Gaussian quadrature

0745ee

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{x_{n,1}, x_{n,2}, \ldots, x_{n,n}\right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{x_{n,1}, x_{n,2}, \ldots, x_{n,n}\right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("0745ee"),
    Formula(Equal(Zeros(LegendrePolynomial(n, z), ForElement(z, CC)), Set(LegendrePolynomialZero(n, k), For(k, 1, n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

ea4754

w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(P'_{n}(x_{n,k})\right)}^{2}}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, n\}

TeX:

w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(P'_{n}(x_{n,k})\right)}^{2}}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, n\}

Definitions:

Fungrim symbol	Notation	Short description
`GaussLegendreWeight`	$w_{n,k}$	Gauss-Legendre quadrature weight
`Pow`	${a}^{b}$	Power
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("ea4754"),
    Formula(Equal(GaussLegendreWeight(n, k), Div(2, Mul(Sub(1, Pow(LegendrePolynomialZero(n, k), 2)), Pow(ComplexDerivative(LegendrePolynomial(n, t), For(t, LegendrePolynomialZero(n, k), 1)), 2))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(k, Range(1, n)))))

47b181

\left|\int_{-1}^{1} f(t) \, 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(t)\right|

Assumptions:

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_{-1}^{1} f(t) \, 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(t)\right|

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`	$\left\|z\right\|$	Absolute value
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sum`	$\sum_{n} f(n)$	Sum
`GaussLegendreWeight`	$w_{n,k}$	Gauss-Legendre quadrature weight
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`Pow`	${a}^{b}$	Power
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	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
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate

Source code for this entry:

Entry(ID("47b181"),
    Formula(Where(LessEqual(Abs(Sub(Integral(f(t), For(t, -1, 1)), Sum(Mul(GaussLegendreWeight(n, k), f(LegendrePolynomialZero(n, k))), For(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)), ForElement(t, BernsteinEllipse(rho)))))),
    Variables(f, n, rho),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(rho, RR), Greater(rho, 1), IsHolomorphic(f(z), ForElement(z, InteriorClosure(BernsteinEllipse(rho)))))),
    References("L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831"))

545987

\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:

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
`Abs`	$\left\|z\right\|$	Absolute value
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sum`	$\sum_{n} f(n)$	Sum
`GaussLegendreWeight`	$w_{n,k}$	Gauss-Legendre quadrature weight
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`Pow`	${a}^{b}$	Power
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	Supremum of a set or function
`BernsteinEllipse`	$\mathcal{E}_{\rho}$	Bernstein ellipse with foci -1,+1 and semi-axis sum rho
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	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"))

Gauss-Legendre quadrature