Math::Symbolic::MiscCalculus(3) Miscellaneous calculus routines (eg Taylor poly)

## SYNOPSIS

use Math::Symbolic qw/:all/;
use Math::Symbolic::MiscCalculus qw/:all/; # not loaded by Math::Symbolic

\$taylor_poly = TaylorPolynomial \$function, \$degree, \$variable;
# or:
\$taylor_poly = TaylorPolynomial \$function, \$degree, \$variable, \$pos;

\$lagrange_error = TaylorErrorLagrange \$function, \$degree, \$variable;
# or:
\$lagrange_error = TaylorErrorLagrange \$function, \$degree, \$variable, \$pos;
# or:
\$lagrange_error = TaylorErrorLagrange \$function, \$degree, \$variable, \$pos,
\$name_for_range_variable;

# This has the same syntax variations as the Lagrange error:
\$cauchy_error = TaylorErrorLagrange \$function, \$degree, \$variable;

## DESCRIPTION

This module provides several subroutines related to calculus such as computing Taylor polynomials and errors the associated errors from Math::Symbolic trees.

Please note that the code herein may or may not be refactored into the OO-interface of the Math::Symbolic module in the future.

## EXPORT

None by default.

You may choose to have any of the following routines exported to the calling namespace. ':all' tag exports all of the following:

```  TaylorPolynomial
TaylorErrorLagrange
TaylorErrorCauchy
```

## TaylorPolynomial

This function (symbolically) computes the nth-degree Taylor Polynomial of a given function. Generally speaking, the Taylor Polynomial is an n-th degree polynomial that approximates the original function. It does so particularly well in the proximity of a certain point x0. (Since my mathematical English jargon is lacking, I strongly suggest you read up on what this is in a book.)

Mathematically speaking, the Taylor Polynomial of the function f(x) looks like this:

```  Tn(f, x, x0) =
sum_from_k=0_to_n(
n-th_total_derivative(f)(x0) / k! * (x-x0)^k
)
```

First argument to the subroutine must be the function to approximate. It may be given either as a string to be parsed or as a valid Math::Symbolic tree. Second argument must be an integer indicating to which degree to approximate. The third argument is the last required argument and denotes the variable to use for approximation either as a string (name) or as a Math::Symbolic::Variable object. That's the 'x' above. The fourth argument is optional and specifies the name of the variable to introduce as the point of approximation. May also be a variable object. It's the 'x0' above. If not specified, the name of this variable will be assumed to be the name of the function variable (the 'x') with '_0' appended.

This routine is for functions of one variable only. There is an equivalent for functions of two variables in the Math::Symbolic::VectorCalculus package.

## TaylorErrorLagrange

TaylorErrorLagrange computes and returns the formula for the Taylor Polynomial's approximation error after Lagrange. (Again, my English terminology is lacking.) It looks similar to this:

```  Rn(f, x, x0) =
n+1-th_total_derivative(f)( x0 + theta * (x-x0) ) / (n+1)! * (x-x0)^(n+1)
```

Please refer to your favourite book on the topic. 'theta' may be any number between 0 and 1.

The calling conventions for TaylorErrorLagrange are similar to those of TaylorPolynomial, but TaylorErrorLagrange takes an extra optional argument specifying the name of 'theta'. If it isn't specified explicitly, the variable will be named 'theta' as in the formula above.

## TaylorErrorCauchy

TaylorErrorCauchy computes and returns the formula for the Taylor Polynomial's approximation error after (guess who!) Cauchy. (Again, my English terminology is lacking.) It looks similar to this:

```  Rn(f, x, x0) = TaylorErrorLagrange(...) * (1 - theta)^n
```

Please refer to your favourite book on the topic and the documentation for TaylorErrorLagrange. 'theta' may be any number between 0 and 1.

The calling conventions for TaylorErrorCauchy are identical to those of TaylorErrorLagrange.

## AUTHOR

Please send feedback, bug reports, and support requests to the Math::Symbolic support mailing list: math-symbolic-support at lists dot sourceforge dot net. Please consider letting us know how you use Math::Symbolic. Thank you.

If you're interested in helping with the development or extending the module's functionality, please contact the developers' mailing list: math-symbolic-develop at lists dot sourceforge dot net.

List of contributors:

```  Steffen MXller, symbolic-module at steffen-mueller dot net
Stray Toaster, mwk at users dot sourceforge dot net
Oliver EbenhXh
```