The **itp** package implements the Interpolate, Truncate, Project (ITP) root-finding algorithm of Oliveira and Takahashi (2021). Each iteration of the algorithm results in a bracketing interval for the root that is narrower than the previous interval. 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.

We use three examples from Section 3 of Oliveira and Takahashi (2021) to illustrate the use of the `itp`

function. Each of these functions has a root in the interval . The function can be supplied either as an R function or as an external pointer to a C++ function.

The Lambert function is continuous.

The `itp`

function finds an estimate of the root, that is, for which is (approximately) equal to 0. The algorithm continues until the length of the interval that brackets the root is smaller than , where is a user-supplied tolerance. The default is .

First, we supply an R function that evaluates the Lambert function.

```
# Lambert, using an R function
lambert <- function(x) x * exp(x) - 1
itp(lambert, c(-1, 1))
#> function: lambert
#> root f(root) iterations
#> 0.5671 2.048e-12 8
```

Now, we create an external pointer to a C++ function that has been provided in the `itp`

package and pass this pointer to the function `itp()`

. For more information see the Overview of the itp package vignette.

```
# Lambert, using an external pointer to a C++ function
lambert_ptr <- xptr_create("lambert")
itp(lambert_ptr, c(-1, 1))
#> function: lambert_ptr
#> root f(root) iterations
#> 0.5671 2.048e-12 8
```

`itp_c`

Also provided is the function `itp_c`

, which is equivalent to `itp`

, but the calculations are performed entirely using C++, and the arguments differ slightly: `itp_c`

has a named required argument `pars`

rather than `...`

and it does not have the arguments `interval`

, `f.a`

or `f.b`

.

The staircase function is discontinuous.

The `itp`

function finds the discontinuity at at which the sign of the function changes. The value of 0.5 returned for the root `res$root`

is the midpoint of the bracketing interval `[res$a, res$b]`

at convergence.

See the Overview of the itp package vignette, which can also be accessed using `vignette("itp-vignette", package = "itp")`

.