1MatrixOps(3) User Contributed Perl Documentation MatrixOps(3)
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6 PDL::MatrixOps -- Some Useful Matrix Operations
7
9 $inv = $x->inv;
10 $det = $x->det;
12 ($lu,$perm,$par) = $x->lu_decomp;
14 $y = lu_backsub($lu,$perm,$z); # solve $x x $y = $z
15
17 PDL::MatrixOps is PDL's built-in matrix manipulation code. It contains
18 utilities for many common matrix operations: inversion, determinant
19 finding, eigenvalue/vector finding, singular value decomposition, etc.
20 PDL::MatrixOps routines are written in a mixture of Perl and C, so that
21 they are reliably present even when there is no FORTRAN compiler or
22 external library available (e.g. PDL::Slatec or any of the PDL::GSL
23 family of modules).
24 Matrix manipulation, particularly with large matrices, is a challenging
26 field and no one algorithm is suitable in all cases. The utilities
27 here use general-purpose algorithms that work acceptably for many cases
28 but might not scale well to very large or pathological (near-singular)
29 matrices.
30 Except as noted, the matrices are PDLs whose 0th dimension ranges over
32 column and whose 1st dimension ranges over row. The matrices appear
33 correctly when printed.
34 These routines should work OK with PDL::Matrix objects as well as with
36 normal PDLs.
37
39 Like most computer languages, PDL addresses matrices in (column,row)
40 order in most cases; this corresponds to (X,Y) coordinates in the
41 matrix itself, counting rightwards and downwards from the upper left
42 corner. This means that if you print a PDL that contains a matrix, the
43 matrix appears correctly on the screen, but if you index a matrix
44 element, you use the indices in the reverse order that you would in a
45 math textbook. If you prefer your matrices indexed in (row, column)
46 order, you can try using the PDL::Matrix object, which includes an
47 implicit exchange of the first two dimensions but should be compatible
48 with most of these matrix operations. TIMTOWDTI.)
49 Matrices, row vectors, and column vectors can be multiplied with the
51 'x' operator (which is, of course, threadable):
52 $m3 = $m1 x $m2;
54 $col_vec2 = $m1 x $col_vec1;
55 $row_vec2 = $row_vec1 x $m1;
56 $scalar = $row_vec x $col_vec;
57 Because of the (column,row) addressing order, 1-D PDLs are treated as
59 _row_ vectors; if you want a _column_ vector you must add a dummy
60 dimension:
61 $rowvec = pdl(1,2); # row vector
63 $colvec = $rowvec->(*1); # 1x2 column vector
64 $matrix = pdl([[3,4],[6,2]]); # 2x2 matrix
65 $rowvec2 = $rowvec x $matrix; # right-multiplication by matrix
66 $colvec = $matrix x $colvec; # left-multiplication by matrix
67 $m2 = $matrix x $rowvec; # Throws an error
68 Implicit threading works correctly with most matrix operations, but you
70 must be extra careful that you understand the dimensionality. In
71 particular, matrix multiplication and other matrix ops need nx1 PDLs as
72 row vectors and 1xn PDLs as column vectors. In most cases you must
73 explicitly include the trailing 'x1' dimension in order to get the
74 expected results when you thread over multiple row vectors.
75 When threading over matrices, it's very easy to get confused about
77 which dimension goes where. It is useful to include comments with every
78 expression, explaining what you think each dimension means:
79 $x = xvals(360)*3.14159/180; # (angle)
81 $rot = cat(cat(cos($x),sin($x)), # rotmat: (col,row,angle)
82 cat(-sin($x),cos($x)));
83
85 MatrixOps includes algorithms and pre-existing code from several
86 origins. In particular, "eigens_sym" is the work of Stephen Moshier,
87 "svd" uses an SVD subroutine written by Bryant Marks, and "eigens" uses
88 a subset of the Small Scientific Library by Kenneth Geisshirt. They
89 are free software, distributable under same terms as PDL itself.
