Function Repository Resource:

# RadicalInverse

Compute the radical inverse of an integer to a given base

Contributed by: Jan Mangaldan
 ResourceFunction["RadicalInverse"][n] gives the base 10 radical inverse of the integer n. ResourceFunction["RadicalInverse"][b,n] gives the base-b radical inverse of the integer n.

## Details

The radical inverse is also known as the van der Corput sequence.
Integer mathematical function, suitable for both symbolic and numerical manipulation.
The base-b radical inverse of n is defined as , where is the base-b expansion of n, and m is IntegerLength[n,b].
The radical inverse is usually used for computing Halton and Hammersley low-discrepancy sequences for quasi-Monte Carlo methods.

## Examples

### Basic Examples (3)

The base-10 radical inverse of 42:

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The base-2 radical inverse of 42:

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Plot the binary radical inverse:

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### Scope (2)

Evaluate for large arguments:

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RadicalInverse automatically threads over lists:

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### Applications (4)

Demonstrate the filling of the unit interval with the decimal radical inverse, also known as the van der Corput sequence:

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Generate a 2D Halton sequence with bases 2 and 3:

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Generate a 2D binary Hammersley sequence:

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Compare the Halton and Hammersley sequences for approximating π by quasi-Monte Carlo integration:

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## Version History

• 1.0.0 – 07 June 2021