,6,8,3,4,5],[5,5,2,8,1,6],[3,2,2,4,2,2],[8,3,2,2,4,1]]);
OUT> -1404
</pre>
The matrix inverse:
<pre>
IN> numeric.inv([[1,2],[3,4]])
OUT> [[ -2, 1],
[ 1.5, -0.5]]
</pre>
The transpose:
<pre>
IN> numeric.transpose([[1,2,3],[4,5,6]])
OUT> [[1,4],
[2,5],
[3,6]]
IN> numeric.transpose([[1,2,3,4,5,6,7,8,9,10,11,12]])
OUT> [[ 1],
[ 2],
[ 3],
[ 4],
[ 5],
[ 6],
[ 7],
[ 8],
[ 9],
[10],
[11],
[12]]
</pre>
You can compute the 2-norm of an Array, which is the square root of the sum of the squares of the entries.
<pre>
IN> numeric.norm2([1,2])
OUT> 2.236
</pre>
Computing the tensor product of two vectors:
<pre>
IN> numeric.tensor([1,2],[3,4,5])
OUT> [[3,4,5],
[6,8,10]]
</pre>
<h1>Data manipulation</h1>
There are also some data manipulation functions. You can parse dates:
<pre>
IN> numeric.parseDate(['1/13/2013','2001-5-9, 9:31']);
OUT> [1.358e12,9.894e11]
</pre>
Parse floating point quantities:
<pre>
IN> numeric.parseFloat(['12','0.1'])
OUT> [12,0.1]
</pre>
Parse CSV files:
<pre>
IN> numeric.parseCSV('a,b,c\n1,2.3,.3\n4e6,-5.3e-8,6.28e+4')
OUT> [[ "a", "b", "c"],
[ 1, 2.3, 0.3],
[ 4e6, -5.3e-8, 62800]]
IN> numeric.toCSV([[1.23456789123,2],[3,4]])
OUT> "1.23456789123,2
3,4
"
</pre>
You can also fetch a URL (a thin wrapper around XMLHttpRequest):
<pre>
IN> numeric.getURL('tools/helloworld.txt').responseText
OUT> "Hello, world!"
</pre>
<h1>Complex linear algebra</h1>
You can also manipulate complex numbers:
<pre>
IN> z = new numeric.T(3,4);
OUT> {x: 3, y: 4}
IN> z.add(5)
OUT> {x: 8, y: 4}
IN> w = new numeric.T(2,8);
OUT> {x: 2, y: 8}
IN> z.add(w)
OUT> {x: 5, y: 12}
IN> z.mul(w)
OUT> {x: -26, y: 32}
IN> z.div(w)
OUT> {x:0.5588,y:-0.2353}
IN> z.sub(w)
OUT> {x:1, y:-4}
</pre>
Complex vectors and matrices can also be handled:
<pre>
IN> z = new numeric.T([1,2],[3,4]);
OUT> {x: [1,2], y: [3,4]}
IN> z.abs()
OUT> {x:[3.162,4.472],y:}
IN> z.conj()
OUT> {x:[1,2],y:[-3,-4]}
IN> z.norm2()
OUT> 5.477
IN> z.exp()
OUT> {x:[-2.691,-4.83],y:[0.3836,-5.592]}
IN> z.cos()
OUT> {x:[-1.528,-2.459],y:[0.1658,-2.745]}
IN> z.sin()
OUT> {x:[0.2178,-2.847],y:[1.163,2.371]}
IN> z.log()
OUT> {x:[1.151,1.498],y:[1.249,1.107]}
</pre>
Complex matrices:
