UPSI Digital Repository (UDRep)
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UPSI Digital Repository (UDRep)
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| Abstract : Universiti Pendidikan Sultan Idris |
| This research aimed to investigate the most efficient iterative method in solving scalar
nonlinear equations. There are three iterative methods that are used to solve the nonlinear
scalar equations that are Bisection, Secant and Newton Raphson’s methods. These
three iterative methods have different order of convergence. Bisection method is linearly
convergence while Secant method is super linear and Newton-Raphson method
has quadratic convergence. It is well known that the method that has a higher order
of convergence, will perform much faster than others. Seven nonlinear scalar equations
are considered based on the combinations of two or three functions and are solved
by the Bisection, Secant and Newton-Raphson methods using Scilab programming language.
The tolerance used is 10−10 and the performances of these methods are based on
number of function evaluation, number of iterations, and computational or CPU time.
Based on the numerical results of the seven nonlinear equations, it is observed that
Newton-Raphson method is still the most efficient method but not for all the equations.
Bisection method has fixed performances on all the nonlinear equations however, the
method failed to converge for the imaginary root. On the other hand, the performance
of Secant method is almost similar to Newton-Raphson method except for the nonlinear
Equations (4.4), and (4.5) on the interval [1.3,2] and [0,1] respectively. In conclusion,
Newton-Raphson method remains the best but not for all nonlinear equations since
there are realistic circumstances that makes Newton-Raphson converges either slower
or identical to Secant method. It is also proven that Secant method can perform faster
than Newton-Raphson method depending on the form of the curve functions that corresponds
to the approximate values. As implications, more than three combinations of
the functions can be investigated and also the research can be extended to system of
nonlinear equations. |
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