Fungrim entry: 47b181

\left|\int_{-1}^{1} f\!\left(t\right) \, dt - \sum_{k=1}^{n} w_{n,k} f\!\left(x_{n,k}\right)\right| \le \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(t\right)\right|

Assumptions:

n \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \rho \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \rho \gt 1 \,\mathbin{\operatorname{and}}\, \operatorname{InteriorClosure}\!\left(\mathcal{E}_{\rho}\right) \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(z\right), z, \mathbb{C}\right)

References:

L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831

TeX:

\left|\int_{-1}^{1} f\!\left(t\right) \, dt - \sum_{k=1}^{n} w_{n,k} f\!\left(x_{n,k}\right)\right| \le \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(t\right)\right|

n \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \rho \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \rho \gt 1 \,\mathbin{\operatorname{and}}\, \operatorname{InteriorClosure}\!\left(\mathcal{E}_{\rho}\right) \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(z\right), z, \mathbb{C}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`GaussLegendreWeight`	$w_{n,k}$	Gauss-Legendre quadrature weight
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`Pow`	${a}^{b}$	Power
`Supremum`	$\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right)$	Supremum of a set or function
`BernsteinEllipse`	$\mathcal{E}_{\rho}$	Bernstein ellipse with foci -1,+1 and semi-axis sum rho
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("47b181"),
    Formula(Where(LessEqual(Abs(Sub(Integral(f(t), Tuple(t, -1, 1)), Sum(Mul(GaussLegendreWeight(n, k), f(LegendrePolynomialZero(n, k))), Tuple(k, 1, n)))), Div(Mul(64, M), Mul(Mul(15, Sub(1, Pow(rho, -2))), Pow(rho, Mul(2, n))))), Equal(M, Supremum(Abs(f(t)), t, Element(t, BernsteinEllipse(rho)))))),
    Variables(f, n, rho),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(rho, RR), Greater(rho, 1), Subset(InteriorClosure(BernsteinEllipse(rho)), HolomorphicDomain(f(z), z, CC)))),
    References("L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831"))

Fungrim entry: 47b181

Topics using this entry