<pre>
IN> A = new numeric.T([[1,2],[3,4]],[[0,1],[2,-1]]);
OUT> {x:[[1, 2],
[3, 4]],
y:[[0, 1],
[2,-1]]}
IN> A.inv();
OUT> {x:[[0.125,0.125],
[ 0.25, 0]],
y:[[ 0.5,-0.25],
[-0.375,0.125]]}
IN> A.inv().dot(A)
OUT> {x:[[1, 0],
[0, 1]],
y:[[0,-2.776e-17],
[0, 0]]}
IN> A.get([1,1])
OUT> {x: 4, y: -1}
IN> A.transpose()
OUT> { x: [[1, 3],
[2, 4]],
y: [[0, 2],
[1,-1]] }
IN> A.transjugate()
OUT> { x: [[ 1, 3],
[ 2, 4]],
y: [[ 0,-2],
[-1, 1]] }
IN> numeric.T.rep([2,2],new numeric.T(2,3));
OUT> { x: [[2,2],
[2,2]],
y: [[3,3],
[3,3]] }
</pre>
<h1>Eigenvalues</h1>
Eigenvalues:
<pre>
IN> A = [[1,2,5],[3,5,-1],[7,-3,5]];
OUT> [[ 1, 2, 5],
[ 3, 5, -1],
[ 7, -3, 5]]
IN> B = numeric.eig(A);
OUT> {lambda:{x:[-4.284,9.027,6.257],y:},
E:{x:[[ 0.7131,-0.5543,0.4019],
[-0.2987,-0.2131,0.9139],
[-0.6342,-0.8046,0.057 ]],
y:}}
IN> C = B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv());
OUT> {x: [[ 1, 2, 5],
[ 3, 5, -1],
[ 7, -3, 5]],
y: }
</pre>
Note that eigenvalues and eigenvectors are returned as complex numbers (type <tt>numeric.T</tt>). This is because
eigenvalues are often complex even when the matrix is real.<br><br>
<h1>Singular value decomposition (Shanti Rao)</h1>
Shanti Rao kindly emailed me an implementation of the Golub and Reisch algorithm:
<pre>
IN> A=[[ 22, 10, 2, 3, 7],
[ 14, 7, 10, 0, 8],
[ -1, 13, -1,-11, 3],
[ -3, -2, 13, -2, 4],
[ 9, 8, 1, -2, 4],
[ 9, 1, -7, 5, -1],
[ 2, -6, 6, 5, 1],
[ 4, 5, 0, -2, 2]];
numeric.svd(A);
OUT> {U:
[[ -0.7071, -0.1581, 0.1768, 0.2494, 0.4625],
[ -0.5303, -0.1581, -0.3536, 0.1556, -0.4984],
[ -0.1768, 0.7906, -0.1768, -0.1546, 0.3967],
[ -1.506e-17, -0.1581, -0.7071, -0.3277, 0.1],
[ -0.3536, 0.1581, 1.954e-15, -0.07265, -0.2084],
[ -0.1768, -0.1581, 0.5303, -0.5726, -0.05555],
[ -7.109e-18, -0.4743, -0.1768, -0.3142, 0.4959],
[ -0.1768, 0.1581, 1.915e-15, -0.592, -0.2791]],
S:
[ 35.33, 20, 19.6, 0, 0],
V:
[[ -0.8006, -0.3162, 0.2887, -0.4191, 0],
[ -0.4804, 0.6325, 7.768e-15, 0.4405, 0.4185],
[ -0.1601, -0.3162, -0.866, -0.052, 0.3488],
[ 4.684e-17, -0.6325, 0.2887, 0.6761, 0.2442],
[ -0.3203, 3.594e-15, -0.2887, 0.413, -0.8022]]}
</pre>
<!--
Some further tests.
