# SpectralToolbox.Spectral1D¶

Functions

 BarycentricWeights(x) BarycentricWeights(): Returns a 1-d array of weights for Lagrange Interpolation FirstPolynomialDerivativeMatrix(x) PolynomialDerivativeMatrix(): Assemble the first Polynomial Derivative Matrix using matrix multiplication. LagrangeInterpolate(x, f, xi) LagrangeInterpolate(): Interpolate function values f from points x to points xi using Lagrange weights LagrangeInterpolationMatrix(x, w, xi) LagrangeInterpolationMatrix(): constructs the Lagrange Interpolation Matrix from points x to points xi LinearInterpolationMatrix(x, xi) LinearInterpolationMatrix(): constructs the Linear Interpolation Matrix from points x to points xi LinearShapeFunction(x, xm, xp, xi) Hat function used for linear interpolation PolynomialDerivativeMatrix(x, k) PolynomialDerivativeMatrix(): Assemble the Polynomial k-th Derivative Matrix using the matrix recursion. SparseLinearInterpolationMatrix(x, xi) LinearInterpolationMatrix(): constructs the Linear Interpolation Matrix from points x to points xi. SparseLinearShapeFunction(x, xm, xp, xi) Hat function used for linear interpolation. cc(N[, norm]) cc(): function for generating 1D Nested Clenshaw-Curtis [-1,1] fej(N[, norm]) fej(): function for generating 1D Nested Fejer’s rule [-1,1] generate(ptype, params) Generate orthogonal basis objects from Spectral1D.AVAIL_POLY. gqn(N) GQN(): function for generating 1D Gaussian quadrature for integral with Gaussian weight (Gauss-Hermite) gqu(N[, norm]) GQU(): function for generating 1D Gaussian quadrature rules for unweighted integral over [-1,1] (Gauss-Legendre) kpn(N) KPN(): function for generating 1D Nested rule for integral with Gaussian weight kpu(N[, norm]) KPU(): function for generating 1D Nested rule for unweighted integral over [-1,1] nestedgauss(N[, norm]) nestedgauss(): function for generating 1D Nested rule for integral with Uniform weight with 2**l scaling nestedlobatto(N[, norm]) nestedlobatto(): function for generating 1D Nested rule for integral with Uniform weight with 2**l scaling

Classes

 Basis() This is an abstract class for 1-d basis ConstantExtendedHermiteProbabilistsFunction([...]) Construction of the Hermite Probabilists’ functions extended with the constant basis ConstantExtendedHermiteProbabilistsRadialBasisFunction(nbasis) Construction of the Hermite Probabilists’ Radial Basis Functions Fourier() GenericOrthogonalPolynomial(mu, endl, endr) Construction of polynomials orthogonal with respect to a generic measure HermitePhysicistsFunction([normalized]) Construction of the Hermite Physiticists’ functions HermitePhysicistsPolynomial([normalized]) Construction of the Hermite Physicists’ polynomials HermiteProbabilistsFunction([normalized]) Construction of the Hermite Probabilists’ functions HermiteProbabilistsPolynomial([normalized]) Construction of the Hermite Probabilists polynomials HermiteProbabilistsRadialBasisFunction(order) Construction of the Hermite Probabilists’ Radial Basis Functions JacobiPolynomial(alpha, beta[, span, normalized]) Construction of Jacobi polynomials LaguerreFunction(alpha[, normalized]) Construction of the Laguerre functions LaguerrePolynomial(alpha[, normalized]) Construction of Laguerre polynomials LinearExtendedHermiteProbabilistsRadialBasisFunction(nbasis) Construction of the Hermite Probabilists’ Radial Basis Functions OrthogonalBasis([normalized]) This is an abstract class for 1-d orthogonal basis OrthogonalPolynomial([normalized]) This is an abstract class for 1-d polynomials Poly1D(poly, params[, sdout]) Initialization of the Polynomial instance.