====== Root findings : equations f(x) = 0 ======

<note tip>
Algorithm used to find roots of an equation use iterations, and a numerical criterion to accept a solution when a sufficiently accurate value is reached. The rate of convergence depends on the used method and the function f(x). Some methods (Newton-Raphson) need the derivative of the function f(x).
</note>

  * Polynomial equations : [[wp>Bairstow's_method|Bairstow's method]] is an efficient algorithm for finding the roots of a real polynomial of arbitrary degree
    * [[https://docs.scipy.org/doc/numpy/reference/routines.polynomials.html|Polynomials in NumPy]]
    * [[https://docs.scipy.org/doc/numpy/reference/routines.polynomials.polynomial.html|polynomial module]], including polyroots(c) to compute the roots of a polynomial.
  * [[wp>Bisection_method|Bisection method]] (dichotomy) : very simple and robust method, but relatively slow. It assumes continuity of the function, and obtain one roots. The algorithm is based on a [[wp>Loop_invariant|loop invariant]] property : an interval [a, b] is said to bracket a root if f(a) and f(b) have opposite signs.
  * [[wp>Secant_method|Secant method]] (retains the last two computed points)
  * [[wp>False_position_method|Regula falsi]] (retains the points which preserve bracketing)
  * [[wp>Newton's_method|Newton-Raphson method]]
  * Chapter 9 in the book "Numerical Recipes" : Root finding an nonlinear sets of equations
    * 9.0 Introduction
    * 9.1 Bracketing and Bisection
    * 9.2 Secant Method, False Position Method, and Ridders' Method
    * 9.4 Newton-Raphson Method Using Derivative
    * 9.5 Roots of Polynomials
  * Python [[https://docs.scipy.org/doc/numpy/|NumPy]] library : [[https://docs.scipy.org/doc/scipy/reference/|SciPy Reference]]
    * [[https://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html|scipy.optimize]] package (root)

===== Applications =====
  * ...

===== Références =====
  * Numerical recipes, The Art of Scientific Computing 3rd Edition, William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, 2007, isbn: 9780521880688
    * [[http://numerical.recipes/]]
      * in C : [[http://apps.nrbook.com/c/index.html]]
    * [[http://www2.units.it/ipl/students_area/imm2/files/Numerical_Recipes.pdf]], p 347...
    * [[http://apps.nrbook.com/empanel/index.html#]]
      * Chapter 9 : Root finding an nonlinear sets of equations