Gauss-Jacobi Mechanical Quadrature

Let "x"₁ < "x"₂ < ... < "x"_n are the zeros of polynomial "p"_n("x")of degree "n". Then there exist real numbers λ₁, λ₂, ... λ_n such that

$int\_a^b\; f(x)\; omega(x)\; ,\; dx\; =\; lambda\_1\; f(x\_1)\; +\; lambda\_2\; f(x\_2)\; +\; ...\; +\; lambda\_n\; f(x\_n)$

for any function "f"("x") that is an arbitrary polynomial of degree 2"n" - 1. Function ω("x") (called the "weight function") is positive on interval ("a", "b"). Numbers λ₁, λ₂, ... λ_n are called Christoffel numbers. The weight function ω("x") and the integer "n" uniquely determine Christoffel numbers. [Weisstein, Eric W."Gauss-Jacobi Mechanical Quadrature" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Gauss-JacobiMechanicalQuadrature.html]

References

* Weisstein, Eric W. "Gauss-Jacobi Mechanical Quadrature." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Gauss-JacobiMechanicalQuadrature.html