The secant method is a root-finding algorithm that uses successive secants of a function to linearly approximate the root. It requires two initial guesses, x0 and x1, to construct a secant line through the points (x0, f(x0)) and (x1, f(x1)). The x-intercept of this line provides the next approximation x2. Repeating this process iteratively refines the approximation until the root is found to within a desired precision. The secant method converges faster than bisection near roots and does not require evaluating derivatives, but it may fail to converge for some functions.

1. Presentation on secant method PRESENTED BY:-RITU NAMDEO M.C.A. 2 ND SEM ‘B’ SEC

3. It is a root finding method.

6. Cont. This new value replaces the oldest x value being used in the calculation. ...

7. First two iteration-

8. Example- Question- Use the secant method to determine root of equation. cos x-x ex=0 solution- Taking the initial approximation as x0=0 ,x1=1 we have for secant method f(0)=1 and f(1)=cos1-e=-2.177979523

10. No need to check for sign.

11. Sometimes it is good to start finding a root using the bisection method then once you know you are close to the root you can switch to the secant method to achieve faster convergence.

13. Another problem of this method that does not know when to stop. It must be performed several times until the f of the current guess is very small.

15. 1.12 Secant Method: Failure The numerical values associated with the “failure” example are:

16. Regulafalsivs secant It is similar to regula falsie except:- Condition f(x1).f(x2)<0 Will convergence always. speed can be slow. No need to check for sign. Begin with a, b, as usual. Regula falsie a variant of the secant method which maintains a bracket around the solution. secant method keeps the most recent two estimates, while the false position method retains the most recent estimate and the next recent one which has an opposite sign in the function value.

17. Fig: comparision between secant and false pasition:

19. The secant method has the same properties as Newton’s method. Convergence is not guaranteed for all xo.

21. Geometrical Derivation

22. 18 By the concept of Similar Triangles in Triangle ABE and CDE It can be written as On rearranging, the secant method is given as