This is the semester project for the course of ‘Numerical Analysis and Computer Applications’ - CS213 for the 2 nd year of Computer and Systems Engineering Department to be delivered to Dr. Wafaa ElHaweet and the teaching assistant Eng. Omar Salaheldine. Our project is developed in Python programming language. We used appJar library to create the interactive GUI.
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The pupose of the app is to find roots of any Equation based on some methods:
- Bracketing Methods.
- Open Methods.
- Bierge Vieta
- General Algorithm.
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the bracketing methods aim to iteratively shorten the interval in order to find the root within a certain level of significance (epsilon).
- The bracketing mechanism is done by finding the function’s value in the middle of the bracketed interval. Then moving to a subproblem of the same type in either the left-halved interval or right-halved interval by checking the bracketing condition’s validation for each.
- The bracketing mechanism is by connecting a line between the two function’s values at the terminals of the interval. The root estimate is the intersection of that line with the abscissa. To get a better estimate, we evaluate the function’s value at the x-coordinate intersection found at that iteration, and run the same bracketing test with the terminals and use the valid interval to the find a better estimate in the next iteration. Regula Falsi, like Bisection, always converges, usually considerably faster than Bisection—but sometimes much slower than Bisection.
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The open points try to find the root in a way that doesn’t require as much initial knowledge as the bracketing methods which rely on having a certain interval where a root is guaranteed to exist within. The open methods for root finding rely mostly on trying to adapt with the function’s behaviour, mostly through its first derivative or its estimate, to guide it to the roots of the equation.
- This method relies on finding the intersection between two functions; y = x and y = g(x). This is why it iterates through this formula of equating them to get g(x) = x and iteratively closer to the solution. We get the function g(x) by separating an x from the function in question f(x). The intersection x-coordinate shall be found to be the x-coordinate of the root.
- The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line (which can be computed using the tools of calculus), and one computes the x-intercept of this tangent line (which is easily done with elementary algebra). This x-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated.
- The secant method is in theory, the same as the Newton-Raphson method explained earlier. However, instead of finding the first derivative through calculus, we use a simple delta y / delta x using the points of previous two iterations. Which introduces the disadvantage of the Secant method which is the fact that it needs two points initially not one, unlike most of the open methods.
- explanation can be found in the Report
- This mode runs by being given a function to compute as much as possible of its roots, it’s a combine of two methods introduced in this report in such a way to come over the disadvantage of each method. Newton and bisection methods are used (An iterative method along with a bracketing one)
a detaield Explanation can be found here Report
- Clone this repo
https://github.com/Magho/Root-finder - cd
Root-finder/appJar - run
appjar.pyfile
