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Maple Working Rules for Newton-Raphson Method

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  Newton-Raphson Method The Newton-Raphson method is based on the principle that if the initial guess of the root of f(x)=0   is at x(i) , then if one draws the tangent to the curve at f(x(i)) , the point x(i+1)   where the tangent crosses the -axis is an improved estimate of the root (Figure 1). Using 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 finds the root within a desirable tolerance. Maple Setup [> NewtonsMethod(x^3 + 4*x - 10, x = 1);                           1.556773264 [> NewtonsMethod(x^3 + 4*x - 10, x = 2, output = sequence); 2, 1.625000000, 1.558650066, 1.556774723, 1.556773264, 1.556773264 [>...
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Maple Program for Secant Method

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Working Rules for Secant Method in Maple 2020  The Secant command numerically approximates the roots of an algebraic function, f(x) , using a technique similar to Newton's method but without the need to evaluate the derivative of f(x) . Given an expression f and an initial approximate a, the Secant command computes a sequence  " p[k],  k=0..n ", of approximations to a root of f(x)=0 , where "n" is the number of iterations taken to reach a stopping criterion. The Secant command is a shortcut for calling the Roots command with the method=secant option The criterion that the approximations must meet before discontinuing the iterations. The following describes each criterion: [> restart; [> with(Student[NumericalAnalysis]); [> f := x^3 + 4*x^2 - 10; [> Secant(f, x = [1, 2], tolerance = 10^(-2));                           1.365211903 [> Secant(f, x = [1, 2], tolerance = 10^(-2), stoppingcriterion =...
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Maple Program for Bisection Method

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  Bisection Method restart;  with(Student[NumericalAnalysis]): f := x^3 + 4*x^2 - 10; Bisection(f, x = [1, 2], tolerance = 10^(-4)); 1.365112304 Bisection(f, x = [1, 2], tolerance = 10^(-4), output = sequence); [1., 2.], [1., 1.500000000], [1.250000000, 1.500000000], [1.250000000, 1.375000000], [1.312500000, 1.375000000], [1.343750000, 1.375000000], [1.359375000, 1.375000000], [1.359375000, 1.367187500], [1.363281250, 1.367187500], [1.363281250, 1.365234375], [1.364257812, 1.365234375] Bisection(f, x = [1, 2], tolerance = 10^(-2), stoppingcriterion = absolute); 1.367187500 Bisection(f, x = [1, 2], output = animation, tolerance = 10^(-3), stoppingcriterion = function_value); Bisection(f, x = [1, 2], output = plot, tolerance = 10^(-3), maxiterations = 15, stoppingcriterion = relative); Bisection(f, x = [1, 2], output = animation, tolerance = 10^(-3), maxiterations = 15, stoppingcriterion = relative); Bisection(f, x = [1, 2], output = information, tolerance = 10^(-9), maxiteratio...
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Introduction to Numerical Analysis/Computations

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  COMSATS University Islamabad,  Pakistan   NUMERICAL COMPUTATIONS What is numerical Computation? Numerical Computation is concerned with the derivation, analysis, and implementation of methods for obtaining reliable numerical answers to complex mathematical problems. In other words, NC is the subject concerned with the construction, analysis, and use of algorithms for the numerical solutions of mathematical problems to the given degree of numerical accuracy. When mathematical problems can be solved analytically, their solution may be exact, but more frequently, there may not be a known method of obtaining its solution. e.g. Many more such examples can be cited for which solutions by analytical means are either impossible or may be so complex that they are quite unsuitable for practical purposes.      During the last century, numerical techniques have witnessed a veritable explosion in research, both in their application to complex mathemati...
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