Overview of the itp package

The Interpolate, Truncate, Project (ITP) root-finding algorithm was developed in Oliveira and Takahashi (2020). It’s performance compares favourably with existing methods on both well-behaved functions and ill-behaved functions while retaining the worst-case reliability of the bisection method. For details see the authors’ Kudos summary and the Wikipedia article ITP method.

The itp function implements the ITP method to find a root \(x^*\) of the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) in the interval \([a, b]\), where \(f(a)f(b) < 0\). If \(f\) is continuous over \([a, b]\) then \(f(x^*) = 0\). If \(f\) is discontinuous over \([a, b]\) then \(x^*\) may be an estimate of a point of discontinuity at which the sign of \(f\) changes, that is, \(f(x^* - \delta)f(x^* + \delta) \leq 0\), where \(0 \leq \delta \leq \epsilon\) for some tolerance value \(\epsilon\).

We use some of the examples presented in Table 1 of Oliveira and Takahashi (2020) to illustrate the use of this function and run the examples using the uniroot function in the stats package as a means of comparison, using a convergence tolerance of \(10^{-10}\) in both cases. The itp function uses the following default values of the tuning parameters: \(\kappa_1 = 0.2 / (b - a)\), \(\kappa_2 = 2\) and \(n_0 = 1\), but these may be changed by the user. See Sections 2 and 3 of Oliveira and Takahashi (2020) for information. The following function prints output from uniroot in the same style as the output from itp.

Well-behaved functions

These functions are infinitely differentiable and contain only one simple root over \([−1, 1]\).

Lambert: \(f(x) = x e^x - 1\)

# Lambert
lambert <- function(x) x * exp(x) - 1
itp(lambert, c(-1, 1))
#>       root     f(root)  iterations  
#>     0.5671   2.048e-12           8
uniroot(lambert, c(-1, 1), tol = 1e-10)
#>       root     f(root)  iterations  
#>     0.5671  -8.623e-12           8

Trigonometric 1: \(f(x) = \tan(x - 1 / 10)\)

# Trigonometric 1
trig1 <- function(x) tan(x - 1 /10)
itp(trig1, c(-1, 1))
#>       root     f(root)  iterations  
#>        0.1  -8.238e-14           8
uniroot(trig1, c(-1, 1), tol = 1e-10)
#>       root     f(root)  iterations  
#>        0.1   6.212e-14           8

Ill-behaved functions

Polynomial 3: \(f(x) = (10^6 x - 1) ^ 3\)

This function has a non-simple root at \(10^{-6}\), with a multiplicity of 3.

# Polynomial 3
poly3 <- function(x) (x * 1e6 - 1) ^ 3
itp(poly3, c(-1, 1))
#>       root     f(root)  iterations  
#>      1e-06   -1.25e-13          36
# Using n0 = 0 leads to (slightly) fewer iterations, in this example
poly3 <- function(x) (x * 1e6 - 1) ^ 3
itp(poly3, c(-1, 1), n0 = 0)
#>       root     f(root)  iterations  
#>      1e-06  -6.739e-14          35
uniroot(poly3, c(-1, 1), tol = 1e-10)
#>       root     f(root)  iterations  
#>      1e-06   1.771e-17          65

Discontinuous

Staircase: \(f(x) = \lceil 10 x - 1 \rceil + 1/2\)

This function has discontinuities, including one at the location of the root.

# Staircase
staircase <- function(x) ceiling(10 * x - 1) + 1 / 2
itp(staircase, c(-1, 1))
#>       root     f(root)  iterations  
#>  7.404e-11         0.5          31
uniroot(staircase, c(-1, 1), tol = 1e-10)
#>       root     f(root)  iterations  
#> -3.739e-11        -0.5          31

Multiple roots

Warsaw: \(f(x) = I(x > -1)\left(1 + \sin\left(\frac{1}{1+x}\right)\right)-1\)

This function has multiple roots, but we only seek one of them.

# Warsaw
warsaw <- function(x) ifelse(x > -1, sin(1 / (x + 1)), -1)
itp(warsaw, c(-1, 1))
#>       root     f(root)  iterations  
#>    -0.6817  -5.472e-11          11
uniroot(warsaw, c(-1, 1), tol = 1e-10)
#>       root     f(root)  iterations  
#>    -0.6817  -3.216e-16           9

In terms of a naive comparison based on the number of iterations itp and uniroot perform similarly, except in the repeated-root “Polynomial 3” example, where itp requires fewer iterations.