# Thread: Secant Method + Interpolation

1. ## Secant Method + Interpolation

So the problem goes,
The Secant Method for finding the root of a function f[x] uses two successive points xn, xn-1 to compute a projected root based on the secant line through xn-1 and xn. By using polynomial interpolation, design a "parabolic method" which fits a quadratic polynomial through three function values xn-2, xn-1 and xn.
Give an explicit formula for xn+1 in terms of (xn,f[xn]), (xn-1,f[xn-1]), (xn-2,f[xn-2])

I appreciate any advices and help!

2. Originally Posted by PianoGal
So the problem goes,
The Secant Method for finding the root of a function f[x] uses two successive points xn, xn-1 to compute a projected root based on the secant line through xn-1 and xn. By using polynomial interpolation, design a "parabolic method" which fits a quadratic polynomial through three function values xn-2, xn-1 and xn.
Give an explicit formula for xn+1 in terms of (xn,f[xn]), (xn-1,f[xn-1]), (xn-2,f[xn-2])

I appreciate any advices and help!
You have three points, write the quadratic equation that goes through these points, and then solve for the roots (keep that nearer to the mid point of the three you are using if possible)

CB

3. I used the Lagrange form to write an interpolating polynomial, just not sure how to present an explicit formula for xn+1. Since I used xn, xn-1, and xn-2 to form that polynomial!
Would the result be any different from using quadratic equation to solve?
Thanks!
Originally Posted by CaptainBlack
You have three points, write the quadratic equation that goes through these points, and then solve for the roots (keep that nearer to the mid point of the three you are using if possible)

CB