Taylor Series Approximation

Description

Local polynomial approximation through Taylor series.

Usage

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taylor(f, x0, n = 4, ...)

Arguments

f

differentiable function.

x0

point where the series expansion will take place.

n

Taylor series order to be used; should be n <= 4.

...

more variables to be passed to function f.

Details

Calculates the first four coefficients of the Taylor series through numerical differentiation and uses some polynomial ‘yoga’.

Value

Vector of length n+1 representing a polynomial of degree n.

Note

TODO: Pade approximation.

See Also

fderiv

Examples

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taylor(sin, 0, 4)  #=> -0.1666666  0.0000000  1.0000000  0.0000000
taylor(exp, 1, 4)  #=>  0.04166657 0.16666673 0.50000000 1.00000000 1.00000000

f <- function(x) log(1+x)
p <- taylor(f, 0, 4)
p                     # log(1+x) = 0 + x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 +- ...
                      # [1] -0.250004  0.333334 -0.500000  1.000000  0.000000

## Not run: 
x <- seq(-1.0, 1.0, length.out=100)
yf <- f(x)
yp <- polyval(p, x)
plot(x, yf, type = "l", col = "gray", lwd = 3)
lines(x, yp, col = "red")
grid()
## End(Not run)

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