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// NEWTON RAPHSON // Finding the root of any given polynomial // // // // // // // //
Algorithm Given the function f(x), solve for the first derivative f'(x). Assume an initial value for f(xi). Repeat: a. Substitute xi to the function f(xi) and f'(xi). b. Solve for the next value of xi: x(i+1) = xi - f(xi)/f'(xi) c. Until abs(f(xi)) < e or abs(x(i+1) - xi) < e.
// // // //
EXAMPLE polynomial = [ 1 5 -6] test point = - 3 root = -6
funcprot(0); //INITIALIZE FUNCTIONS // DERIVATIVE OF A POLYNOMIAL TEST FUNCTION function [newpoly]=derivativepoly(polynomial) terms = length(polynomial); //DETERMINE NUMBER OF TERMS base = terms - 1; //BASE TERM element = 1; //ELEMENT NUMBER while element ~= terms newpoly(1,element) = polynomial(1,element) * base; element = element + 1; //INCREMENT ELEMENT base = base - 1; //DECREMENT BASE TERM end endfunction //POLYNOMIAL EVALUATION FUNCTION function [sumpoly]=evalpoly(polynomial, point) terms = length(polynomial); //DETERMINE NUMBER OF TERMS power = terms - 1; //HIGHEST POWER WOULD BE element = 1; sumpoly = 0; //INITIALIZE SUM = 0; while element ~= terms + 1 base = point**power; //POWER OF THE VARIABLE sumpoly = sumpoly + polynomial(1,element) * base; element = element + 1; //INC ELEMENT power = power - 1; //DEC POWER end endfunction // NEWTON'S METHOD OF ROOT FINDING MAIN PROG disp("NEWTON''S METHOD OF ROOT FINDING a.k.a Newton Raphson") polynomial = input("Enter polynomial coefficients: "); //ENTER POLYNOMIAL x = input("Enter test point: "); //TEST POINT root = evalpoly(polynomial,x); //FIRST EVALUATION newpoly = derivativepoly(polynomial); //FIRST DERIVATIVE while abs(root) > 10**-15 //LOOP STARTS UNTIL ROOT ~ 0 num = evalpoly(polynomial,x) //NUMERATOR den = evalpoly(newpoly,x); //DENOMINATOR x = x - num/den; //NEW X root = evalpoly(polynomial,x); //EVALUATE NEW X end disp(x, "One root is: ")
