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https://github.com/jacobwilliams/roots-fortran
A modern Fortran library for finding the roots of continuous scalar functions of a single real variable, using derivative-free methods.
https://github.com/jacobwilliams/roots-fortran
bisection brent-dekker fortran fortran-package-manager muller muller-s-method regula-falsi root-finding zeroin
Last synced: about 6 hours ago
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A modern Fortran library for finding the roots of continuous scalar functions of a single real variable, using derivative-free methods.
- Host: GitHub
- URL: https://github.com/jacobwilliams/roots-fortran
- Owner: jacobwilliams
- License: other
- Created: 2020-01-11T19:33:52.000Z (almost 5 years ago)
- Default Branch: master
- Last Pushed: 2024-01-07T03:17:29.000Z (10 months ago)
- Last Synced: 2024-01-07T15:55:15.694Z (10 months ago)
- Topics: bisection, brent-dekker, fortran, fortran-package-manager, muller, muller-s-method, regula-falsi, root-finding, zeroin
- Language: Fortran
- Homepage:
- Size: 1.68 MB
- Stars: 31
- Watchers: 6
- Forks: 4
- Open Issues: 8
-
Metadata Files:
- Readme: README.md
- License: LICENSE.md
Awesome Lists containing this project
README
============
**roots-fortran**: root solvers for modern Fortran
[](https://github.com/topics/fortran)
[](https://github.com/jacobwilliams/roots-fortran/releases/latest)
[](https://github.com/jacobwilliams/roots-fortran/actions)
[](https://codecov.io/gh/jacobwilliams/roots-fortran)
[](https://github.com/jacobwilliams/roots-fortran/commits/master)
## Description
A modern Fortran library for finding the roots of continuous scalar functions of a single real variable.
## Compiling
A `fpm.toml` file is provided for compiling roots-fortran with the [Fortran Package Manager](https://github.com/fortran-lang/fpm). For example, to build:
```
fpm build --profile release
```
By default, the library is built with double precision (`real64`) real values. Explicitly specifying the real kind can be done using the following processor flags:
Preprocessor flag | Kind | Number of bytes
----------------- | ----- | ---------------
`REAL32` | `real(kind=real32)` | 4
`REAL64` | `real(kind=real64)` | 8
`REAL128` | `real(kind=real128)` | 16
For example, to build a single precision version of the library, use:
```
fpm build --profile release --flag "-DREAL32"
```
To run the unit tests:
```
fpm test
```
To use `roots-fortran` within your fpm project, add the following to your `fpm.toml` file:
```toml
[dependencies]
roots-fortran = { git="https://github.com/jacobwilliams/roots-fortran.git" }
```
or, to use a specific version:
```toml
[dependencies]
roots-fortran = { git="https://github.com/jacobwilliams/roots-fortran.git", tag = "1.0.0" }
```
To generate the documentation using [ford](https://github.com/Fortran-FOSS-Programmers/ford), run: ```ford roots-fortran.md```
## Usage
### Methods
The module contains the following methods:
Procedure | Description | Reference
--- | --- | ---
[`bisection`](https://jacobwilliams.github.io/roots-fortran/proc/bisection.html) | Classic bisection method | Bolzano (1817)
[`regula_falsi`](https://jacobwilliams.github.io/roots-fortran/proc/regula_falsi.html) | Classic regula falsi method | ?
