Class GaussianOrthogonalPolynomial

Namespace

NanoQL.Numerics

Assembly

NanoQL.dll

An orthogonal polynomial for Gaussian quadratures, defined by the three-term recurrence P_{k+1}(x) = (x − α_k)·P_k(x) − β_k·P_{k−1}(x) and μ₀ = ∫ w(x) dx: G.H. Golub and J.H. Welsch, "Calculation of Gauss quadrature rules", Math. Comput. 23 (1969), 221-230; "Numerical Recipes in C", 2nd edition, Press, Teukolsky, Vetterling, Flannery.

public abstract class GaussianOrthogonalPolynomial

Inheritance

object

GaussianOrthogonalPolynomial

Derived

GaussHermitePolynomial

GaussHyperbolicPolynomial

GaussJacobiPolynomial

GaussLaguerrePolynomial

MomentBasedGaussianPolynomial

Inherited Members

object.GetType()

object.MemberwiseClone()

object.ToString()

object.Equals(object)

object.Equals(object, object)

object.ReferenceEquals(object, object)

object.GetHashCode()

Constructors

GaussianOrthogonalPolynomial()

protected GaussianOrthogonalPolynomial()

Methods

Alpha(int)

The recurrence's α_i.

public abstract double Alpha(int i)

Parameters

i int

The index, non-negative.

Returns

double

α_i.

Exceptions

ArgumentOutOfRangeException

i is negative.

Beta(int)

The recurrence's β_i.

public abstract double Beta(int i)

Parameters

i int

The index, non-negative.

Returns

double

β_i.

Exceptions

ArgumentOutOfRangeException

i is negative.

LogW(double)

log(w(x)), kept finite past the range where w(x) itself underflows to zero. The default is log(w(x)); a polynomial whose w can underflow while log(w) stays representable overrides it.

public virtual double LogW(double x)

Parameters

x double

The point.

Returns

double

log(w(x)).

Mu_0()

μ₀, the integral of the weight function.

public abstract double Mu_0()

Returns

double

μ₀.

Value(int, double)

The polynomial P_n at x, by the recurrence.

public double Value(int n, double x)

Parameters

n int

The degree, non-negative.

x double

The point.

Returns

double

P_n(x).

Exceptions

ArgumentOutOfRangeException

n is negative.

W(double)

The weight function w(x).

public abstract double W(double x)

Parameters

x double

The point.

Returns

double

w(x).

WeightedValue(int, double)

The polynomial weighted by the square root of the weight, √w(x)·P_n(x).

public double WeightedValue(int n, double x)

Parameters

n int

The degree, non-negative.

x double

The point.

Returns

double

√w(x)·P_n(x).

Exceptions

ArgumentOutOfRangeException

n is negative.