Gauss-Hermite Quadrature Calculator

Enter an integrand f(x) and order n to numerically evaluate ∫[-∞,∞] f(x)e^(-x²)dx by Gauss-Hermite quadrature, using the roots of the Hermite polynomial as nodes with standard weights.

Input

Enter the integrand f(x) and the order n to evaluate ∫[-∞,∞] f(x)e^(-x²)dx by Gauss-Hermite quadrature. The weight e^(-x²) is built into the rule, so do not include e^(-x²) in f(x).

Integrand f(x)

e.g. cos(x), x^2, 1/(1+x^2). The weight e^(-x²) is applied automatically. See the notes for available functions and constants.

Order n

Number of nodes (1-128)

Result

Integral ∫ f(x) e^(-x²) dx

1.380388447

Interval (-∞, ∞), weight function e^(-x²)

Order n

Nodes

Nodes, weights, and function values

Roots x_i of the Hermite polynomial H_n, weights w_i, and function values f(x_i).

#	Node x_i	Weight w_i	f(x_i)
1	-0.27348105	0.50792948	0.96283655
2	-0.82295145	0.28064746	0.6800603
3	-1.38025854	0.08381004	0.18938698
4	-1.95178799	0.01288031	-0.37184121
5	-2.54620216	0.00093228	-0.82792956
6	-3.17699916	2.711860e-5	-0.99937326
7	-3.8694479	2.320981e-7	-0.74660296
8	-4.68873894	2.654807e-10	-0.02364784
9	4.68873894	2.654807e-10	-0.02364784
10	3.8694479	2.320981e-7	-0.74660296
11	3.17699916	2.711860e-5	-0.99937326
12	2.54620216	0.00093228	-0.82792956
13	1.95178799	0.01288031	-0.37184121
14	1.38025854	0.08381004	0.18938698
15	0.82295145	0.28064746	0.6800603
16	0.27348105	0.50792948	0.96283655

How it works

Gauss-Hermite quadrature approximates the weighted infinite-interval integral ∫[-∞,∞] f(x)e^(-x²)dx by the finite sum Σ w_i f(x_i). The weight function e^(-x²) is built into the rule, so do not include e^(-x²) in your f(x) (e.g. to evaluate ∫ cos(x)e^(-x²)dx, enter f(x)=cos(x)).
The nodes x_i are the n roots of the (physicists') Hermite polynomial H_n(x) of degree n. H_n is evaluated via the recurrence H_{k+1}(x)=2x H_k(x)-2k H_{k-1}(x) (with H_0=1, H_1=2x) and its roots are found by Newton's method. Since H_n is even or odd, the nodes are symmetric about the origin.
Each weight is computed as w_i = 2^(n-1) n! √π / ( n² H_{n-1}(x_i)² ). To avoid n! overflow for large n, this tool evaluates the weights in the logarithmic domain.
The rule integrates polynomials of degree up to 2n-1 exactly (algebraic degree 2n-1). For smooth integrands with at most exponential growth, increasing the order n improves accuracy.
f(x) supports power ^, the four arithmetic operations, the functions sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, log (=ln), log10, sqrt, cbrt, abs, and the constants pi and e. Only x may be used as a variable. Expressions are parsed by a custom recursive-descent parser, never eval.
Order n must be an integer from 1 to 128. Note this differs from the probabilists' variant, which uses the weight e^(-x²/2).

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