1. MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante IMSP, UPLB 2 nd Sem AY 2012-2013

2. Question: What if we do not have means of getting an initial bracket? Let’s start with SECANT METHOD… Suppose that f is a continuous function. Pick two initial points (not necessarily forming a bracket), then do linear interpolation (not inverse).

3. 4 th Method: SECANT METHOD Initial points: Interpolating line: Use x-intercept:

4. 4 th Method: SECANT METHOD Approximate: -Same as the formula for Regula Falsi -In an iteration, if the points form a bracket, then the method is similar to Regula Falsi; else the method uses extrapolation. In whatever case, the new points will be Old x 2

5. 4 th Method: SECANT METHOD In short: Pick any two distinct points, draw the secant through them, and use the x-intercept (x 3 ) of that secant line as the new estimate of the zero of the function. For the next iteration, discard the oldest point and add (x 3, f(x 3 )) as the new point. WE DO NOT NEED IZT (IVT) ANYMORE!

11. 4 th Method: SECANT METHOD Secant method can be considerably faster than the previous methods. However, it may fail to converge. Example: if f(x 1 )=f(x 2 ), then what would happen?

12. 4 th Method: SECANT METHOD Notice that can be written as

13. and can also be written as You can use any of these formulas for Regula Falsi and Secant Method 4 th Method: SECANT METHOD

14. For Secant Method (not for Regula Falsi), we can generalize the formulas as follows (k=1,2,3,…)

15. 4 th Method: SECANT METHOD Example: Find a zero of Use 0 & 1 as initial values. 0 10.5 0.636363636 0.690052356 0.68202042 0.682325781 0.682327804 =(A1*(A2^3+A2-1)- A2*(A1^3+A1- 1))/((A2^3+A2-1)-(A1^3+A1- 1)) =B2

16. 4 th Method: SECANT METHOD Assuming that the secant method converges to the root, the order of convergence of the method is SUPERLINEAR!!! (but not yet quadratic)

17. 4 th Method: SECANT METHOD Stopping criterion: You can use tol=10^(-m): accurate at least up to m decimal places

18. 5 th Method: Newton’s Method/Newton-Raphson Iteration What if we make the secant line is a tangent line? Hence, we only need one initial point. But we add another assumption: f should be differentiable!

19. 5 th Method: Newton’s Method/Newton-Raphson Iteration From Secant Method: If we use tangent lines: as x k-2 approaches x k-1

20. 5 th Method: Newton’s Method/Newton-Raphson Iteration Hence:

25. Newton’s Method: To be continued… Assignment: List the advantages and disadvantages of the discussed methods. Research other disadvantages that we did not mention in the class.