demuller¶
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sherpa.utils.demuller(fcn, xa, xb, xc, fa=None, fb=None, fc=None, args=(), maxfev=32, tol=1e-06)[source]¶ A root-finding algorithm using Muller’s method [1].
p( x ) = f( xc ) + A ( x - xc ) + B ( x - xc ) ( x - xb )
Notes
The general case:
2 f( x ) n x = x - ---------------------------------------- n+1 n C + sgn( C ) sqrt( C^2 - 4 f( x ) B ) n n n n n 1 ( f( x ) - f( x ) f( x ) - f( x ) ) ( n n-1 n-1 n-2 ) B = ------- ( ------------------ - ------------------- ) n x - x ( x - x x - x ) n n-2 ( n n-1 n-1 n-2 ) f( x ) - f( x ) n n-1 A = ----------------- n x - x n n-1 C = A + B ( x - x ) n n n n n-1
The convergence rate for Muller’s method can be shown to be the real root of the cubic x - x^3, that is:
p = (a + 4 / a + 1) / 3 a = (19 + 3 sqrt(33))^1/3
In other words: O(h^p) where p is approximately 1.839286755.
References
[1] http://en.wikipedia.org/wiki/Muller%27s_method