Math::Utils(3pm)

1Math::Utils(3)        User Contributed Perl Documentation       Math::Utils(3)
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4

NAME

6       Math::Utils - Useful mathematical functions not in Perl.
7

SYNOPSIS

9           use Math::Utils qw(:utility);    # Useful functions
10
11           #
12           # Base 10 and base 2 logarithms.
13           #
14           $scale = log10($pagewidth);
15           $bits = log2(1/$probability);
16
17           #
18           # Two uses of sign().
19           #
20           $d = sign($z - $w);
21
22           @ternaries = sign(@coefficients);
23
24           #
25           # Using copysign(), $dist will be doubled negative or
26           # positive $offest, depending upon whether ($from - $to)
27           # is positive or negative.
28           #
29           my $dist = copysign(2 * $offset, $from - $to);
30
31           #
32           # Change increment direction if goal is negative.
33           #
34           $incr = flipsign($incr, $goal);
35
36           #
37           # floor() and ceil() functions.
38           #
39           $point = floor($goal);
40           $limit = ceil($goal);
41
42           #
43           # gcd() and lcm() functions.
44           #
45           $divisor = gcd(@multipliers);
46           $numerator = lcm(@multipliers);
47
48           #
49           # Safer summation.
50           #
51           $tot = fsum(@inputs);
52
53           #
54           # The remainders of n after successive divisions of b, or
55           # remainders after a set of divisions.
56           #
57           @rems = moduli($n, $b);
58
59       or
60
61           use Math::Utils qw(:compare);    # Make comparison functions with tolerance.
62
63           #
64           # Floating point comparison function.
65           #
66           my $fltcmp = generate_fltmcp(1.0e-7);
67
68           if (&$fltcmp($x0, $x1) < 0)
69           {
70               add_left($data);
71           }
72           else
73           {
74               add_right($data);
75           }
76
77           #
78           # Or we can create single-operation comparison functions.
79           #
80           # Here we are only interested in the greater than and less than
81           # comparison functions.
82           #
83           my(undef, undef,
84               $approx_gt, undef, $approx_lt) = generate_relational(1.5e-5);
85
86       or
87
88           use Math::Utils qw(:polynomial);    # Basic polynomial ops
89
90           #
91           # Coefficient lists run from 0th degree upward, left to right.
92           #
93           my @c1 = (1, 3, 5, 7, 11, 13, 17, 19);
94           my @c2 = (1, 3, 1, 7);
95           my @c3 = (1, -1, 1)
96
97           my $c_ref = pl_mult(\@c1, \@c2);
98           $c_ref = pl_add($c_ref, \@c3);
99

