Page - 187 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

7.2 Newton’sMethod 187 thesolutionandthenumberof functioncalls.Themaincostofamethodforsolving f(x)=0equationsisusually theevaluationoff(x)andf ′(x), so the totalnumber ofcalls to thesefunctionsisan interestingmeasureof thecomputationalwork.Note that in functionNewton there is an initial call tof(x) and then one call tof and one tof ′ ineach iteration. RunningNewtons_method.py,we get the followingprintouton thescreen: Number of function calls: 25 A solution is: 3.000000 TheNewtonschemewillworkbetter if thestartingvalue isclose to thesolution. Agoodstartingvaluemayoftenmakethedifferenceas towhether thecodeactually finds a solution or not. Because of its speed (and when speed matters), Newton’s methodisoftenthemethodoffirstchoiceforsolvingnonlinearalgebraicequations, even if the scheme is not guaranteed to work. In cases where the initial guess may be far fromthesolution,agoodstrategy is to runa fewiterationswith the bisection method(see Sect.7.4) to narrowdownthe regionwheref is close to zeroand then switch toNewton’smethodfor fast convergenceto thesolution. Using sympy to Find the Derivative Newton’s method requires the analytical expression for the derivative f ′(x). Derivation of f ′(x) is not always a reliable process by hand if f(x) is a complicated function. However, Python has the symbolicpackageSymPy, which we may use to create the requireddfdx function. Withoursampleproblem,weget: import sympy as sym x = sym.symbols(’x’) f_expr = x**2 - 9 # symbolic expression for f(x) dfdx_expr = sym.diff(f_expr, x) # compute f’(x) symbolically # turn f_expr and dfdx_expr into plain Python functions f = sym.lambdify([x], # argument to f f_expr) # symbolic expression to be evaluated dfdx = sym.lambdify([x], dfdx_expr) print(f(3), dfdx(3)) # will print 0 and 6 The nice feature of this code snippet is that dfdx_expr is the exact analytical expression for the derivative, 2*x (seen if you print it out). This is a symbolic expression,sowecannotdonumericalcomputingwithit.However,withlambdify, such symbolic expression are turned into callable Python functions, as seen here withfanddfdx. The next method is the secant method, which is usually slower than Newton’s method,but itdoesnot requireanexpressionforf ′(x), and ithasonlyonefunction callper iteration.