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 Tay‐
7       lor 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
37       None by default.
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39       You may choose to have any of the following routines exported to the
40       calling namespace. ':all' tag exports all of the following:
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42         TaylorPolynomial
43         TaylorErrorLagrange
44         TaylorErrorCauchy
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SUBROUTINES

47       TaylorPolynomial
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49       This function (symbolically) computes the nth-degree Taylor Polynomial
50       of a given function. Generally speaking, the Taylor Polynomial is an
51       n-th degree polynomial that approximates the original function. It does
52       so particularily well in the proximity of a certain point x0.  (Since
53       my mathematical English jargon is lacking, I strongly suggest you read
54       up on what this is in a book.)
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56       Mathematically speaking, the Taylor Polynomial of the function f(x)
57       looks like this:
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59         Tn(f, x, x0) =
60           sum_from_k=0_to_n(
61               n-th_total_derivative(f)(x0) / k! * (x-x0)^k
62           )
63
64       First argument to the subroutine must be the function to approximate.
65       It may be given either as a string to be parsed or as a valid
66       Math::Symbolic tree.  Second argument must be an integer indicating to
67       which degree to approximate.  The third argument is the last required
68       argument and denotes the variable to use for approximation either as a
69       string (name) or as a Math::Symbolic::Variable object. That's the 'x'
70       above.  The fourth argument is optional and specifies the name of the
71       variable to introduce as the point of approximation. May also be a
72       variable object.  It's the 'x0' above. If not specified, the name of
73       this variable will be assumed to be the name of the function variable
74       (the 'x') with '_0' appended.
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76       This routine is for functions of one variable only. There is an equiva‐
77       lent for functions of two variables in the Math::Symbolic::VectorCalcu‐
78       lus package.
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80       TaylorErrorLagrange
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82       TaylorErrorLagrange computes and returns the formula for the Taylor
83       Polynomial's approximation error after Lagrange. (Again, my English
84       terminology is lacking.) It looks similar to this:
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86         Rn(f, x, x0) =
87           n+1-th_total_derivative(f)( x0 + theta * (x-x0) ) / (n+1)! * (x-x0)^(n+1)
88
89       Please refer to your favourite book on the topic. 'theta' may be any
90       number between 0 and 1.
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92       The calling conventions for TaylorErrorLagrange are similar to those of
93       TaylorPolynomial, but TaylorErrorLagrange takes an extra optional argu‐
94       ment specifying the name of 'theta'. If it isn't specified explicitly,
95       the variable will be named 'theta' as in the formula above.
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97       TaylorErrorCauchy
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99       TaylorErrorCauchy computes and returns the formula for the Taylor Poly‐
100       nomial's approximation error after (guess who!) Cauchy.  (Again, my
101       English terminology is lacking.) It looks similar to this:
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103         Rn(f, x, x0) = TaylorErrorLagrange(...) * (1 - theta)^n
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105       Please refer to your favourite book on the topic and the documentation
106       for TaylorErrorLagrange. 'theta' may be any number between 0 and 1.
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108       The calling conventions for TaylorErrorCauchy are identical to those of
109       TaylorErrorLagrange.
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AUTHOR

112       Please send feedback, bug reports, and support requests to the
113       Math::Symbolic support mailing list: math-symbolic-support at lists dot
114       sourceforge dot net. Please consider letting us know how you use
115       Math::Symbolic. Thank you.
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117       If you're interested in helping with the development or extending the
118       module's functionality, please contact the developers' mailing list:
119       math-symbolic-develop at lists dot sourceforge dot net.
120
121       List of contributors:
122
123         Steffen Müller, symbolic-module at steffen-mueller dot net
124         Stray Toaster, mwk at users dot sourceforge dot net
125         Oliver Ebenhöh
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NAME

SYNOPSIS

DESCRIPTION

SUBROUTINES

AUTHOR

SEE ALSO