EXPORT

101       All functions can be exported by name, or by using the tag that they're
102       grouped under.
103
104   utility tag
105       Useful, general-purpose functions, including those that originated in
106       FORTRAN and were implemented in Perl in the module Math::Fortran, by J.
107       A. R. Williams.
108
109       There is a name change -- copysign() was known as sign() in
110       Math::Fortran.
111
112       log10()
113
114           $xlog10 = log10($x);
115           @xlog10 = log10(@x);
116
117       Return the log base ten of the argument. A list form of the function is
118       also provided.
119
120       log2()
121
122           $xlog2 = log2($x);
123           @xlog2 = log2(@x);
124
125       Return the log base two of the argument. A list form of the function is
126       also provided.
127
128       sign()
129
130           $s = sign($x);
131           @valsigns = sign(@values);
132
133       Returns -1 if the argument is negative, 0 if the argument is zero, and
134       1 if the argument is positive.
135
136       In list form it applies the same operation to each member of the list.
137
138       copysign()
139
140           $ms = copysign($m, $n);
141           $s = copysign($x);
142
143       Take the sign of the second argument and apply it to the first. Zero is
144       considered part of the positive signs.
145
146           copysign(-5, 0);  # Returns 5.
147           copysign(-5, 7);  # Returns 5.
148           copysign(-5, -7); # Returns -5.
149           copysign(5, -7);  # Returns -5.
150
151       If there is only one argument, return -1 if the argument is negative,
152       otherwise return 1. For example, copysign(1, -4) and copysign(-4) both
153       return -1.
154
155       flipsign()
156
157           $ms = flipsign($m, $n);
158
159       Multiply the signs of the arguments and apply it to the first. As with
160       copysign(), zero is considered part of the positive signs.
161
162       Effectively this means change the sign of the first argument if the
163       second argument is negative.
164
165           flipsign(-5, 0);  # Returns -5.
166           flipsign(-5, 7);  # Returns -5.
167           flipsign(-5, -7); # Returns 5.
168           flipsign(5, -7);  # Returns -5.
169
170       If for some reason flipsign() is called with a single argument, that
171       argument is returned unchanged.
172
173       floor()
174
175           $b = floor($a/2);
176
177           @ilist = floor(@numbers);
178
179       Returns the greatest integer less than or equal to its argument.  A
180       list form of the function also exists.
181
182           floor(1.5, 1.87, 1);        # Returns (1, 1, 1)
183           floor(-1.5, -1.87, -1);     # Returns (-2, -2, -1)
184
185       ceil()
186
187           $b = ceil($a/2);
188
189           @ilist = ceil(@numbers);
190
191       Returns the lowest integer greater than or equal to its argument.  A
192       list form of the function also exists.
193
194           ceil(1.5, 1.87, 1);        # Returns (2, 2, 1)
195           ceil(-1.5, -1.87, -1);     # Returns (-1, -1, -1)
196
197       fsum()
198
199       Return a sum of the values in the list, done in a manner to avoid
200       rounding and cancellation errors. Currently this is done via Kahan's
201       summation algorithm
202       <https://en.wikipedia.org/wiki/Kahan_summation_algorithm>.
203
204       softmax()
205
206       Return a list of values as probabilities.
207
208       The function takes the list, and creates a new list by raising e to
209       each value. The function then returns each value divided by the sum of
210       the list. Each value in the new list is now a set of probabilities that
211       sum to 1.0.
212
213       The summation is performed using fsum() above.
214
215       See Softmax function <https://en.wikipedia.org/wiki/Softmax_function>
216       at Wikipedia.
217
218       uniform_scaling
219
220       uniform_01scaling
221
222       Uniformly, or linearly, scale a number either from one range to another
223       range ("uniform_scaling()"), or to a default range of [0 .. 1]
224       ("uniform_01scaling()").
225
226           @v = uniform_scaling(\@original_range, \@new_range, @oldvalues);
227
228       For example, these two lines are equivalent, and both return 0:
229
230           $y = uniform_scaling([50, 100], [0, 1], 50);
231
232           $y = uniform_01scaling([50, 100], 50);
233
234       They may also be called with a list or array of numbers:
235
236           @cm_measures = uniform_scaling([0, 10000], [0, 25400], @in_measures);
237
238           @melt_centigrade = uniform_scaling([0, 2000], [-273.15, 1726.85], \@melting_points);
239
