Function Repository Resource:

PaduaIntegrate

Numerically integrate a function using the Padua points

Contributed by: Jan Mangaldan

ResourceFunction["PaduaIntegrate"][f,{x,x_min,x_max},{y,y_min,y_max}]

gives a numerical approximation to the multiple integral by evaluating f at the Padua points.

Details and Options

The Padua points are a unisolvent point set that can be used to interpolate and integrate a bivariate function over a rectangular domain. ResourceFunction["PaduaIntegrate"] evaluates f at the Padua points, and integrates the interpolating polynomial passing through them.

ResourceFunction["PaduaIntegrate"] takes the following options:

InterpolationOrder

order of the interpolating polynomial generated

"PaduaType"

type of Padua points to use

WorkingPrecision

MachinePrecision

the precision used in internal computations

Possible types of Padua points are 1, 2, 3 and 4.

Examples

Basic Examples (2)

Numerically integrate the function Exp[-x²-y²]:

In[1]:=

$ResourceFunction["PaduaIntegrate"][ Exp[-x^2 - y^2], {x, -1, 1}, {y, -1, 1}]$

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Compare with the exact answer:

In[2]:=

$% - \[Pi] Erf[1]^2$

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Scope (3)

A test function due to Franke:

In[3]:=

franke[x_, y_] := 3/4 Exp[-(((9 x - 2)^2 + (9 y - 2)^2)/4)] + 3/4 Exp[-((9 x + 1)^2/49) - (9 y + 1)/10] + 1/2 Exp[-(((9 x - 7)^2 + (9 y - 3)^2)/4)] - 1/5 Exp[-(9 x - 4)^2 - (9 y - 7)^2]

Integrate the function over a rectangular domain:

In[4]:=

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Compare with the result of NIntegrate:

In[5]:=

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Options (3)

InterpolationOrder (1)

Use a degree 25 interpolant for integrating the Dixon–Szegö function:

In[6]:=

$ResourceFunction[ "PaduaIntegrate"][(4 - 2.1 x^2 + x^4/3) x^2 + x y + 4 (y^2 - 1) y^2, {x, -2, 2}, {y, -1.25, 1.25}, InterpolationOrder -> 25]$

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PaduaType (1)

Use type-3 Padua points in integrating the Dixon–Szegö function:

In[7]:=

$ResourceFunction[ "PaduaIntegrate"][(4 - 2.1 x^2 + x^4/3) x^2 + x y + 4 (y^2 - 1) y^2, {x, -2, 2}, {y, -1.25, 1.25}, "PaduaType" -> 3]$

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WorkingPrecision (1)

Use 25-digit precision for the numerical integration:

In[8]:=

$ResourceFunction[ "PaduaIntegrate"][(4 - 21/10 x^2 + x^4/3) x^2 + x y + 4 (y^2 - 1) y^2, {x, -2, 2}, {y, -(5/4), 5/4}, WorkingPrecision -> 25]$

Out[8]=

Properties and Relations (2)

If the input function is a polynomial of degree k, PaduaIntegrate gives the exact answer provided k is less than or equal to the setting for InterpolationOrder:

In[9]:=

$ResourceFunction[ "PaduaIntegrate"][-(x^2 y - x - 1)^2 - (x^2 - 1)^2, {x, -3, 3}, {y, -3, 3}, InterpolationOrder -> 6]$

Out[9]=

In[10]:=

$% - Integrate[-(x^2 y - x - 1)^2 - (x^2 - 1)^2, {x, -3, 3}, {y, -3, 3}]$

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A test function due to Franke:

In[11]:=

Integrate the function by integrating the interpolant obtained from the resource function PaduaInterpolation:

In[12]:=

pin = ResourceFunction["PaduaInterpolation"][ franke[x, y], {x, 0, 1}, {y, 0, 1}, InterpolationOrder -> 19, WorkingPrecision -> 30]; Integrate[Integrate[pin[x, y], {x, 0, 1}], {y, 0, 1}]

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The result of directly using PaduaIntegrate is more accurate:

In[14]:=

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Version History

1.0.0 – 22 April 2021

Source Metadata

Citation:
- Bos L., Caliari M., De Marchi S., Vianello M., Xu Y., "Bivariate Lagrange interpolation at the Padua points: the generating curve approach." Journal of Approximation Theory, vol. 143, no. 1, 15–25, 2006. DOI: 10.1016/j.jat.2006.03.008
- Caliari M., De Marchi S., Vianello M., "Bivariate polynomial interpolation on the square at new nodal sets." Applied Mathematics and Computation, vol. 165, no. 2, 261–274, 2005. DOI: 10.1016/j.amc.2004.07.001
- Caliari M., De Marchi S., Vianello M., "Bivariate Lagrange interpolation at the Padua points: Computational aspects." Journal of Computational and Applied Mathematics, vol. 221, no. 2, 284–292, 2008. DOI: 10.1016/j.cam.2007.10.027

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License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

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PaduaIntegrate

Details and Options