interpmat {rjacobi}R Documentation

Interpolation Matrix of Gauss-Jacobi Quadrature

Description

Calculates the Interpolation matrix of the Q point Gauss-Jacobi quadrature.

Usage

Imat = interpmatGJ(zp, x, a=0.0, b=0.0)
Imat = interpmatGLJ(zp, x, a=0.0, b=0.0)
Imat = interpmatGRJM(zp, x, a=0.0, b=0.0)
Imat = interpmatGRJP(zp, x, a=0.0, b=0.0)

Arguments

zp Points where to calculate the interpolation
x Quadrature node calculated with zerosGJ (depending on quadrature type)
a α parameter of Jacobi polynomial. Defaults to 0
b β parameter of Jacobi polynomial. Defaults to 0

Details

The interpolation matrix is used to interpolate functions known at quadrature nodes to other points. It is actually a polynomial interpolation using Lagrange interpolants through quadrature nodes. The interpolation matrix is given by:

I_{ij} = h_j(x_i)

where x_i are the interpolation points and h_j(x) are the Lagrangean interpolators associated to the quadrature points.

Different types of quadrature are possible:

With this matrix, a function known at quadrature points can be calculated with

[f]_{interp} = {I_{ij} }[f]_{quad}

df = D %*% f(x) where x is a vector containning the quadrature nodes.

Value

The interpolation matrix

References

Abramowitz, Milton and Stegun, Irene (editors); "Handbook of Mathematical functions", Dover Publications 1965.

Karniadakis, George Em and Sherwin, Spencer; "Spectral/hp Element Methods for Computational Fluid Dynamics", Oxford Science Publications, 2nd edition, 2005.

See Also

quadrature xinterp zerosGJ diffmatGJ weightsGJ

Examples


## Interpolates the Runge function in the interval (-1,1)

runge <- function(x){1 / (1 + 25*x^2)}

z <- zerosGLJ(13)
x <- seq(-1, 1, by=0.02)

x2 <- seq(-1, 1, len=201)

fe <- runge(x2)

f <- runge(z)
Imat <- interpmatGLJ(x, z)

finterp <- Imat %*% f

plot(x2, fe, ty='l', xlab='x', ylab='y',
main='Runge function interpolation')

points(z, f)
lines(x, finterp, lty=2)



[Package rjacobi version 0.9.2 Index]