90
92 This is intended as a general-purpose linear algebra package for small-
93 to-mid sized matrices. The algorithms may not scale well to large
94 matrices (hundreds by hundreds) or to near singular matrices.
95 If there is something you want that is not here, please add and
97 document it!
98
100 identity
101 Signature: (n; [o]a(n,n))
102 Return an identity matrix of the specified size. If you hand in a
104 scalar, its value is the size of the identity matrix; if you hand in a
105 dimensioned PDL, the 0th dimension is the first two dimensions of the
106 matrix, with higher dimensions preserved.
107 stretcher
109 Signature: (a(n); [o]b(n,n))
110 $mat = stretcher($eigenvalues);
112 Return a diagonal matrix with the specified diagonal elements
114 inv
116 Signature: (a(m,m); sv opt )
117 $a1 = inv($a, {$opt});
119 Invert a square matrix.
121 You feed in an NxN matrix in $a, and get back its inverse (if it
123 exists). The code is inplace-aware, so you can get back the inverse in
124 $a itself if you want -- though temporary storage is used either way.
125 You can cache the LU decomposition in an output option variable.
126 "inv" uses "lu_decomp" by default; that is a numerically stable
128 (pivoting) LU decomposition method.
129 OPTIONS:
131 • s
133 Boolean value indicating whether to complain if the matrix is
135 singular. If this is false, singular matrices cause inverse to
136 barf. If it is true, then singular matrices cause inverse to return
137 undef.
138 • lu (I/O)
140 This value contains a list ref with the LU decomposition,
142 permutation, and parity values for $a. If you do not mention the
143 key, or if the value is undef, then inverse calls "lu_decomp". If
144 the key exists with an undef value, then the output of "lu_decomp"
145 is stashed here (unless the matrix is singular). If the value
146 exists, then it is assumed to hold the LU decomposition.
147 • det (Output)
149 If this key exists, then the determinant of $a get stored here,
151 whether or not the matrix is singular.
152 det
154 Signature: (a(m,m); sv opt)
155 $det = det($a,{opt});
157 Determinant of a square matrix using LU decomposition (for large
159 matrices)
160 You feed in a square matrix, you get back the determinant. Some
162 options exist that allow you to cache the LU decomposition of the
163 matrix (note that the LU decomposition is invalid if the determinant is
164 zero!). The LU decomposition is cacheable, in case you want to re-use
165 it. This method of determinant finding is more rapid than recursive-
166 descent on large matrices, and if you reuse the LU decomposition it's
167 essentially free.
168 OPTIONS:
170 • lu (I/O)
172 Provides a cache for the LU decomposition of the matrix. If you
174 provide the key but leave the value undefined, then the LU
175 decomposition goes in here; if you put an LU decomposition here, it
176 will be used and the matrix will not be decomposed again.
177 determinant
179 Signature: (a(m,m))
180 $det = determinant($x);
182 Determinant of a square matrix, using recursive descent (threadable).
184 This is the traditional, robust recursive determinant method taught in
186 most linear algebra courses. It scales like "O(n!)" (and hence is
187 pitifully slow for large matrices) but is very robust because no
188 division is involved (hence no division-by-zero errors for singular
189 matrices). It's also threadable, so you can find the determinants of a
190 large collection of matrices all at once if you want.
191 Matrices up to 3x3 are handled by direct multiplication; larger
193 matrices are handled by recursive descent to the 3x3 case.
194 The LU-decomposition method "det" is faster in isolation for single
196 matrices larger than about 4x4, and is much faster if you end up
197 reusing the LU decomposition of $a (NOTE: check performance and
198 threading benchmarks with new code).
199 eigens_sym
201 Signature: ([phys]a(m); [o,phys]ev(n,n); [o,phys]e(n))
202 Eigenvalues and -vectors of a symmetric square matrix. If passed an
204 asymmetric matrix, the routine will warn and symmetrize it, by taking
205 the average value. That is, it will solve for 0.5*($a+$a->transpose).