<pre>
IN> n = 31; A = numeric.random([n,n]); B = numeric.eig(A); !(B.E.dot(numeric.T.diag(B.lambda).dot(B.E.inv())).sub(A).norm2()>1e-12)
OUT> true
IN> n = 32; A = numeric.random([n,n]); B = numeric.eig(A); !(B.E.dot(numeric.T.diag(B.lambda).dot(B.E.inv())).sub(A).norm2()>1e-12)
OUT> true
IN> n = 33; A = numeric.random([n,n]); B = numeric.eig(A); !(B.E.dot(numeric.T.diag(B.lambda).dot(B.E.inv())).sub(A).norm2()>1e-12)
OUT> true
IN> n = 34; A = numeric.random([n,n]); B = numeric.eig(A); !(B.E.dot(numeric.T.diag(B.lambda).dot(B.E.inv())).sub(A).norm2()>1e-12)
OUT> true
IN> n = 35; A = numeric.random([n,n]); B = numeric.eig(A); !(B.E.dot(numeric.T.diag(B.lambda).dot(B.E.inv())).sub(A).norm2()>1e-12)
OUT> true
IN> m = 17; n = 12; A = numeric.random([m,n]); B = numeric.svd(A); U = new numeric.T(B.U); V = new numeric.T(B.V); !(U.dot(numeric.T.diag(B.S)).dot(V.transpose()).sub(A).norm2()>1e-12)
OUT> true
IN> m = 21; n = 19; A = numeric.random([m,n]); B = numeric.svd(A); U = new numeric.T(B.U); V = new numeric.T(B.V); !(U.dot(numeric.T.diag(B.S)).dot(V.transpose()).sub(A).norm2()>1e-12)
OUT> true
IN> m = 33; n = 33; A = numeric.random([m,n]); B = numeric.svd(A); U = new numeric.T(B.U); V = new numeric.T(B.V); !(U.dot(numeric.T.diag(B.S)).dot(V.transpose()).sub(A).norm2()>1e-12)
OUT> true
IN> m = 59; n = 42; A = numeric.random([m,n]); B = numeric.svd(A); U = new numeric.T(B.U); V = new numeric.T(B.V); !(U.dot(numeric.T.diag(B.S)).dot(V.transpose()).sub(A).norm2()>1e-12)
OUT> true
IN> numeric.eig([[1, 0, 0], [0, 0.7181, -0.6960], [0, 0.6960, 0.7181]]) // This was a bug found by bdmartin
OUT> {lambda:
{x:
[ 1, 0.7181, 0.7181],
y:
[ 0, 0.696, -0.696]},
E:
{x:
[[ 1, 0, 0],
[ 0, 0, 0.7071],
[ 0, -0.7071, 0]],
y:
[[ 0, 0, 0],
[ 0, -0.7071, 0],
[ 0, 0, 0.7071]]}}
</pre>
-->
<h1>Sparse linear algebra</h1>
Sparse matrices are matrices that have a lot of zeroes. In numeric, sparse matrices are stored in the
"compressed column storage" ordering. Example:
<pre>
IN> A = [[1,2,0],
[0,3,0],
[2,0,5]];
SA = numeric.ccsSparse(A);
OUT> [[0,2,4,5],
[0,2,0,1,2],
[1,2,2,3,5]]
</pre>
The relation between A and its sparse representation SA is:
<pre >
A[i][SA[1][k]] = SA[2][k] with SA[0][i] ≤ k < SA[0][i+1]
</pre >
In other words, SA[2] stores the nonzero entries of A; SA[1] stores the row numbers and SA[0] stores the
offsets of the columns. See <i>I. DUFF, R. GRIMES, AND J. LEWIS, Sparse matrix test problems, ACM Trans. Math. Soft., 15 (1989), pp. 1-14.</i>
<pre>
IN> A = numeric.ccsSparse([[ 3, 5, 8,10, 8],
[ 7,10, 3, 5, 3],
[ 6, 3, 5, 1, 8],
[ 2, 6, 7, 1, 2],
[ 1, 2, 9, 3, 9]]);
OUT> [[0,5,10,15,20,25],
[0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4],
[3,7,6,2,1,5,10,3,6,2,8,3,5,7,9,10,5,1,1,3,8,3,8,2,9]]
IN> numeric.ccsFull(A);
OUT> [[ 3, 5, 8,10, 8],
[ 7,10, 3, 5, 3],
[ 6, 3, 5, 1, 8],
[ 2, 6, 7, 1, 2],
[ 1, 2, 9, 3, 9]]
IN> numeric.ccsDot(numeric.ccsSparse([[1,2,3],[4,5,6]]),numeric.ccsSparse([[7,8],[9,10],[11,12]]))
OUT> [[0,2,4],
[0,1,0,1],
[58,139,64,154]]
IN> M = [[0,1,3,6],[0,0,1,0,1,2],[3,-1,2,3,-2,4]];
