Function Repository Resource:

# ThieleExpansion

Expand a function into a Thiele continued fraction

Contributed by: Jan Mangaldan
 ResourceFunction["ThieleExpansion"][f,{x,x0,n}] generates a Thiele continued fraction expansion for f about the point x=x0 up to the nth order convergent, where n is an explicit integer.

## Details and Options

ResourceFunction["ThieleExpansion"] can find the Thiele continued fraction expansion about the point x=x0 only when it can evaluate power series at that point.
ResourceFunction["ThieleExpansion"] takes the following option:
 Method Automatic method to use
Possible settings for Method include "Viscovatov" and "WynnOmega", with the default being Automatic.

## Examples

### Basic Examples (2)

Compute the Thiele continued fraction for the exponential function around x=0:

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Thiele continued fraction of an arbitrary function around x=a:

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### Options (2)

#### Method (2)

Use different methods for generating the Thiele expansion:

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The two methods will give results that are mathematically equivalent but may differ in form:

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### Applications (1)

Plot successive Thiele continued fraction approximations to exp(x):

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### Properties and Relations (1)

The Thiele expansion of order n is equivalent to the result of PadeApproximant of order {Ceiling[n/2],Floor[n/2]}:

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### Possible Issues (2)

Thiele expansion is not defined for even or odd functions:

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Transform the variable to get a continued fraction expansion:

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## Version History

• 1.0.0 – 13 January 2021