206 It's threadable, so if $a is 3x3x100, it's treated as 100 separate 3x3
208 matrices, and both $ev and $e get extra dimensions accordingly.
209 If called in scalar context it hands back only the eigenvalues.
211 Ultimately, it should switch to a faster algorithm in this case (as
212 discarding the eigenvectors is wasteful).
213 The algorithm used is due to J. vonNeumann, which was a rediscovery of
215 Jacobi's Method
216 <http://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm> .
217 The eigenvectors are returned in COLUMNS of the returned PDL. That
219 makes it slightly easier to access individual eigenvectors, since the
220 0th dim of the output PDL runs across the eigenvectors and the 1st dim
221 runs across their components.
222 ($ev,$e) = eigens_sym $x; # Make eigenvector matrix
224 $vector = $ev->($n); # Select nth eigenvector as a column-vector
225 $vector = $ev->(($n)); # Select nth eigenvector as a row-vector
226 ($ev, $e) = eigens_sym($x); # e-vects & e-values
228 $e = eigens_sym($x); # just eigenvalues
229 eigens_sym ignores the bad-value flag of the input ndarrays. It will
231 set the bad-value flag of all output ndarrays if the flag is set for
232 any of the input ndarrays.
233 eigens
235 Signature: ([phys]a(m); [o,phys]ev(l,n,n); [o,phys]e(l,n))
236 Real eigenvalues and -vectors of a real square matrix.
238 (See also "eigens_sym", for eigenvalues and -vectors of a real,
240 symmetric, square matrix).
241 The eigens function will attempt to compute the eigenvalues and
243 eigenvectors of a square matrix with real components. If the matrix is
244 symmetric, the same underlying code as "eigens_sym" is used. If
245 asymmetric, the eigenvalues and eigenvectors are computed with
246 algorithms from the sslib library. If any imaginary components exist
247 in the eigenvalues, the results are currently considered to be invalid,
248 and such eigenvalues are returned as "NaN"s. This is true for
249 eigenvectors also. That is if there are imaginary components to any of
250 the values in the eigenvector, the eigenvalue and corresponding
251 eigenvectors are all set to "NaN". Finally, if there are any repeated
252 eigenvectors, they are replaced with all "NaN"s.
253 Use of the eigens function on asymmetric matrices should be considered
255 experimental! For asymmetric matrices, nearly all observed matrices
256 with real eigenvalues produce incorrect results, due to errors of the
257 sslib algorithm. If your assymmetric matrix returns all NaNs, do not
258 assume that the values are complex. Also, problems with memory access
259 is known in this library.
260 Not all square matrices are diagonalizable. If you feed in a non-
262 diagonalizable matrix, then one or more of the eigenvectors will be set
263 to NaN, along with the corresponding eigenvalues.
264 "eigens" is threadable, so you can solve 100 eigenproblems by feeding
266 in a 3x3x100 array. Both $ev and $e get extra dimensions accordingly.
267 If called in scalar context "eigens" hands back only the eigenvalues.
269 This is somewhat wasteful, as it calculates the eigenvectors anyway.
270 The eigenvectors are returned in COLUMNS of the returned PDL (ie the
272 the 0 dimension). That makes it slightly easier to access individual
273 eigenvectors, since the 0th dim of the output PDL runs across the
274 eigenvectors and the 1st dim runs across their components.
275 ($ev,$e) = eigens $x; # Make eigenvector matrix
277 $vector = $ev->($n); # Select nth eigenvector as a column-vector
278 $vector = $ev->(($n)); # Select nth eigenvector as a row-vector
279 DEVEL NOTES:
281 For now, there is no distinction between a complex eigenvalue and an
283 invalid eigenvalue, although the underlying code generates complex
284 numbers. It might be useful to be able to return complex eigenvalues.
