ridder#
- scipy.optimize.ridder(f, a, b, args=(), xtol=2e-12, rtol=np.float64(8.881784197001252e-16), maxiter=100, full_output=False, disp=True)[source]#
- Find a root of a function in an interval using Ridder’s method. - Parameters:
- ffunction
- Python function returning a number. f must be continuous, and f(a) and f(b) must have opposite signs. 
- ascalar
- One end of the bracketing interval [a,b]. 
- bscalar
- The other end of the bracketing interval [a,b]. 
- xtolnumber, optional
- The computed root - x0will satisfy- np.isclose(x, x0, atol=xtol, rtol=rtol), where- xis the exact root. The parameter must be positive.
- rtolnumber, optional
- The computed root - x0will satisfy- np.isclose(x, x0, atol=xtol, rtol=rtol), where- xis the exact root. The parameter cannot be smaller than its default value of- 4*np.finfo(float).eps.
- maxiterint, optional
- If convergence is not achieved in maxiter iterations, an error is raised. Must be >= 0. 
- argstuple, optional
- Containing extra arguments for the function f. f is called by - apply(f, (x)+args).
- full_outputbool, optional
- If full_output is False, the root is returned. If full_output is True, the return value is - (x, r), where x is the root, and r is a- RootResultsobject.
- dispbool, optional
- If True, raise RuntimeError if the algorithm didn’t converge. Otherwise, the convergence status is recorded in any - RootResultsreturn object.
 
- Returns:
- rootfloat
- Root of f between a and b. 
- rRootResults(present iffull_output = True)
- Object containing information about the convergence. In particular, - r.convergedis True if the routine converged.
 
 - See also - brentq,- brenth,- bisect,- newton
- 1-D root-finding 
- fixed_point
- scalar fixed-point finder 
- elementwise.find_root
- efficient elementwise 1-D root-finder 
 - Notes - Uses [Ridders1979] method to find a root of the function f between the arguments a and b. Ridders’ method is faster than bisection, but not generally as fast as the Brent routines. [Ridders1979] provides the classic description and source of the algorithm. A description can also be found in any recent edition of Numerical Recipes. - The routine used here diverges slightly from standard presentations in order to be a bit more careful of tolerance. - As mentioned in the parameter documentation, the computed root - x0will satisfy- np.isclose(x, x0, atol=xtol, rtol=rtol), where- xis the exact root. In equation form, this terminating condition is- abs(x - x0) <= xtol + rtol * abs(x0).- The default value - xtol=2e-12may lead to surprising behavior if one expects- ridderto always compute roots with relative error near machine precision. Care should be taken to select xtol for the use case at hand. Setting- xtol=5e-324, the smallest subnormal number, will ensure the highest level of accuracy. Larger values of xtol may be useful for saving function evaluations when a root is at or near zero in applications where the tiny absolute differences available between floating point numbers near zero are not meaningful.- References - Examples - >>> def f(x): ... return (x**2 - 1) - >>> from scipy import optimize - >>> root = optimize.ridder(f, 0, 2) >>> root 1.0 - >>> root = optimize.ridder(f, -2, 0) >>> root -1.0