A proof in Numerical Analysis

[pinodk]

Hello there!
yet another proof, that i need help on
I am supposed to prove that the following statement holds for the secant method
dk+1/ek -> -1 for k->Infinity
where
dk+1 is the next change and ek is the error.

I have this idea, but i want to hear whether its a valid proof.

i use the expression for the secant method

xk+1 = xk - f(xk) * ( xk-xk-1/f(xk)-f(xk-1) )

and derive that
dk+1 = xk+1 - xk = - f(xk) * ( xk-xk-1/f(xk)-f(xk-1) ) (1)

I then use an expression in the lecture book, saying that
f(xk) = ek* ( f(xk)-f(xk-1)/xk-xk-1 ) - (ek-1*ek * f''(xa)/2 )

My argument is then that for k->Infinity, i will get that - (ek-1*ek * f''(xa)/2 ) goes towards zero. xa is in the interval between the exact solution and the current x, xk.
This is the part that I am not sure if I am right about, can i argue like this?

I then get the following expression

f(xk) = ek* ( f(xk)-f(xk-1)/xk-xk-1 )

Where I use the expression (1) and get
f(xk) = ek* (- f(xk) /dk+1)
Ánd from this I get
dk+1/ek = -1

Cheers
-Daniel

PS: How do you make those javascript math expressions I've seen in some of the posts?

 

[Nate810]

Yes, your proof is valid. To make a math expression, you can use LaTeX notation. For example: \frac{2}{3} will output $\frac{2}{3}$.

 

[quicknote]

Hello Daniel,

Your proof looks valid to me. You have correctly used the expression for the secant method and derived an expression for dk+1. Your argument for f''(xa) going towards zero as k->Infinity seems reasonable, as xa is in the interval between the exact solution and the current xk. Your final expression, dk+1/ek = -1, satisfies the statement you were asked to prove.

As for the javascript math expressions, they are typically created using a scripting language such as MathJax or LaTeX. These languages allow for the formatting of mathematical equations and symbols. There are many tutorials and resources available online to learn how to use them. I hope this helps!