285 ($ev, $e) = eigens($x); # e'vects & e'vals
287 $e = eigens($x); # just eigenvalues
288 eigens ignores the bad-value flag of the input ndarrays. It will set
290 the bad-value flag of all output ndarrays if the flag is set for any of
291 the input ndarrays.
292 svd
294 Signature: (a(n,m); [o]u(n,m); [o,phys]z(n); [o]v(n,n))
295 ($u, $s, $v) = svd($x);
297 Singular value decomposition of a matrix.
299 "svd" is threadable.
301 Given an m x n matrix $a that has m rows and n columns (m >= n), "svd"
303 computes matrices $u and $v, and a vector of the singular values $s.
304 Like most implementations, "svd" computes what is commonly referred to
305 as the "thin SVD" of $a, such that $u is m x n, $v is n x n, and there
306 are <=n singular values. As long as m >= n, the original matrix can be
307 reconstructed as follows:
308 ($u,$s,$v) = svd($x);
310 $ess = zeroes($x->dim(0),$x->dim(0));
311 $ess->slice("$_","$_").=$s->slice("$_") foreach (0..$x->dim(0)-1); #generic diagonal
312 $a_copy = $u x $ess x $v->transpose;
313 If m==n, $u and $v can be thought of as rotation matrices that convert
315 from the original matrix's singular coordinates to final coordinates,
316 and from original coordinates to singular coordinates, respectively,
317 and $ess is a diagonal scaling matrix.
318 If n>m, "svd" will barf. This can be avoided by passing in the
320 transpose of $a, and reconstructing the original matrix like so:
321 ($u,$s,$v) = svd($x->transpose);
323 $ess = zeroes($x->dim(1),$x->dim(1));
324 $ess->slice("$_","$_").=$s->slice("$_") foreach (0..$x->dim(1)-1); #generic diagonal
325 $x_copy = $v x $ess x $u->transpose;
326 EXAMPLE
328 The computing literature has loads of examples of how to use SVD.
330 Here's a trivial example (used in PDL::Transform::map) of how to make a
331 matrix less, er, singular, without changing the orientation of the
332 ellipsoid of transformation:
333 { my($r1,$s,$r2) = svd $x;
335 $s++; # fatten all singular values
336 $r2 *= $s; # implicit threading for cheap mult.
337 $x .= $r2 x $r1; # a gets r2 x ess x r1
338 }
339 svd ignores the bad-value flag of the input ndarrays. It will set the
341 bad-value flag of all output ndarrays if the flag is set for any of the
342 input ndarrays.
343 lu_decomp
345 Signature: (a(m,m); [o]lu(m,m); [o]perm(m); [o]parity)
346 LU decompose a matrix, with row permutation
348 ($lu, $perm, $parity) = lu_decomp($x);
350 $lu = lu_decomp($x, $perm, $par); # $perm and $par are outputs!
352 lu_decomp($x->inplace,$perm,$par); # Everything in place.
354 "lu_decomp" returns an LU decomposition of a square matrix, using
356 Crout's method with partial pivoting. It's ported from Numerical
357 Recipes. The partial pivoting keeps it numerically stable but means a
358 little more overhead from threading.
359 "lu_decomp" decomposes the input matrix into matrices L and U such that
361 LU = A, L is a subdiagonal matrix, and U is a superdiagonal matrix. By
362 convention, the diagonal of L is all 1's.
363 The single output matrix contains all the variable elements of both the
365 L and U matrices, stacked together. Because the method uses pivoting
366 (rearranging the lower part of the matrix for better numerical
367 stability), you have to permute input vectors before applying the L and
368 U matrices. The permutation is returned either in the second argument
369 or, in list context, as the second element of the list. You need the
370 permutation for the output to make any sense, so be sure to get it one
371 way or the other.
372 LU decomposition is the answer to a lot of matrix questions, including
374 inversion and determinant-finding, and "lu_decomp" is used by "inv".
