Numerical integration approximates definite integrals using finitely many arithmetic operations and function evaluations.
An -node quadrature rule already exact through degree is exact through degree exactly when its nodal polynomialis orthogonal to every polynomial of degree at most .
No -node quadrature rule for a positive interval weight can be exact through degree , because it evaluates the nonnegative nodal square as zero although its integral is positive.
If an -node quadrature rule for a positive interval weight is exact through degree , every weight is positive. Indeed, applying the rule to the square of the corresponding degree- Lagrange cardinal polynomial isolates that weight.
If each -node quadrature rule is exact through degree and has positive weights, then it converges on every continuous function. The Weierstrass approximation theorem reduces the error to the uniform polynomial-approximation error, while exactness on constants controls the sum of the weights.
A quadrature rule is exact through degree precisely when its node weights reproduce the first moments of the integration measure.
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Numerical integration is a computational technique used to estimate the value of a definite integral when an analytical solution is difficult or impossible to obtain. It involves approximating the area under a curve defined by a mathematical function over a specified interval. This is particularly useful for functions that are complex, have no closed-form antiderivative, or are only known through discrete data points. There are various methods of numerical integration, each with its own advantages and limitations.