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The false position method

False position method

Introduction:

The False Position method in Numerical Analysis is a method to find the root which involves ideas of both Bisection and Secant method.

Diagrammatic Explanation

  • In this diagram which is shown above the red curve shows the function f(x) and the blue lines are the secants.
  • Just like the case of Bisection method, in the false position method also we start with two points a0 and b0 such that f(a0) and f(b0) are of opposite signs, which implies by the intermediate value theorem that the function f has a root in the interval [a0, b0].
  • The formula used above is also used in the secant method, but the secant method always retains the last two computed points, while the false position method retains two points which certainly bracket a root.
  • The only difference between the false position method and the bisection method is that the latter uses cn = (an + bn) / 2.

Description

  • If the initial end-points a0 and b0 are chosen such that f(a0) and f(b0) are of opposite signs, then one of the end-points will converge to a root of f.
  • Asymptotically, the other end-point will remain fixed for all subsequent iterations while the converging endpoint becomes updated.
  • As a result, unlike the bisection method, the width of the bracket does not tend to zero. As a consequence, the linear approximation to f(x), which is used to pick the false position, does not improve in its quality.

Newton's method

Introduction:

  • In the 19th cent. Two famous scientists namely Issac Newton and Joseph Raphson collectively made a method of finding roots of a equation called Newton-Raphson method.
  • This method is till now the best known method for finding successively better approximations to the zeroes (or roots) of a function in the subject of Numerical Analysis.
  • Correct implementations of this method embed it in a routine that also detects and perhaps overcomes possible convergence failures.
  • Interestingly ,it can be deduced that an alternative application of Newton-Raphson division, is to quickly find the reciprocal of a number using only multiplication and subtraction.

Diagrammatic Justification

  • Here we have show an illustration of one iteration of Newton's method (the function ƒ is shown in blue and the tangent line is in red). In this case xn+1 is a better approximation than xn for the root x of the function f.
  • 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 process is started off with some arbitrary initial value x0. (The closer to the zero, the better. But, in the absence of any intuition about where the zero might lie, a "hit and trial" method might narrow the possibilities to a reasonably small interval by appealing to the intermediate value theorem.)
  • The method will usually converge, provided this initial guess is close enough to the unknown zero, and that ƒ'(x0) ?0.
  • Furthermore, for a zero of multiplicity1, the convergence is at least quadratic (see rate of convergence) in a neighbourhood of the zero, which intuitively means that the number of correct digits roughly at least doubles in every step.

Conclusion

The root finding methods discussed in this section are quite successful for calculating roots of any given equation. The False Position Method is useful for finding the real roots of an equation. The Newton Raphson's method however can be used to calculate both real and complex roots. Newton Raphson's method is useful in cases when the function f(x) graph is nearly vertical while crossing the x-axis.


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