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Given $f:R^n\rightarrow R^n$ continuously differentiable in some convex open set $D$ and $x,x+p\in D$, taylor's theorem is given as:

$f(x+p)=f(x)+\int_0^1 J(x+tp)p \, dt,\text{ where }J\text{ is the jacobian of }f.$

I'm trying to follow a given proof which states that given $f(x^*)=0$ for some $x^*\in D$ and $\{x_k\}$ defined as the sequence of Newton's Iterations to approximate $x^*$, we have

$f(x_k)=f(x_k)-f(x^*)=J(x_k)(x_k-x^*)+\int_0^1[J(x_k+t(x^*-x_k))-J(x_k)](x_k-x^*) \, dt$

However, when I try to derive the right hand side of this equation, I obtain:

$f(x_k)-f(x^*)=\int_{x_k}^{x^*}J(z) \, dz$ which with the change of variables $z=x_k+t(x_k-x^*)$, I only obtain

$f(x_k)-f(x^*)=\int_0^1 J(x_k+t(x_k-x^*))(x_k-x^*) \, dt$

I'm not sure how to obtain the other terms. I'm sure there's something that I'm missing, but I'm not sure what that is. Any help would be greatly appreciated! :)

1 Answers 1

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If you want to verify the identity then you are almost there. You can use the adding 0 trick.

First observe that $\int_{0}^1 J(x_k)(x_k-x^*)dt=J(x_k)(x_k-x^*)$ since the integrand is constant with respect to $t$. Then observe that $\int_{0}^1 J(x_k+t(x_k-x^*))(x_k-x^*)dt$ $=J(x_k)(x_k-x^*)-J(x_k)(x_k-x^*) + \int_{0}^1 J(x_k+t(x_k-x^*))(x_k-x^*)dt$ $=J(x_k)(x_k-x^*) + \int_{0}^1 J(x_k+t(x_k-x^*))(x_k-x^*)-J(x_k)(x_k-x^*)dt$ $=J(x_k)(x_k-x^*) + \int_{0}^1 [J(x_k+t(x_k-x^*))-J(x_k)](x_k-x^*)dt$.