b = [9,3,2];
x = numeric.ccsTSolve(M,b);
OUT> [3.167,2,0.5]
IN> numeric.ccsDot(M,[[0,3],[0,1,2],x])
OUT> [[0,3],[0,1,2],[9,3,2]]
</pre>
We provide an LU=PA decomposition:
<pre>
IN> A = [[0,5,10,15,20,25],
[0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4],
[3,7,6,2,1,5,10,3,6,2,8,3,5,7,9,10,5,1,1,3,8,3,8,2,9]];
LUP = numeric.ccsLUP(A);
OUT> {L:[[0,5,9,12,14,15],
[0,2,4,1,3,1,3,4,2,2,4,3,3,4,4],
[1,0.1429,0.2857,0.8571,0.4286,1,-0.1282,-0.5641,-0.1026,1,0.8517,0.7965,1,-0.67,1]],
U:[[0,1,3,6,10,15],
[0,0,1,0,1,2,0,1,2,3,0,1,2,3,4],
[7,10,-5.571,3,2.429,8.821,5,-3.286,1.949,5.884,3,5.429,9.128,0.1395,-3.476]],
P:[1,2,4,0,3],
Pinv:[3,0,1,4,2]}
IN> numeric.ccsFull(numeric.ccsDot(LUP.L,LUP.U))
OUT> [[ 7,10, 3, 5, 3],
[ 6, 3, 5, 1, 8],
[ 1, 2, 9, 3, 9],
[ 3, 5, 8,10, 8],
[ 2, 6, 7, 1, 2]]
IN> x = numeric.ccsLUPSolve(LUP,[96,63,82,51,89])
OUT> [3,1,4,1,5]
IN> X = numeric.trunc(numeric.ccsFull(numeric.ccsLUPSolve(LUP,A)),1e-15); // Solve LUX = PA
OUT> [[1,0,0,0,0],
[0,1,0,0,0],
[0,0,1,0,0],
[0,0,0,1,0],
[0,0,0,0,1]]
IN> numeric.ccsLUP(A,0.4).P;
OUT> [0,2,1,3,4]
</pre>
The LUP decomposition uses partial pivoting and has an optional thresholding argument.
With a threshold of 0.4, the pivots are [0,2,1,3,4] (only rows 1 and 2 have been exchanged) instead of the
"full partial pivoting" order above which was [1,2,4,0,3]. Threshold=0 gives no pivoting
and threshold=1 gives normal partial pivoting. Note that a small or zero threshold can result in numerical
instabilities and is normally used when the matrix A is already in some order that minimizes fill-in.
<!-- Stress test:
<pre>
IN> result = "Sparse LUP Stress Test OK";
for(k=0;k<1000;++k) {
A = numeric.ccsSparse(numeric.random([10,10]));
LUP = numeric.ccsLUP(A);
foo = numeric.ccsFull(numeric.ccsDot(LUP.L,LUP.U));
PA = numeric.ccsFull(numeric.ccsGetBlock(A,LUP.P));
res = numeric.norminf(numeric.sub(foo,PA));
if(!isFinite(res) || res>1e-6) {
result = {
code: "Failed during 1000 sparse LUP",
k:k,A:A,LUP:LUP,res:res
};
break;
};
};
result;
OUT> "Sparse LUP Stress Test OK"
</pre>
-->
We also support arithmetic on CCS matrices:
<pre>
IN> A = numeric.ccsSparse([[1,2,0],[0,3,0],[0,0,5]]);
B = numeric.ccsSparse([[2,9,0],[0,4,0],[-2,0,0]]);
numeric.ccsadd(A,B);
OUT> [[0,2,4,5],
[0,2,0,1,2],
[3,-2,11,7,5]]
</pre>
We also have scatter/gather functions
<pre>
IN> X = [[0,0,1,1,2,2],[0,1,1,2,2,3],[1,2,3,4,5,6]];
SX = numeric.ccsScatter(X);
OUT> [[0,1,3,5,6],
[0,0,1,1,2,2],
[1,2,3,4,5,6]]
IN> numeric.ccsGather(SX)
OUT> [[0,0,1,1,2,2],[0,1,1,2,2,3],[1,2,3,4,5,6]]
</pre>
<h1>Coordinate matrices</h1>
We also provide a banded matrix implementation using the coordinate encoding.<br><br>
LU decomposition:
<pre>
IN> lu = numeric.cLU([[0,0,1,1,1,2,2],[0,1,0,1,2,1,2],[2,-1,-1,2,-1,-1,2]])
OUT> {U:[[ 0, 0, 1, 1, 2 ],
[ 0, 1, 1, 2, 2 ],
[ 2, -1, 1.5, -1, 1.333]],
L:[[ 0, 1, 1, 2, 2 ],
[ 0, 0, 1, 1, 2 ],
[ 1, -0.5, 1,-0.6667, 1 ]]}
IN> numeric.cLUsolve(lu,[5,-8,13])
OUT> [3,1,7]
</pre>
Note that <tt>numeric.cLU()</tt> does not have any pivoting.