240       A number that is outside the original bounds will be proportionally
241       changed to be outside of the new bounds, but then again having a number
242       outside the original bounds is probably an error that should be checked
243       before calling this function.
244
245       <https://stats.stackexchange.com/q/281164>
246
247       gcd
248
249       hcf
250
251       Return the greatest common divisor (also known as the highest common
252       factor) of a list of integers. These are simply synomyms:
253
254           $factor = gcd(@numbers);
255           $factor = hcf(@numbers);
256
257       lcm
258
259       Return the least common multiple of a list of integers.
260
261           $factor = lcm(@values);
262
263       moduli()
264
265       Return the moduli of an integer after repeated divisions. The
266       remainders are returned in a list from left to right.
267
268           @digits = moduli(1899, 10);   # Returns (9, 9, 8, 1)
269           @rems = moduli(29, 3);        # Returns (2, 0, 0, 1)
270
271   compare tag
272       Create comparison functions for floating point (non-integer) numbers.
273
274       Since exact comparisons of floating point numbers tend to be iffy, the
275       comparison functions use a tolerance chosen by you. You may then use
276       those functions from then on confident that comparisons will be
277       consistent.
278
279       If you do not provide a tolerance, a default tolerance of 1.49012e-8
280       (approximately the square root of an Intel Pentium's machine epsilon
281       <https://en.wikipedia.org/wiki/Machine_epsilon>) will be used.
282
283       generate_fltcmp()
284
285       Returns a comparison function that will compare values using a
286       tolerance that you supply. The generated function will return -1 if the
287       first argument compares as less than the second, 0 if the two arguments
288       compare as equal, and 1 if the first argument compares as greater than
289       the second.
290
291           my $fltcmp = generate_fltcmp(1.5e-7);
292
293           my(@xpos) = grep {&$fltcmp($_, 0) == 1} @xvals;
294
295       generate_relational()
296
297       Returns a list of comparison functions that will compare values using a
298       tolerance that you supply. The generated functions will be the
299       equivalent of the equal, not equal, greater than, greater than or
300       equal, less than, and less than or equal operators.
301
302           my($eq, $ne, $gt, $ge, $lt, $le) = generate_relational(1.5e-7);
303
304           my(@approx_5) = grep {&$eq($_, 5)} @xvals;
305
306       Of course, if you were only interested in not equal, you could use:
307
308           my(undef, $ne) = generate_relational(1.5e-7);
309
310           my(@not_around5) = grep {&$ne($_, 5)} @xvals;
311
312   polynomial tag
313       Perform some polynomial operations on plain lists of coefficients.
314
315           #
316           # The coefficient lists are presumed to go from low order to high:
317           #
318           @coefficients = (1, 2, 4, 8);    # 1 + 2x + 4x**2 + 8x**3
319
320       In all functions the coeffcient list is passed by reference to the
321       function, and the functions that return coefficients all return
322       references to a coefficient list.
323
324       It is assumed that any leading zeros in the coefficient lists have
325       already been removed before calling these functions, and that any
326       leading zeros found in the returned lists will be handled by the
327       caller. This caveat is particularly important to note in the case of
328       "pl_div()".
329
330       Although these functions are convenient for simple polynomial
331       operations, for more advanced polynonial operations Math::Polynomial is
332       recommended.
333
334       pl_evaluate()
335
336       Returns either a y-value for a corresponding x-value, or a list of
337       y-values on the polynomial for a corresponding list of x-values, using
338       Horner's method.
339
340           $y = pl_evaluate(\@coefficients, $x);
341           @yvalues = pl_evaluate(\@coefficients, @xvalues);
342
343           @ctemperatures = pl_evaluate([-160/9, 5/9], @ftemperatures);
344
345       The list of X values may also include X array references:
346
347           @yvalues = pl_evaluate(\@coefficients, @xvalues, \@primes, $x, [-1, -10, -100]);
348
349       pl_dxevaluate()
350
351           ($y, $dy, $ddy) = pl_dxevaluate(\@coefficients, $x);
352
353       Returns p(x), p'(x), and p"(x) of the polynomial for an x-value, using
354       Horner's method. Note that unlike "pl_evaluate()" above, the function
355       can only use one x-value.
356
357       If the polynomial is a linear equation, the second derivative value
