Bisection Method: Please explain Step 6 for me in DETAILS, here is the algorithm

Question

Bisection Method:

Please explain Step 6 for me in DETAILS, here is the algorithm:

Step 1    Set i = 1

                 FA = f(a)

Step 2 While i<=No do Steps 3-6

           Step 3 Set p = a + (b - a )/2; ( Compute pi.)

                                FP = f(p).

           Step 4 If FP = 0 or (b-a)/2 < TOL then

                            OUTPUT (p); (Procedure completed successfully.)

                            STOP.

            Step 5 Set i = i + 1.

            Step 6 if FA*FP > 0 then set a = p; (Compute ai, bi.)

                                                                     FA = FP

                                                 else set b = p. (FA is unchanged.)

            Step 7 OUTPUT ('Method failed after No iterations, No =', No);

                          (The procedure was unsuccessful.)

                         STOP.

PLEASE EXPLAIN STEP 6 FOR ME IN DETAILS FOR NEWTON'S METHOD:

Newton's Method:

INPUT initial approximation p0; tolerance TOL; maximum number of iterations No.

OUTPUT approximate solution p or message of failure.

Step 1 Set i = 1.

Step 2 While i<= No do Steps 3-6.

           Step 3 Set p = po - f(po)/f '(po).   (Compute pi.)

          Step 4   If |p - po | < TOL then

                              OUTPUT (p); (The procedure was successful.)

                               STOP.

        Step 5   Set i = i + 1.

          Step 6 Set po = p. (Update po.)

Step 7 OUTPUT (' The method failed after No iteration, No =', No);

             (The procedure was unsuccessful.)

              STOP.

Explanation / Answer

Bisection method

Consider a transcendental equation f (x) = 0 which has a zero in the interval [a,b] and f (a) * f (b) < 0. Bisection scheme computes the zero, say c, by repeatedly halving the interval [a,b]. That is, starting with

c = (a+b) / 2

the interval [a,b] is replaced either with [c,b] or with [a,c] depending on the sign of f (a) * f (c) . This process is continued until the zero is obtained. Since the zero is obtained numerically the value of c may not exactly match with all the decimal places of the analytical solution of f (x) = 0 in the interval [a,b].

FA*FP > 0 means that lower bound and uper bound are of same sign. either negative or positive.
i.e, in [a,b], both are either positive or negative. As such 0 doesnot lie between a and b. So we try to find a new interval [e,f] where 0 lies between lower and upper bound.


Newton Method

The Newton

Related Questions

Navigate

• Browse (All)
• Browse B
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.