Assumptions:
References:
- L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831
TeX:
\left|\int_{a}^{b} f(t) \, dt - \frac{b - a}{2} \sum_{k=1}^{n} w_{n,k} f\!\left(\frac{b - a}{2} x_{n,k} + \frac{a + b}{2}\right)\right| \le \frac{\left|b - a\right|}{2} \frac{64 M}{15 \left(1 - {\rho}^{-2}\right) {\rho}^{2 n}}\; \text{ where } M = \mathop{\operatorname{sup}}\limits_{t \in \mathcal{E}_{\rho}} \left|f\!\left(\frac{b - a}{2} t + \frac{a + b}{2}\right)\right| a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \rho \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \rho > 1 \;\mathbin{\operatorname{and}}\; f(z) \text{ is holomorphic on } z \in \operatorname{InteriorClosure}\!\left(\mathcal{E}_{\rho}\right)
Definitions:
Fungrim symbol | Notation | Short description |
---|---|---|
Abs | Absolute value | |
Integral | Integral | |
Sum | Sum | |
GaussLegendreWeight | Gauss-Legendre quadrature weight | |
LegendrePolynomialZero | Legendre polynomial zero | |
Pow | Power | |
Supremum | Supremum of a set or function | |
BernsteinEllipse | Bernstein ellipse with foci -1,+1 and semi-axis sum rho | |
CC | Complex numbers | |
ZZGreaterEqual | Integers greater than or equal to n | |
RR | Real numbers | |
IsHolomorphic | Holomorphic predicate |
Source code for this entry:
Entry(ID("545987"), Formula(Where(LessEqual(Abs(Sub(Integral(f(t), For(t, a, b)), Mul(Div(Sub(b, a), 2), Sum(Mul(GaussLegendreWeight(n, k), f(Add(Mul(Div(Sub(b, a), 2), LegendrePolynomialZero(n, k)), Div(Add(a, b), 2)))), For(k, 1, n))))), Mul(Div(Abs(Sub(b, a)), 2), Div(Mul(64, M), Mul(Mul(15, Sub(1, Pow(rho, -2))), Pow(rho, Mul(2, n)))))), Equal(M, Supremum(Abs(f(Add(Mul(Div(Sub(b, a), 2), t), Div(Add(a, b), 2)))), ForElement(t, BernsteinEllipse(rho)))))), Variables(f, a, b, n, rho), Assumptions(And(Element(a, CC), Element(b, CC), Element(n, ZZGreaterEqual(1)), Element(rho, RR), Greater(rho, 1), IsHolomorphic(f(z), ForElement(z, Subset(InteriorClosure(BernsteinEllipse(rho))))))), References("L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831"))