Gauss-Kronrod Quadrature (G7-K15) Calculator

Enter an integrand f(x) and an interval [a, b] to evaluate the definite integral with the 15-point Kronrod rule and estimate the error from the 7-point Gauss value.

Input

Enter the integrand f(x) and the interval [a, b] to evaluate the definite integral with Gauss-Kronrod (G7-K15) quadrature and estimate the error from the Gauss value.

Integrand f(x)

e.g. exp(-x^2), sin(x)/x, x^2 + 1. See the method section for supported functions, constants and operators.

Lower limit a

Upper limit b

Result

Kronrod approximation K15 ∫ f(x) dx

0.7468241328

Interval [0, 1]

Gauss value G7

0.7468241328

Estimated error

1.981138e-15

Nodes, weights and values

Shows the nodes x_i (mapped to [a, b]), Kronrod weights w_i, function values f(x_i) and the node type.

#	Node x_i	Weight w_i	f(x_i)	Type
1	0.00427231	0.01146766	0.99998175	Kronrod
2	0.02544604	0.03154605	0.99935271	Gauss
3	0.06756779	0.05239501	0.995445	Kronrod
4	0.12923441	0.07032663	0.98343717	Gauss
5	0.20695638	0.08450236	0.95807334	Kronrod
6	0.29707742	0.09517529	0.91552739	Gauss
7	0.39610752	0.10221647	0.85478853	Kronrod
8	0.5	0.10474107	0.77880078	Gauss
9	0.60389248	0.10221647	0.69441458	Kronrod
10	0.70292258	0.09517529	0.61011968	Gauss
11	0.79304362	0.08450236	0.53316828	Kronrod
12	0.87076559	0.07032663	0.46849366	Gauss
13	0.93243221	0.05239501	0.41919049	Kronrod
14	0.97455396	0.03154605	0.38683563	Gauss
15	0.99572769	0.01146766	0.37102953	Kronrod

How it works

The 15-point Kronrod rule (K15) is the main approximation; the error is estimated from the difference with the 7-point Gauss rule (G7), whose nodes are a subset of the Kronrod nodes.
Standard-interval nodes and weights are mapped to [a, b] via the linear transform x ＝ (b − a)/2 t ＋ (a ＋ b)/2.
The estimated error comes from the absolute difference between K15 and G7, sharpened in the QUADPACK style. Smaller values indicate higher confidence.
The integrand is evaluated with a recursive-descent parser supporting functions such as sin, cos, exp, log and sqrt, the constants pi and e, the four basic operations, the power operator ＾ and implicit multiplication (e.g. 2x).
If a singularity lies inside the interval and a node value is not finite, the calculation fails; split the interval and try again.

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