[`muller`](https://jacobwilliams.github.io/roots-fortran/proc/muller.html) | Improved Muller method (for real roots only) | [Muller (1956)](https://www.ams.org/journals/mcom/1956-10-056/S0025-5718-1956-0083822-0/S0025-5718-1956-0083822-0.pdf)
[`brent`](https://jacobwilliams.github.io/roots-fortran/proc/brent.html) | Classic Brent's method (a.k.a. Zeroin) | [Brent (1971)](https://maths-people.anu.edu.au/~brent/pd/rpb005.pdf)
[`brenth`](https://jacobwilliams.github.io/roots-fortran/proc/brenth.html) | SciPy variant of `brent` | [SciPy](https://github.com/scipy/scipy/)
[`brentq`](https://jacobwilliams.github.io/roots-fortran/proc/brentq.html) | SciPy variant of `brent` | [SciPy](https://github.com/scipy/scipy/)
[`illinois`](https://jacobwilliams.github.io/roots-fortran/proc/illinois.html) | Illinois method | [Dowell & Jarratt (1971)](https://personal.math.ubc.ca/~loew/mech2/Dowell+Jarratt.pdf)
[`pegasus`](https://jacobwilliams.github.io/roots-fortran/proc/pegasus.html) | Pegasus method | [Dowell & Jarratt (1972)](https://link.springer.com/article/10.1007/BF01932959)
[`anderson_bjorck`](https://jacobwilliams.github.io/roots-fortran/proc/anderson_bjorck.html) | Anderson-Bjorck method | [King (1973)](https://link.springer.com/article/10.1007/BF01933405)
[`anderson_bjorck_king`](https://jacobwilliams.github.io/roots-fortran/proc/anderson_bjorck_king.html) | a [variant](https://link.springer.com/content/pdf/bbm%3A978-3-642-05175-3%2F1.pdf) of `anderson_bjorck` | [King (1973)](https://link.springer.com/article/10.1007/BF01933405)
[`ridders`](https://jacobwilliams.github.io/roots-fortran/proc/ridders.html) | Classic Ridders method | [Ridders (1979)](https://cs.fit.edu/~dmitra/SciComp/Resources/RidderMethod.pdf)
[`toms748`](https://jacobwilliams.github.io/roots-fortran/proc/toms748.html) | Algorithm 748 | [Alefeld, Potra, Shi (1995)](https://dl.acm.org/doi/abs/10.1145/210089.210111)
[`chandrupatla`](https://jacobwilliams.github.io/roots-fortran/proc/chandrupatla.html) | Hybrid quadratic/bisection algorithm | [Chandrupatla (1997)](https://dl.acm.org/doi/10.1016/S0965-9978%2896%2900051-8)
[`bdqrf`](https://jacobwilliams.github.io/roots-fortran/proc/bdqrf.html) | Bisected Direct Quadratic Regula Falsi | [Gottlieb & Thompson (2010)](http://www.m-hikari.com/ams/ams-2010/ams-13-16-2010/gottliebAMS13-16-2010.pdf)
[`zhang`](https://jacobwilliams.github.io/roots-fortran/proc/zhang.html) | Zhang's method (with corrections from Stage) | [Zhang (2011)](https://www.cscjournals.org/download/issuearchive/IJEA/Volume2/IJEA_V2_I1.pdf)
[`itp`](https://jacobwilliams.github.io/roots-fortran/proc/itp.html) | Interpolate Truncate and Project method | [Oliveira & Takahashi (2020)](https://dl.acm.org/doi/abs/10.1145/3423597)
[`barycentric`](https://jacobwilliams.github.io/roots-fortran/proc/barycentric.html) | Barycentric interpolation method | [Mendez & Castillo (2021)](https://www.researchgate.net/publication/352162661_A_highly_efficient_numerical_method_to_solve_non-linear_functions_using_barycentric_interpolation)
[`blendtf`](https://jacobwilliams.github.io/roots-fortran/proc/blendtf.html) | Blended method of trisection and false position | [Badr, Almotairi, Ghamry (2021)](https://www.mdpi.com/2227-7390/9/11/1306/htm)
In general, all the methods are guaranteed to converge. Some will be more efficient (in terms of number of function evaluations) than others for various problems. The methods can be broadly classified into three groups:
* Simple classical methods (`bisection`, `regula_falsi`, `illinois`, `ridders`).
* Newfangled methods (`zhang`, `barycentric`, `blendtf`, `bdqrf`, `anderson_bjorck_king`). These rarely or ever seem to be better than the best methods.
* Best methods (`anderson_bjorck`, `muller`, `pegasus`, `toms748`, `brent`, `brentq`, `brenth`, `chandrupatla`, `itp`). Generally, one of these will be the most efficient method.
Note that some of the implementations in this library contain additional checks for robustness, and so may behave better than naive implementations of the same algorithms. In addition, all methods have an option to fall back to bisection if the method fails to converge.
### Functional Interface Example
```fortran
program main
use root_module, wp => root_module_rk
implicit none
real(wp) :: x, f
integer :: iflag
call root_scalar('bisection',func,-9.0_wp,31.0_wp,x,f,iflag)
write(*,*) 'f(',x,') = ', f
write(*,*) 'iflag = ', iflag
contains
function func(x) result(f)
implicit none
real(wp),intent(in) :: x
real(wp) :: f
f = -200.0_wp * x * exp(-3.0_wp*x)
end function func
end program main
```
### Object Oriented Interface Example
```fortran
program main
use root_module, wp => root_module_rk
implicit none
type,extends(bisection_solver) :: my_solver
end type my_solver
real(wp) :: x, f
integer :: iflag
type(my_solver) :: solver
call solver%initialize(func)
call solver%solve(-9.0_wp,31.0_wp,x,f,iflag)
write(*,*) 'f(',x,') = ', f
write(*,*) 'iflag = ', iflag
contains
function func(me,x)
class(root_solver),intent(inout) :: me
real(wp),intent(in) :: x
real(wp) :: f
f = -200.0_wp * x * exp(-3.0_wp*x)
end function func
end program main
```
### Result
```
f( -2.273736754432321E-013 ) = 4.547473508867743E-011
iflag = 0
```
## Notes
* Originally this was an idea for the Fortran [stdlib](https://github.com/fortran-lang/stdlib). See: [#87: Optimization, Root finding, and Equation Solvers](https://github.com/fortran-lang/stdlib/issues/87). Eventually, it may be merged into the other one in one form or another.
