Often one of the points gets “stuck,” and several variants such as the Illinois or Pegasus methods and variations are used to “unstick” it. In the Regula Falsi variation we start with initial guesses ( a, f ( a ) ) and ( b, f ( b ) ) such that f ( a ) f ( b ) < 0 after an iteration similar to the above we replace either a or b by the new value x i + 1 depending on which of f ( a ) or f ( b ) has the same sign as f ( x i + 1 ). X i + 1 = x i - f i x i - x i - 1 f i - f i - 1 ( i = 1, 2, 3, … )where ( x 0, f 0 ), ( x 1, f 1 ) are initial guesses. Two applications of Muller's method gives the iterates 0.732166 and 0.732244. Example 8.4įor the equation 2 x − 5 x + 2 = 0 with x 0 = 0 and x 1 = 1 the secant method (see Example 8.2) gives x 2 = 0.15. If this occurs, we can arrange to drop the point x 2 and continue the search for the root by applying the bisection or regula falsi method. If | x 1 − x 0 | approaches zero, the effect of rounding error can perturb the coefficients of the polynomial p 2 sufficiently to create complex roots instead of the predicted real roots. It is necessary to add one further set of instructions to this algorithm. We repeat the process with the new set until our stopping criterion is satisfied, as mentioned above. This completes one iteration of the method. 8.4 we replace x 0 by x 2 and x 2 by x 3. Next, we replace either x 0 or x 1 by x 2, exactly as in the regula falsi method, and replace x 2 by x 3. Only one of these, denoted by x 3, will be inside. The polynomial equation p 2( x) = 0 must then have real roots. We could select x 2 as the midpoint of or as the number obtained by one iteration of the secant method. This time we begin with numbers x 0 < x 1 such that f(x 0) f(x 1) < 0 and select x 2 as some number in. To avoid the possibility of the polynomial equation p 2( x) = 0 having complex roots, we may adopt a variant of the above method in which the root is bracketed at any stage.
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