Math::Symbolic::MiscCalculus(3pm)

1Math::Symbolic::MiscCalUcsuelrusC(o3n)tributed Perl DocuMmaetnht:a:tSiyomnbolic::MiscCalculus(3)
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NAME

6       Math::Symbolic::MiscCalculus - Miscellaneous calculus routines (eg
7       Taylor poly)
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SYNOPSIS

10         use Math::Symbolic qw/:all/;
11         use Math::Symbolic::MiscCalculus qw/:all/; # not loaded by Math::Symbolic
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13         $taylor_poly = TaylorPolynomial $function, $degree, $variable;
14         # or:
15         $taylor_poly = TaylorPolynomial $function, $degree, $variable, $pos;
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17         $lagrange_error = TaylorErrorLagrange $function, $degree, $variable;
18         # or:
19         $lagrange_error = TaylorErrorLagrange $function, $degree, $variable, $pos;
20         # or:
21         $lagrange_error = TaylorErrorLagrange $function, $degree, $variable, $pos,
22                                               $name_for_range_variable;
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24         # This has the same syntax variations as the Lagrange error:
25         $cauchy_error = TaylorErrorLagrange $function, $degree, $variable;
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DESCRIPTION

28       This module provides several subroutines related to calculus such as
29       computing Taylor polynomials and errors the associated errors from
30       Math::Symbolic trees.
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32       Please note that the code herein may or may not be refactored into the
33       OO-interface of the Math::Symbolic module in the future.
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35   EXPORT
36       None by default.
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38       You may choose to have any of the following routines exported to the
39       calling namespace. ':all' tag exports all of the following:
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41         TaylorPolynomial
42         TaylorErrorLagrange
43         TaylorErrorCauchy
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SUBROUTINES

46   TaylorPolynomial
47       This function (symbolically) computes the nth-degree Taylor Polynomial
48       of a given function. Generally speaking, the Taylor Polynomial is an
49       n-th degree polynomial that approximates the original function. It does
50       so particularly well in the proximity of a certain point x0.  (Since my
51       mathematical English jargon is lacking, I strongly suggest you read up
52       on what this is in a book.)
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54       Mathematically speaking, the Taylor Polynomial of the function f(x)
55       looks like this:
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57         Tn(f, x, x0) =
58           sum_from_k=0_to_n(
59               n-th_total_derivative(f)(x0) / k! * (x-x0)^k
60           )
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62       First argument to the subroutine must be the function to approximate.
63       It may be given either as a string to be parsed or as a valid
64       Math::Symbolic tree.  Second argument must be an integer indicating to
65       which degree to approximate.  The third argument is the last required
66       argument and denotes the variable to use for approximation either as a
67       string (name) or as a Math::Symbolic::Variable object. That's the 'x'
68       above.  The fourth argument is optional and specifies the name of the
69       variable to introduce as the point of approximation. May also be a
70       variable object.  It's the 'x0' above. If not specified, the name of
71       this variable will be assumed to be the name of the function variable
72       (the 'x') with '_0' appended.
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74       This routine is for functions of one variable only. There is an
75       equivalent for functions of two variables in the
76       Math::Symbolic::VectorCalculus package.
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78   TaylorErrorLagrange
79       TaylorErrorLagrange computes and returns the formula for the Taylor
80       Polynomial's approximation error after Lagrange. (Again, my English
81       terminology is lacking.) It looks similar to this:
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83         Rn(f, x, x0) =
84           n+1-th_total_derivative(f)( x0 + theta * (x-x0) ) / (n+1)! * (x-x0)^(n+1)
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86       Please refer to your favourite book on the topic. 'theta' may be any
87       number between 0 and 1.
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89       The calling conventions for TaylorErrorLagrange are similar to those of
90       TaylorPolynomial, but TaylorErrorLagrange takes an extra optional
91       argument specifying the name of 'theta'. If it isn't specified
92       explicitly, the variable will be named 'theta' as in the formula above.
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94   TaylorErrorCauchy
95       TaylorErrorCauchy computes and returns the formula for the Taylor
96       Polynomial's approximation error after (guess who!) Cauchy.  (Again, my
97       English terminology is lacking.) It looks similar to this:
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99         Rn(f, x, x0) = TaylorErrorLagrange(...) * (1 - theta)^n
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101       Please refer to your favourite book on the topic and the documentation
102       for TaylorErrorLagrange. 'theta' may be any number between 0 and 1.
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104       The calling conventions for TaylorErrorCauchy are identical to those of
105       TaylorErrorLagrange.
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AUTHOR

108       Please send feedback, bug reports, and support requests to the
109       Math::Symbolic support mailing list: math-symbolic-support at lists dot
110       sourceforge dot net. Please consider letting us know how you use
111       Math::Symbolic. Thank you.
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113       If you're interested in helping with the development or extending the
114       module's functionality, please contact the developers' mailing list:
115       math-symbolic-develop at lists dot sourceforge dot net.
116
117       List of contributors:
118
119         Steffen Müller, symbolic-module at steffen-mueller dot net
120         Stray Toaster, mwk at users dot sourceforge dot net
121         Oliver Ebenhöh
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NAME

SYNOPSIS

DESCRIPTION

SUBROUTINES

AUTHOR

SEE ALSO