ChebCoefficients: ChebCoefficients

In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

ChebCoefficients

Description

ChebCoefficients(values) converts function values, say values = f(x), evaluated at Chebyshev points of the 2nd kind, say n-dimensional x, to n-dimensional Chebyshev coefficients.

Usage

ChebCoefficients(values)

Arguments

values

function values evaluated at Chebyshev points of the 2nd kind

Value

c = ChebCoefficients(values) returns the n-dimensional vector of coefficients such that f(x) = C(1)*T_{0}(x) + C(2)*T_{1}(x)+ ... + C(N)*T_{n-1}(x) (where T_{k}(x) denotes the k-th 1st-kind Chebyshev polynomial) interpolates the data [values(1); ... ; values(n)] at Chebyshev points x = [x(1); ... ; x(n)] of the 2nd kind.

If values is an (n x m)-matrix, then C = ChebCoefficients(values) returns the (n x m)-matrix of coefficients C.

Note

Ver.: 15-Nov-2021 17:33:19 (consistent with Matlab CharFunTool v1.3.0, 28-May-2021 14:28:24).

See Also

Other Utility Function: ChebPoints(), ChebPolyValues(), ChebPoly(), ChebValues(), GammaLog(), GammaMultiLog(), GammaMulti(), GammaZX(), Hypergeom1F1MatApprox(), Hypergeom1F1Mat(), Hypergeom2F1Mat(), Hypergeom2F1(), HypergeompFqMat(), InterpChebValues(), hypergeom1F1(), interpBarycentric()

Examples

## EXAMPLE1 (Chebyshev coefficients of the Sine function on (-pi,pi))
   # n      <- 2^5+1
   # domain <- c(-pi,pi)
   # x      <- ChebPoints(n,domain)
   # f      <-list( sin(x[[1]]))
   # coeffs <- ChebCoefficients(f)
   # print(list(x[[1]], coeffs))
   # x      <- seq(-pi,pi,length.out=100)
   # pval   <- ChebPolyValues(coeffs,x,domain)
   # plotReIm(function(x)
   #         pval,x,
   # xlab="x",
   # ylab="Chebyshev polynomial",
   # title="Chebyshev Polynomial Values Specified by its Coefficients")

## EXAMPLE2 (Chebyshev coefficients of the Sine and the Cosine on (-pi,pi))
   n      <- 2^5+1
   domain <- c(-pi,pi)
   x      <- ChebPoints(n,domain)
   f      <-list(sin(x[[1]]), cos(x[[1]]))
   coeffs <- ChebCoefficients(f)
   print(list(x[[1]], coeffs))
   x      <- seq(-pi,pi,length.out=100)
   pval   <- ChebPolyValues(coeffs,x,domain)
   pval1<-list()
   for (i in 1:length(pval[1,])) {

    pval1[[i]]<-pval[,i]
   }
  # matplot(x, pval,type="o",col=c("red","green"),lty=c(1,1),
   #        xlab="x",
    #       ylab="Chebyshev polynomial",
     #      main="Chebyshev Polynomial Values Specified by its Coefficients")
   plotReIm3( pval1,
              x,
              title="Chebyshev Polynomial Values Specified by its Coefficients",
              xlab = "x",
              ylab ="Chebyshev polynomial" )

gajdosandrej/CharFunToolR documentation built on June 3, 2024, 7:46 p.m.