Computes the Taylor series of functions or characters.
taylor(
f,
var,
params = list(),
order = 1,
accuracy = 4,
stepsize = NULL,
zero = 1e-07
)character, or function returning a numeric scalar value.
vector giving the variable names with respect to which the derivatives are to be computed and/or the point where the derivatives are to be evaluated (the center of the Taylor series). See derivative.
list of additional parameters passed to f.
the order of the Taylor approximation.
degree of accuracy for numerical derivatives.
finite differences stepsize for numerical derivatives. It is based on the precision of the machine by default.
tolerance used for deciding which derivatives are zero. Absolute values less than this number are set to zero.
list with components:
the Taylor series.
the approximation order.
data.frame containing the variables, coefficients and degrees of each term in the Taylor series.
Guidotti E (2022). "calculus: High-Dimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 1-37. doi:10.18637/jss.v104.i05
Other polynomials:
hermite()
Other derivatives:
derivative()
### univariate taylor series (in x=0)
taylor("exp(x)", var = "x", order = 2)
#> $f
#> [1] "(1) * 1 + (1) * x^1 + (0.5) * x^2"
#>
#> $order
#> [1] 2
#>
#> $terms
#> var coef degree
#> 0 1 1.0 0
#> 1 x^1 1.0 1
#> 2 x^2 0.5 2
#>
### univariate taylor series of user-defined functions (in x=0)
f <- function(x) exp(x)
taylor(f = f, var = c(x=0), order = 2)
#> $f
#> [1] "(1) * 1 + (0.999999999996052) * x^1 + (0.499999999979692) * x^2"
#>
#> $order
#> [1] 2
#>
#> $terms
#> var coef degree
#> 0 1 1.0 0
#> 1 x^1 1.0 1
#> 2 x^2 0.5 2
#>
### multivariate taylor series (in x=0 and y=1)
taylor("x*(y-1)", var = c(x=0, y=1), order = 4)
#> $f
#> [1] "(1) * x^1*(y-1)^1"
#>
#> $order
#> [1] 4
#>
#> $terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 (y-1)^1 0 1
#> 1,0 x^1 0 1
#> 0,2 (y-1)^2 0 2
#> 2,0 x^2 0 2
#> 1,1 x^1*(y-1)^1 1 2
#> 0,3 (y-1)^3 0 3
#> 3,0 x^3 0 3
#> 1,2 x^1*(y-1)^2 0 3
#> 2,1 x^2*(y-1)^1 0 3
#> 0,4 (y-1)^4 0 4
#> 4,0 x^4 0 4
#> 1,3 x^1*(y-1)^3 0 4
#> 3,1 x^3*(y-1)^1 0 4
#> 2,2 x^2*(y-1)^2 0 4
#>
### multivariate taylor series of user-defined functions (in x=0 and y=1)
f <- function(x,y) x*(y-1)
taylor(f, var = c(x=0, y=1), order = 4)
#> $f
#> [1] "(0.999999999999994) * x^1*(y-1)^1"
#>
#> $order
#> [1] 4
#>
#> $terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 (y-1)^1 0 1
#> 1,0 x^1 0 1
#> 0,2 (y-1)^2 0 2
#> 2,0 x^2 0 2
#> 1,1 x^1*(y-1)^1 1 2
#> 0,3 (y-1)^3 0 3
#> 3,0 x^3 0 3
#> 1,2 x^1*(y-1)^2 0 3
#> 2,1 x^2*(y-1)^1 0 3
#> 0,4 (y-1)^4 0 4
#> 4,0 x^4 0 4
#> 1,3 x^1*(y-1)^3 0 4
#> 3,1 x^3*(y-1)^1 0 4
#> 2,2 x^2*(y-1)^2 0 4
#>
### vectorized interface
f <- function(x) prod(x)
taylor(f, var = c(0,0,0), order = 3)
#> $f
#> [1] "(1) * x1^1*x2^1*x3^1"
#>
#> $order
#> [1] 3
#>
#> $terms
#> var coef degree
#> 0,0,0 1 0 0
#> 0,0,1 x3^1 0 1
#> 0,1,0 x2^1 0 1
#> 1,0,0 x1^1 0 1
#> 0,0,2 x3^2 0 2
#> 0,2,0 x2^2 0 2
#> 2,0,0 x1^2 0 2
#> 0,1,1 x2^1*x3^1 0 2
#> 1,0,1 x1^1*x3^1 0 2
#> 1,1,0 x1^1*x2^1 0 2
#> 0,0,3 x3^3 0 3
#> 0,3,0 x2^3 0 3
#> 3,0,0 x1^3 0 3
#> 0,1,2 x2^1*x3^2 0 3
#> 0,2,1 x2^2*x3^1 0 3
#> 1,0,2 x1^1*x3^2 0 3
#> 1,2,0 x1^1*x2^2 0 3
#> 2,0,1 x1^2*x3^1 0 3
#> 2,1,0 x1^2*x2^1 0 3
#> 1,1,1 x1^1*x2^1*x3^1 1 3
#>