375 If you pass in $perm and $parity, they either must be predeclared PDLs
377 of the correct size ($perm is an n-vector, $parity is a scalar) or
378 scalars.
379 If the matrix is singular, then the LU decomposition might not be
381 defined; in those cases, "lu_decomp" silently returns undef. Some
382 singular matrices LU-decompose just fine, and those are handled OK but
383 give a zero determinant (and hence can't be inverted).
384 "lu_decomp" uses pivoting, which rearranges the values in the matrix
386 for more numerical stability. This makes it really good for large and
387 even near-singular matrices. There is a non-pivoting version
388 "lu_decomp2" available which is from 5 to 60 percent faster for typical
389 problems at the expense of failing to compute a result in some cases.
390 Now that the "lu_decomp" is threaded, it is the recommended LU
392 decomposition routine. It no longer falls back to "lu_decomp2".
393 "lu_decomp" is ported from Numerical Recipes to PDL. It should probably
395 be implemented in C.
396 lu_decomp2
398 Signature: (a(m,m); [o]lu(m,m))
399 LU decompose a matrix, with no row permutation
401 ($lu, $perm, $parity) = lu_decomp2($x);
403 $lu = lu_decomp2($x,$perm,$parity); # or
405 $lu = lu_decomp2($x); # $perm and $parity are optional
406 lu_decomp($x->inplace,$perm,$parity); # or
408 lu_decomp($x->inplace); # $perm and $parity are optional
409 "lu_decomp2" works just like "lu_decomp", but it does no pivoting at
411 all. For compatibility with "lu_decomp", it will give you a
412 permutation list and a parity scalar if you ask for them -- but they
413 are always trivial.
414 Because "lu_decomp2" does not pivot, it is numerically unstable -- that
416 means it is less precise than "lu_decomp", particularly for large or
417 near-singular matrices. There are also specific types of non-singular
418 matrices that confuse it (e.g. ([0,-1,0],[1,0,0],[0,0,1]), which is a
419 90 degree rotation matrix but which confuses "lu_decomp2").
420 On the other hand, if you want to invert rapidly a few hundred thousand
422 small matrices and don't mind missing one or two, it could be the
423 ticket. It can be up to 60% faster at the expense of possible failure
424 of the decomposition for some of the input matrices.
425 The output is a single matrix that contains the LU decomposition of $a;
427 you can even do it in-place, thereby destroying $a, if you want. See
428 "lu_decomp" for more information about LU decomposition.
429 "lu_decomp2" is ported from Numerical Recipes into PDL.
431 lu_backsub
433 Signature: (lu(m,m); perm(m); b(m))
434 Solve A x = B for matrix A, by back substitution into A's LU
436 decomposition.
