Numerical Analysis (Numerical Computations)

    Assignment No 1 1.       Use Newton’s method to find solutions accurate to within 10 -3 for the following problems.                  2.       Find all the zeros of the following function by using the fixed-point iteration method for an appropriate iteration function g . Find the zeros accurate to within 10 -3 .                 3.       Find an approximation to correct to within 10 -2 using the Bisection Algorithm.   4.       Use Secant and False-Position Methods to approximate the root of         Submission Date: 17-10-22 Lab Assignment No 1 Make a Maple Program for the following given numerical technique to approximate the solution of  f(x)=x-0.8-0.2sin(x) with predefined ac...

Newton-Raphson method is an open method. in this method we start with only one initial approximation  x(i) , the approximation  x(i+1)   is the  x -intercept of the tangent line to the graph of  f  at  ( x(i) ,  f ( x(i+1) )) . The  approximation x (i+2)   is the  x -intercept of the tangent line to the graph of  f  at  ( x(i+1) ,  f ( x(i+1) ))  and  so on.  From fig. 1 uses the definition of the slope of a function ,  at  x=x(i)                                               Equation (1) is called the Newton-Raphson formula for solving nonlinear equations of the form  f(x)=0 .  So starting with an initial guess ,   x(i) , one can find the next guess ,   x(i+1) , by using Equation (1).  One can repeat this process until one fi...

  The bisection method is one of the first numerical methods developed to find the root of a nonlinear equation f(x)=0 , also called the  binary-search method.  In the Bisection method, we start with two initial approximations ‘a’ and ‘b’, and we suppose that f is a continuous function defined on the interval [ a , b ], with f(a) and f(b) of opposite sign . Furthermore if f(a)*f(b)<0, has at least one root between a and b, and if f(a)*f(b)>0, there may or may not be any root between a and b.  If f(a)*f(b)>0, then there may be more than one root between a and b.  So the method only guarantees one root between a and b. Since the method is based on finding the root between two points, the method falls under the category of bracketing methods. Since the root is bracketed between two points, a and b, one can find the mid-point,  Choose a and b as two guesses for the root such that f(a)*f(b)<0 , or in other words, f(x) changes the sign betw...

Numerical methods provide estimates that are very close to an exact analytical solution, obviously an error is introduced into the computation therefore it is important to understand the errors. Numerical computation can also be defined as a vehicle to study errors in computations.                 Error Error is the difference between the actual value and the computed value. Let x be the actual value of a certain problem and   be its approximate value then mathematically error can be written as                  Let x=2 be the actual value and be its approximate solution then an error is 0.01   Three Important steps in Error Analysis   It is the study and evaluation of errors. An error analysis generally consists ...