## Documentation
The latest API documentation for the `master` branch can be found [here](https://jacobwilliams.github.io/roots-fortran/). This was generated from the source code using [FORD](https://github.com/Fortran-FOSS-Programmers/ford).
## License
The roots-fortran source code and related files and documentation are distributed under a permissive free software [license](https://github.com/jacobwilliams/roots-fortran/blob/master/LICENSE.md) (BSD-style).
## Related Fortran libraries
* [polyroots-fortran](https://github.com/jacobwilliams/polyroots-fortran) -- For finding all the roots of polynomials with real or complex coefficients.
## Similar libraries in other programming languages
Language | Library
--- | ---
C | [GSL](https://www.gnu.org/software/gsl/doc/html/roots.html)
C++ | [Boost Math Toolkit](https://www.boost.org/doc/libs/1_80_0/libs/math/doc/html/root_finding.html)
Julia | [Roots.jl](https://github.com/JuliaMath/Roots.jl)
R | [uniroot](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/uniroot.html)
Rust | [roots](https://docs.rs/roots/latest/roots/)
MATLAB | [fzero](https://www.mathworks.com/help/matlab/ref/fzero.html)
Python | [scipy.optimize.root_scalar](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root_scalar.html)
## References
* D. E. Muller, "[A Method for Solving Algebraic Equations Using an Automatic Computer](https://www.ams.org/journals/mcom/1956-10-056/S0025-5718-1956-0083822-0/S0025-5718-1956-0083822-0.pdf)", Mathematical Tables and Other Aids to Computation, 10 (1956), 208-215.
* M. Dowell, P. Jarratt, "[A modified regula falsi method for computing the root of an equation](https://personal.math.ubc.ca/~loew/mech2/Dowell+Jarratt.pdf)", BIT 11 (1971), 168-174.
* R. P. Brent, "[An algorithm with guaranteed convergence for finding a zero of a function](http://maths-people.anu.edu.au/~brent/pd/rpb005.pdf)", The Computer Journal, Vol 14, No. 4., 1971.
* R. P. Brent, "[Algorithms for minimization without derivatives](http://maths-people.anu.edu.au/~brent/pub/pub011.html)", Prentice-Hall, Inc., 1973.
* Ridders, C., "[A new algorithm for computing a single root of a real continuous function](https://cs.fit.edu/~dmitra/SciComp/Resources/RidderMethod.pdf)", IEEE Trans. on Circuits and Systems, Vol 26, Issue 11, Nov 1979.
* G. E. Alefeld, F. A. Potra and Yixun Shi, "[Algorithm 748: Enclosing Zeros of Continuous Functions](https://dl.acm.org/doi/abs/10.1145/210089.210111)", ACM Transactions on Mathematical Software, Vol. 21. No. 3. September 1995. Pages 327-344.
* G.E. Mullges & F. Uhlig, "Numerical Algorithms with Fortran", Springer, 1996. Section 2.8.1, p 32-34.
* T.R. Chandrupatla, "[A new hybrid quadratic/bisection algorithm for finding the zero of a nonlinear function without derivatives](https://dl.acm.org/doi/10.1016/S0965-9978%2896%2900051-8)", Advances in Engineering Software, Vol 28, 1997, pp. 145-149.
* R. G. Gottlieb, B. F. Thompson, "[Bisected Direct Quadratic Regula Falsi](https://www.researchgate.net/publication/228712261_Bisected_Direct_Quadratic_Regula_Falsi)", Applied Mathematical Sciences, Vol. 4, 2010, no. 15, 709-718.
* A. Zhang, "[An Improvement to the Brent's Method](https://www.cscjournals.org/download/issuearchive/IJEA/Volume2/IJEA_V2_I1.pdf)", International Journal of Experimental Algorithms (IJEA), Volume (2) : Issue (1) : 2011.
* E Badr, S Almotairi, A El Ghamry, "[A Comparative Study among New Hybrid Root Finding Algorithms and Traditional Methods](https://www.mdpi.com/2227-7390/9/11/1306)", Mathematics 2021, 9, 1306.
## See also
* [Code coverage statistics](https://app.codecov.io/gh/jacobwilliams/roots-fortran) [codecov.io]