Gauss–Legendre Quadrature

An n-point Gauss–Legendre rule chooses nodes and weights to integrate polynomials through degree 2n−1 exactly. The compact relation ∫₋₁¹ f(x)dx ≈ Σ w_if(x_i) shows how the defined quantities interact; Nodes and interval are dimensionless on [−1,1]; weights are dimensionless; the integral has the integrand's unit on this normalized interval. Interpretation depends on the model context: Approximate integrals with high polynomial exactness using few function evaluations. Also inspect whether a quantity enters linearly, quadratically, inversely or through another function. This lets you predict how the result changes before calculating. The equation is therefore both a calculation rule and a statement about the model's structure.

Assign each given value to a symbol and convert units consistently first. The two-point rule has nodes ±1/√3 and weights 1. For f(x)=x² it gives 1/3+1/3=2/3. Check: ∫₋₁¹x²dx=[x³/3]₋₁¹=2/3; degree 2 is within the exactness limit 3. Then verify the result independently: The n-node Gauss–Legendre rule is exact for polynomials through degree 2n−1 and approximate for general functions Compare the final unit and order of magnitude with the context, and round only after completing the calculation. This check can reveal not only arithmetic mistakes but also a wrong model or a missed factor. When several quantities enter the relation, vary one mentally and ask whether the result follows the expected trend.

The key step is: The n-node Gauss–Legendre rule is exact for polynomials through degree 2n−1 and approximate for general functions. This leads to ∫₋₁¹ f(x)dx ≈ Σ w_if(x_i). The derivation separates definitions, subject-specific assumptions and the final algebraic step; correct algebra cannot replace a missing condition. The two-point rule has nodes ±1/√3 and weights 1. Nodes are roots of the nth Legendre polynomial, with matching weights Approximate integrals with high polynomial exactness using few function evaluations. Reconstruct the central idea before memorizing the compact form. That makes it easier to recognize when a similar-looking formula does not apply.

The n-node Gauss–Legendre rule is exact for polynomials through degree 2n−1 and approximate for general functions. Nodes are roots of the nth Legendre polynomial, with matching weights. To use [a,b], map x=(a+b)/2+(b−a)t/2 and multiply the integral by (b−a)/2. Singularities or behavior poorly represented by polynomials may require splitting the interval or choosing another method. Also check Nodes and interval are dimensionless on [−1,1]; weights are dimensionless; the integral has the integrand's unit on this normalized interval. Violating the validity range can produce a precise but substantively wrong result. Verify that assumptions about geometry, smoothness, independence, symmetry or distribution actually hold in the problem. If a condition is not met, label the result as an approximation or choose a more general method. Extra decimal places cannot repair a model mismatch. This check matters especially when a neighbouring formula uses different assumptions.

The two relations are neighbours in the subject, but answer different questions. Gauss–Legendre Quadrature: An n-point Gauss–Legendre rule chooses nodes and weights to integrate polynomials through degree 2n−1 exactly. Composite Simpson's Rule: Composite Simpson's rule approximates an integral on a uniform grid by piecewise quadratic parabolas; the number n of subintervals must be even. Choose based on the unknown and on which assumptions the data satisfy. Approximate integrals with high polynomial exactness using few function evaluations. As a check, use the example The two-point rule has nodes ±1/√3 and weights 1 and identify which extra quantity the companion relation would need. This clarifies when one equation is enough and when both belong in a solution.