358       will be zero.  Similarly, if the polynomial is a simple constant, the
359       first derivative value will be zero.
360
361       pl_translate()
362
363           $x = [8, 3, 1];
364           $y = [3, 1];
365
366           #
367           # Translating C<x**2 + 3*x + 8> by C<x + 3> returns [26, 9, 1]
368           #
369           $z = pl_translate($x, $y);
370
371       Returns a polynomial transformed by substituting a polynomial variable
372       with another polynomial.  For example, a simple linear translation by 1
373       to the polynomial "x**3 + x**2 + 4*x + 4" would be accomplished by
374       setting x = (y - 1); resulting in "x**3 - 2*x**2 + 5*x".
375
376           $x = [4, 4, 1, 1];
377           $y = [-1, 1];
378           $z = pl_translate($x, $y);         # Becomes [0, 5, -2, 1]
379
380       pl_add()
381
382           $polyn_ref = pl_add(\@m, \@n);
383
384       Add two lists of numbers as though they were polynomial coefficients.
385
386       pl_sub()
387
388           $polyn_ref = pl_sub(\@m, \@n);
389
390       Subtract the second list of numbers from the first as though they were
391       polynomial coefficients.
392
393       pl_div()
394
395           ($q_ref, $r_ref) = pl_div(\@numerator, \@divisor);
396
397       Synthetic division for polynomials. Divides the first list of
398       coefficients by the second list.
399
400       Returns references to the quotient and the remainder.
401
402       Remember to check for leading zeros (which are rightmost in the list)
403       in the returned values. For example,
404
405           my @n = (4, 12, 9, 3);
406           my @d = (1, 3, 3, 1);
407
408           my($q_ref, $r_ref) = pl_div(\@n, \@d);
409
410       After division you will have returned "(3)" as the quotient, and "(1,
411       3, 0)" as the remainder. In general, you will want to remove the
412       leading zero, or for that matter values within epsilon of zero, in the
413       remainder.
414
415           my($q_ref, $r_ref) = pl_div($f1, $f2);
416
417           #
418           # Remove any leading zeros (i.e., numbers smaller in
419           # magnitude than machine epsilon) in the remainder.
420           #
421           my @remd = @{$r_ref};
422           pop @remd while (@remd and abs($remd[$#remd]) < $epsilon);
423
424           $f1 = $f2;
425           $f2 = [@remd];
426
427       If $f1 and $f2 were to go through that bit of code again, not removing
428       the leading zeros would lead to a divide-by-zero error.
429
430       If either list of coefficients is empty, pl_div() returns undefs for
431       both quotient and remainder.
432
433       pl_mult()
434
435           $m_ref = pl_mult(\@coefficients1, \@coefficients2);
436
437       Returns the reference to the product of the two multiplicands.
438
439       pl_derivative()
440
441           $poly_ref = pl_derivative(\@coefficients);
442
443       Returns the derivative of a polynomial.
444
445       pl_antiderivative()
446
447           $poly_ref = pl_antiderivative(\@coefficients);
448
449       Returns the antiderivative of a polynomial. The constant value is
450       always set to zero and will need to be changed by the caller if a
451       different constant is needed.
452
453         my @coefficients = (1, 2, -3, 2);
454         my $integral = pl_antiderivative(\@coefficients);
455
456         #
457         # Integral needs to be 0 at x = 1.
458         #
459         my @coeff1 = @{$integral};
460         $coeff1[0] = - pl_evaluate($integral, 1);
461

AUTHOR

463       John M. Gamble, "<jgamble at cpan.org>"
464

BUGS

484       Please report any bugs or feature requests to "bug-math-util at
485       rt.cpan.org", or through the web interface at
486       <http://rt.cpan.org/NoAuth/ReportBug.html?Queue=Math-Utils>.  I will be
487       notified, and then you'll automatically be notified of progress on your
488       bug as I make changes.
489

SUPPORT

491       This module is on Github at <https://github.com/jgamble/Math-Utils>.
492
493       You can also look for information at:
494
495       •   RT: CPAN's request tracker (report bugs here)
496
497           <http://rt.cpan.org/NoAuth/Bugs.html?Dist=Math-Utils>
498
499       •   MetaCPAN
500
501           <https://metacpan.org/release/Math-Utils>
502

ACKNOWLEDGEMENTS

504       To J. A. R. Williams who got the ball rolling with Math::Fortran.
505

LICENSE AND COPYRIGHT

507       Copyright (c) 2017 John M. Gamble. All rights reserved. This program is
508       free software; you can redistribute it and/or modify it under the same
509       terms as Perl itself.
510
511
512
513perl v5.34.0                      2021-07-22                    Math::Utils(3)