# triangle_fekete_rule

triangle_fekete_rule, a MATLAB code which can return information defining any of seven Fekete rules for high order interpolation and quadrature over the interior of a triangle in 2D.

Fekete points can be defined for any region OMEGA. To define the Fekete points for a given region, let Poly(N) be some finite dimensional vector space of polynomials, such as all polynomials of degree less than L, or all polynomials whose monomial terms have total degree less than some value L.

Let P(1:M) be any basis for Poly(N). For this basis, the Fekete points are defined as those points Z(1:M) which maximize the determinant of the corresponding Vandermonde matrix:

```        V = [ P1(Z1)  P1(Z2)  ... P1(ZM) ]
[ P2(Z1)  P2(Z2)  ... P2(ZM) ]
...
[ PM(ZM)  P2(ZM)  ... PM(ZM) ]
```

The seven rules have the following orders and precisions:
RuleOrderPrecision
1 10 3
2 28 6
3 55 9
4 9112
5 9112
613615
719018

On the triangle, it is known that some Fekete points will lie on the boundary, and that on each side of the triangle, these points will correspond to a set of Gauss-Lobatto points.

### Languages:

triangle_fekete_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

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triangle_nco_rule, a MATLAB code which defines newton-cotes open quadrature rules over the interior of a triangle in 2d.

triangle_symq_rule, a MATLAB code which returns efficient symmetric quadrature rules, with exactness up to total degree 50, over the interior of an arbitrary triangle in 2d, by hong xiao and zydrunas gimbutas.

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### Reference:

1. SF Bockman,
Generalizing the Formula for Areas of Polygons to Moments,
American Mathematical Society Monthly,
Volume 96, Number 2, February 1989, pages 131-132.
2. Hermann Engels,
ISBN: 012238850X,
LC: QA299.3E5.
3. Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
4. Mark Taylor, Beth Wingate, Rachel Vincent,
An Algorithm for Computing Fekete Points in the Triangle,
SIAM Journal on Numerical Analysis,
Volume 38, Number 5, 2000, pages 1707-1720.
5. Stephen Wandzura, Hong Xiao,
Symmetric Quadrature Rules on a Triangle,
Computers and Mathematics with Applications,
Volume 45, 2003, pages 1829-1840.

### Source Code:

Last revised on 05 April 2019.