MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante IMSP, UPLB 2 nd Sem AY

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).

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

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

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!

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?

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

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

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

4 th Method: SECANT METHOD Example: Find a zero of Use 0 & 1 as initial values =(A1*(A2^3+A2-1)- A2*(A1^3+A1- 1))/((A2^3+A2-1)-(A1^3+A1- 1)) =B2

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)

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

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!

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

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

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.