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Fungrim entry: 545987

abf(t)dtba2k=1nwn,kf ⁣(ba2xn,k+a+b2)ba264M15(1ρ2)ρ2n   where M=suptEρf ⁣(ba2t+a+b2)\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|
Assumptions:aCandbCandnZ1andρRandρ>1andf(z) is holomorphic on zInteriorClosure ⁣(Eρ)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)
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
Absz\left|z\right| Absolute value
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sumnf(n)\sum_{n} f(n) Sum
GaussLegendreWeightwn,kw_{n,k} Gauss-Legendre quadrature weight
LegendrePolynomialZeroxn,kx_{n,k} Legendre polynomial zero
Powab{a}^{b} Power
SupremumsupxSf(x)\mathop{\operatorname{sup}}\limits_{x \in S} f(x) Supremum of a set or function
BernsteinEllipseEρ\mathcal{E}_{\rho} Bernstein ellipse with foci -1,+1 and semi-axis sum rho
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
RRR\mathbb{R} Real numbers
IsHolomorphicf(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c 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"))

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2019-10-05 13:11:19.856591 UTC