437 ($lu,$perm,$par) = lu_decomp($A);
439 $x = lu_backsub($lu,$perm,$par,$A); # or
441 $x = lu_backsub($lu,$perm,$B); # $par is not required for lu_backsub
442 lu_backsub($lu,$perm,$B->inplace); # modify $B in-place
444 $x = lu_backsub(lu_decomp($A),$B); # (ignores parity value from lu_decomp)
446 # starting from square matrix A and columns matrix B, with mathematically
448 # correct dimensions
449 $A = identity(4) + ones(4, 4);
450 $A->slice('2,0') .= 0; # break symmetry to see if need transpose
451 $B = sequence(2, 4);
452 # all these functions take B as rows, interpret as though notional columns
453 # mathematically confusing but can't change as back-compat and also
454 # familiar to Fortran users, so just transpose inputs and outputs
455 # using lu_backsub
457 ($lu,$perm,$par) = lu_decomp($A);
458 $x = lu_backsub($lu,$perm,$par, $B->transpose)->transpose;
459 # or with Slatec LINPACK
461 use PDL::Slatec;
462 gefa($lu=$A->copy, $ipiv=null, $info=null);
463 # 1 = do transpose because Fortran's idea of rows vs columns
464 gesl($lu, $ipiv, $x=$B->transpose->copy, 1);
465 $x = $x->inplace->transpose;
466 # or with LAPACK
468 use PDL::LinearAlgebra::Real;
469 getrf($lu=$A->copy, $ipiv=null, $info=null);
470 getrs($lu, 1, $x=$B->transpose->copy, $ipiv, $info=null); # again, need transpose
471 $x=$x->inplace->transpose;
472 # or with GSL
474 use PDL::GSL::LINALG;
475 LU_decomp(my $lu=$A->copy, my $p=null, my $signum=null);
476 # $B and $x, first dim is because GSL treats as vector, higher dims thread
477 # so we transpose in and back out
478 LU_solve($lu, $p, $B->transpose, my $x=null);
479 $x=$x->inplace->transpose;
480 # proof of the pudding is in the eating:
482 print $A x $x;
483 Given the LU decomposition of a square matrix (from "lu_decomp"),
485 "lu_backsub" does back substitution into the matrix to solve "a x = b"
486 for given vector "b". It is separated from the "lu_decomp" method so
487 that you can call the cheap "lu_backsub" multiple times and not have to
488 do the expensive LU decomposition more than once.
489 "lu_backsub" acts on single vectors and threads in the usual way, which
491 means that it treats $y as the transpose of the input. If you want to
492 process a matrix, you must hand in the transpose of the matrix, and
493 then transpose the output when you get it back. that is because pdls
494 are indexed by (col,row), and matrices are (row,column) by convention,
495 so a 1-D pdl corresponds to a row vector, not a column vector.
496 If $lu is dense and you have more than a few points to solve for, it is
498 probably cheaper to find "a^-1" with "inv", and just multiply "x = a^-1
499 b".) in fact, "inv" works by calling "lu_backsub" with the identity
500 matrix.
501 "lu_backsub" is ported from section 2.3 of Numerical Recipes. It is
503 written in PDL but should probably be implemented in C.
504 simq
506 Signature: ([phys]a(n,n); [phys]b(n); [o,phys]x(n); int [o,phys]ips(n); int flag)
507 Solution of simultaneous linear equations, "a x = b".
509 $a is an "n x n" matrix (i.e., a vector of length "n*n"), stored row-
511 wise: that is, "a(i,j) = a[ij]", where "ij = i*n + j".
512 While this is the transpose of the normal column-wise storage, this
514 corresponds to normal PDL usage. The contents of matrix a may be
515 altered (but may be required for subsequent calls with flag = -1).
516 $y, $x, $ips are vectors of length "n".
518 Set "flag=0" to solve. Set "flag=-1" to do a new back substitution for
520 different $y vector using the same a matrix previously reduced when
521 "flag=0" (the $ips vector generated in the previous solution is also
522 required).
523 See also "lu_backsub", which does the same thing with a slightly less
525 opaque interface.
526 simq ignores the bad-value flag of the input ndarrays. It will set the
528 bad-value flag of all output ndarrays if the flag is set for any of the
529 input ndarrays.
530 squaretotri
532 Signature: (a(n,n); b(m))
533 Convert a symmetric square matrix to triangular vector storage.
535 squaretotri does not process bad values. It will set the bad-value
537 flag of all output ndarrays if the flag is set for any of the input
538 ndarrays.
539
541 Copyright (C) 2002 Craig DeForest (deforest@boulder.swri.edu), R.J.R.
542 Williams (rjrw@ast.leeds.ac.uk), Karl Glazebrook
543 (kgb@aaoepp.aao.gov.au). There is no warranty. You are allowed to
544 redistribute and/or modify this work under the same conditions as PDL
545 itself. If this file is separated from the PDL distribution, then the
546 PDL copyright notice should be included in this file.
547perl v5.34.0 2021-08-16 MatrixOps(3)