<h1>Solving PDEs</h1>
The functions <tt>numeric.cgrid()</tt> and <tt>numeric.cdelsq()</tt> can be used to obtain a
numerical Laplacian for a domain.
<pre>
IN> g = numeric.cgrid(5)
OUT>
[[-1,-1,-1,-1,-1],
[-1, 0, 1, 2,-1],
[-1, 3, 4, 5,-1],
[-1, 6, 7, 8,-1],
[-1,-1,-1,-1,-1]]
IN> coordL = numeric.cdelsq(g)
OUT>
[[ 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8],
[ 1, 3, 0, 0, 2, 4, 1, 1, 5, 2, 0, 4, 6, 3, 1, 3, 5, 7, 4, 2, 4, 8, 5, 3, 7, 6, 4, 6, 8, 7, 5, 7, 8],
[-1,-1, 4,-1,-1,-1, 4,-1,-1, 4,-1,-1,-1, 4,-1,-1,-1,-1, 4,-1,-1,-1, 4,-1,-1, 4,-1,-1,-1, 4,-1,-1, 4]]
IN> L = numeric.sscatter(coordL); // Just to see what it looks like
OUT>
[[ 4, -1, , -1],
[ -1, 4, -1, , -1],
[ , -1, 4, , , -1],
[ -1, , , 4, -1, , -1],
[ , -1, , -1, 4, -1, , -1],
[ , , -1, , -1, 4, , , -1],
[ , , , -1, , , 4, -1],
[ , , , , -1, , -1, 4, -1],
[ , , , , , -1, , -1, 4]]
IN> lu = numeric.cLU(coordL); x = numeric.cLUsolve(lu,[1,1,1,1,1,1,1,1,1]);
OUT> [0.6875,0.875,0.6875,0.875,1.125,0.875,0.6875,0.875,0.6875]
IN> numeric.cdotMV(coordL,x)
OUT> [1,1,1,1,1,1,1,1,1]
IN> G = numeric.rep([5,5],0); for(i=0;i<5;i++) for(j=0;j<5;j++) if(g[i][j]>=0) G[i][j] = x[g[i][j]]; G
OUT>
[[ 0 , 0 , 0 , 0 , 0 ],
[ 0 , 0.6875, 0.875 , 0.6875, 0 ],
[ 0 , 0.875 , 1.125 , 0.875 , 0 ],
[ 0 , 0.6875, 0.875 , 0.6875, 0 ],
[ 0 , 0 , 0 , 0 , 0 ]]
IN> workshop.html('<img src="'+numeric.imageURL(numeric.mul([G,G,G],200))+'" width=100 />');
OUT>
<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAUAAAAFCAIAAAACDbGyAAAAcElEQVQIHQARAO7/AAAAAAAAAAAAAAAAAAAAAAAAEADv/wAAAIqKiq+vr4mJiQAAAAAAEADv/wAAAK+vr+Hh4a+vrwAAAAAAEADv/wAAAIqKiq+vr4qKigAAAAABEADv/wAAAAAAAAAAAAAAAAAAAACRjRFNqL3leAAAAABJRU5ErkJggg==" width=100 />
</pre>
You can also work on an L-shaped or arbitrary-shape domain:
<pre>
IN> numeric.cgrid(6,'L')
OUT>
[[-1,-1,-1,-1,-1,-1],
[-1, 0, 1,-1,-1,-1],
[-1, 2, 3,-1,-1,-1],
[-1, 4, 5, 6, 7,-1],
[-1, 8, 9,10,11,-1],
[-1,-1,-1,-1,-1,-1]]
IN> numeric.cgrid(5,function(i,j) { return i!==2 || j!==2; })
OUT>
[[-1,-1,-1,-1,-1],
[-1, 0, 1, 2,-1],
[-1, 3,-1, 4,-1],
[-1, 5, 6, 7,-1],