AbstractWeights

Abstract weights for numerical interpolation, differentiation, and integration. The purpose of this abstract type is to construct the type system.

PolynomialInterpolationWeights

Weights for polynomial interpolation.

PolynomialDifferentiationWeights

Weights for polynomial differentiation.

PolynomialIntegrationWeights

Weights for polynomial integration.

BackwardDifferentiationWeights

Weights for backward differentiation.

GregoryIntegrationWeights

Weights for Gregory integration.

BoundaryConvolutionWeights

Weights for boundary convolution.

calc_poly_interpolation(k::I64)

Calculate polynomial interpolation weights.

References :

See [NESSi] Eq.~(80).

calc_poly_differentiation(k::I64, Wi::Matrix{F64})

Calculate polynomial differentiation weights.

References :

See [NESSi] Eq.~(85).

calc_poly_integration(k::I64, Wi::Matrix{F64})

Calculate polynomial integration weights.

References :

See [NESSi] Eq.~(89).

calc_backward_differentiation(k::I64, Wi::Matrix{F64})

Calculate backward differentiation weights.

References :

See [NESSi] Eq.~(90).

calc_gregory_integration(k::I64, Wt::Array{F64,3})

Calculate Gregory integration weights.

References :

See Remarks.

calc_gregory_weights(k::I64, n::I64)

Try to calculate the integration weights based on Gregory quadrature rule for 𝑘-order and 𝑛 + 1 nodes. This function only works for 𝑛 > 𝑘. It just returns a vector with 𝑛 + 1 elements (They are all Rational numbers).

References :

See Remarks.

calc_boundary_convolution(k::I64, Wi::Matrix{F64})

Calculate integration weights for boundary convolution.

References :

See [NESSi] Eq.~(104).

trapezoid(n::I64)

Return integration weights based on the trapezoid rule. Note that the return value of this function is a Rational number.

References :

See [REVIEW] Eq.~(A9).

Λ(k::I64)

Try to calculate the Laplace coefficients Λ. This a recursive function. Note that the return value is a Rational number.

References :

See [MABOOK] Eqs.~(8.4.37), (8.4.50), and (8.4.54).

γⱼ(k::I64)

Try to calculate the 𝑘th order coefficients γⱼ, where 𝑗 ∈ [0,𝑘].

References :

See [QUADRATURE] Eqs.~(2.5) and (2.6).

𝐑(m::I64, a::I64, b::I64)

Try to calculate integration ∫^{m}{0} dx (m-x)ᵃxᵇ. We note that this integration appears in [NESSi] Eq.~(104). This function is called by `calcboundary_convolution()` internally. It should not be exported.

References :

See [NESSi] Eq.~(104).

Γ(n::I64)

Try to calculate Γ function. This function should not be exported. Note that the function name for the calculation of Γ function is :tgamma, instead of :gamma. This is quite strange.

References :

See [MATABLE] Section 6.21.