Muller's method
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Muller's method is a root-finding algorithm, a numerical method for solving equations of the form f(x) = 0. It was first presented by David E. Muller in 1956.
Muller's method proceeds according to a third-order recurrence relation similar to the second-order recurrence relation of the secant method. Whereas the secant method proceeds by constructing a line through two points on the graph of f corresponding to the last two iterative approximations and then uses the line's root as the next approximation at every iteration, by contrast, Muller's method uses three points corresponding to the last three iterative approximations, constructs a parabola through these three points, and then uses a root of the parabola as the next approximation at every iteration.
Derivation
[edit]Muller's method uses three initial approximations of the root, x0, x1 and x2, and determines the next approximation x3 by considering the intersection of the x-axis with the parabola through (x0, f(x0)), (x1, f(x1)) and (x2, f(x2)).
Consider the quadratic polynomial
1 |
that passes through (x0, f(x0)), (x1, f(x1)) and (x2, f(x2)). Define the differences
and
Substituting each of the three points (x0, f(x0)), (x1, f(x1)) and (x2, f(x2)) into equation (1) and solving simultaneously for a, b, and c gives
The quadratic formula is then applied to (1) to determine x3 as
The sign preceding the radical term is chosen to match the sign of b to ensure the next iterate is closest to x2, giving
Once x3 is determined, the process is repeated. Note that due to the radical expression in the denominator, iterates can be complex even when the previous iterates are all real. This is in contrast with other root-finding algorithms like the secant method, Sidi's generalized secant method or Newton's method, whose iterates will remain real if one starts with real numbers. Having complex iterates can be an advantage (if one is looking for complex roots) or a disadvantage (if it is known that all roots are real), depending on the problem.
Speed of convergence
[edit]For well-behaved functions, the order of convergence of Muller's method is approximately 1.839 and exactly the tribonacci constant. This can be compared with approximately 1.618, exactly the golden ratio, for the secant method and with exactly 2 for Newton's method. So, the secant method makes less progress per iteration than Muller's method and Newton's method makes more progress.
More precisely, if ξ denotes a single root of f (so f(ξ) = 0 and f'(ξ) ≠ 0), f is three times continuously differentiable, and the initial guesses x0, x1, and x2 are taken sufficiently close to ξ, then the iterates satisfy
where μ ≈ 1.84 is the positive solution of , the defining equation for the tribonacci constant.
Generalizations and related methods
[edit]Muller's method fits a parabola, i.e. a second-order polynomial, to the last three obtained points f(xk-1), f(xk-2) and f(xk-3) in each iteration. One can generalize this and fit a polynomial pk,m(x) of degree m to the last m+1 points in the kth iteration. Our parabola yk is written as pk,2 in this notation. The degree m must be 1 or larger. The next approximation xk is now one of the roots of the pk,m, i.e. one of the solutions of pk,m(x)=0. Taking m=1 we obtain the secant method whereas m=2 gives Muller's method.
Muller calculated that the sequence {xk} generated this way converges to the root ξ with an order μm where μm is the positive solution of .
The method is much more difficult though for m>2 than it is for m=1 or m=2 because it is much harder to determine the roots of a polynomial of degree 3 or higher. Another problem is that there seems no prescription of which of the roots of pk,m to pick as the next approximation xk for m>2.
These difficulties are overcome by Sidi's generalized secant method which also employs the polynomial pk,m. Instead of trying to solve pk,m(x)=0, the next approximation xk is calculated with the aid of the derivative of pk,m at xk-1 in this method.
Computational example
[edit]Below, Muller's method is implemented in the Python programming language. It takes as parameters the three initial estimates of the root, as well as the desired decimals places of accuracy and the maximum number of iterations. The program is then applied to find a root of the function f(x) = x2 − 612.
from cmath import sqrt # Use the complex sqrt as we may generate complex numbers
def func(x):
return (x ** 2) - 612
def muller(x0, x1, x2, decimal_places, maximum_iterations):
iteration_counter = 0
iterates = [x0, x1, x2]
solution_found = False
while not solution_found and iteration_counter < maximum_iterations:
final_index = len(iterates)-1
h0 = iterates[final_index - 1] - iterates[final_index - 2]
h1 = iterates[final_index] - iterates[final_index - 1]
f_x0 = func(iterates[final_index - 2])
f_x1 = func(iterates[final_index - 1])
f_x2 = func(iterates[final_index])
delta0 = (f_x1 - f_x0) / h0
delta1 = (f_x2 - f_x1) / h1
coeff_a = (delta1 - delta0) / (h1 + h0)
coeff_b = coeff_a*h1 + delta1
coeff_c = f_x2
sqrt_delta = sqrt(pow(coeff_b,2) - 4*coeff_a*coeff_c)
denominators = [coeff_b - sqrt_delta, coeff_b + sqrt_delta]
# Take the higher-magnitude denominator
next_iterate = iterates[final_index] - (2 * coeff_c)/max(denominators, key=abs)
iterates.append(next_iterate)
solution_found = abs(func(next_iterate)) < pow(10, -decimal_places)
iteration_counter = iteration_counter + 1
if solution_found:
print("Solution found: {}".format(next_iterate))
else:
print("No solution found.")
muller(10, 20, 30, 9, 20) # Solution found: (24.73863375370596+0j)
See also
[edit]- Halley's method, with cubic convergence
- Householder's method, includes Newton's, Halley's and higher-order convergence
References
[edit]- Atkinson, Kendall E. (1989). An Introduction to Numerical Analysis, 2nd edition, Section 2.4. John Wiley & Sons, New York. ISBN 0-471-50023-2.
- Burden, R. L. and Faires, J. D. Numerical Analysis, 4th edition, pages 77ff.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 9.5.2. Muller's Method". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
Further reading
[edit]- A bracketing variant with global convergence: Costabile, F.; Gualtieri, M.I.; Luceri, R. (March 2006). "A modification of Muller's method". Calcolo. 43 (1): 39–50. doi:10.1007/s10092-006-0113-9. S2CID 124772103.