STEP 8

NONLINEAR EQUATIONS 3

Method of false position

As mentioned in the Prologue, the method of false position dates back to the ancient Egyptians. It remains an effective alternative to the bisection method for solving the equation f(x) = 0 for a real root between a and b, given that f (x) is continuous and f (a) and f(b) have opposite signs. The algorithm is suitable for automatic computation (pseudo.code)

1. PROCEDURE

The curve y = f(x) is not generally a straight line. However, one may join the points (a,f(a)) and (b,f(b)) by the straight line

Thus straight line cuts the x-axis at (X, 0) where

so that

Suppose that f(a) is negative and f(b) is positive. As in the bisection method, there are the three possibilities :

  1. f(X) = 0, when case X is the root ;
  2. f(X) < 0, when the root lies between X and b ;
  3. f(X)>0, when the root lies between X and a.

Again, in Case 1, the process is terminated, in either Case 2 or Case 3, the process can be repeated until the root is obtained to the desired accuracy. In Fig. 6, the successive points where the straight lines cut the axis are denoted by x1, x2, x3.

2. EFFECTIVENESS AND THE SECANT METHOD

Like the bisection method, the method of false position has almost assured convergence, and it may converge to a root faster. However, it may happen that most or all the calculated values of X are on the same side of the root, in which case convergence may be slow (Fig. 7). This is avoided in the secant method, which resembles the method of false position except that no attempt is made to ensure that the root is enclosed; starting with two approximations to the root (x0, x1), further approximations x2, x3,… are computed from

There is no longer convergence, but the process is simpler (the sign of f(xn+1) is not tested) and often converges faster.

With respect to speed of convergence of the secant method, one has at the (n+1)th step:

Hence, expanding in terms of the Taylor series,

where we have used the fact that f(a)=0. Thus we see that en+1 is proportional to enen-1, which may be expressed in mathematical notation as

We seek k such that

Hence the speed of convergence is faster than linear (k =1 ), but slower than quadratic (k=2). This rate of convergence is sometimes referred to as superlinear convergence.

3. EXAMPLE

Use the method of false position to solve

Then

f (x1) =f (0.257637) = 3 x 0.257637 0 0.772875 = 0.772912 - 0.772875 = 0.000036.

The student may verify that doing one more iteration of the method of false position yields an estimate x2 = 0.257628 for which the function value is less than 5*10-6. Since x1 and x2 agree to 4D, we conclude that the root is 0.2576, correct to 4D.

Checkpoint

  1. When may the method of false position be used to find a root of the equation f(x) = 0?
  2. On what geometric construction is the method of false position based?

EXERCISES

  1. Use the method of false position to find the smallest root of the equation f (x) = 2 sin x + x - 2 = 0, stopping when

  2. Compare the results obtained when you use
    1. the bisection method,
    2. the method of false position, and
    3. the secant method

      with starting values 0.7 and 0.9 to solve the equation

      3sin x = x + 1/x.

    4. Use the method of false position to find the root of the equation

    5. Each equation in Exercises 2(a)-2(c) of Step 6 has only one root. Use the method of false position to find each root stopping when

Answers

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