Function Repository Resource:

Get a numerically sorted list of abscissas for Chebyshev equal-weight quadrature

Contributed by: Jan Mangaldan
 ResourceFunction["ChebyshevQuadratureAbscissas"][n,{a,b}] gives a list of the n abscissas xi of the n-point Chebyshev equal-weight formula for quadrature on the interval a to b. ResourceFunction["ChebyshevQuadratureAbscissas"][n,{a,b},prec] uses the working precision prec.

## Details

Chebyshev equal-weight quadrature approximates the value of an integral as a linear combination of values of the integrand evaluated at optimal abscissas xi: .
In Chebyshev quadrature, the abscissas are chosen such that the equation above is exact for all polynomials up to degree n if n is odd, and up to degree n+1 if n is even.

## Examples

### Basic Examples (2)

The abscissas for a 9-point Chebyshev quadrature on a given interval:

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Use the specified precision:

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### Applications (4)

Use Chebyshev quadrature to approximate the area under a curve:

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Plot the curve over a given interval:

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Approximate the area under the curve using n-point Chebyshev quadrature:

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Compare to the output of NIntegrate:

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### Properties and Relations (2)

Chebyshev abscissas are purely real-valued only for n7 and n=9:

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For other values of n, complex-valued Chebyshev abscissas appear:

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### Neat Examples (1)

Plot the 100-point Chebyshev quadrature abscissas in the complex plane:

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## Version History

• 1.0.0 – 26 January 2021