So I'm really kinda bummed out because I just don't get how to solve numerical differentiation problems with linear algebra. As an example :
Estimate f ' (-1) if f (1) = 3, f (0) = -2, f (-2) = -18.
I'd like to say I have an idea on how to at least set up the problem, but to be honest I have no idea. I would think we to use the transpose of the vandermond matrix like we would in numerical integration. But I'm not positive.
Sorry, I know I'm essentially asking you to guide me through step by step from the beginning, but I don't know what else to do as I've read the section in my textbook multiple times and I'm still lost.
in our book it's called numerical differentiation. Here I'll write down what my book says.
"suppose that f is a differentiable function and we wish to estimate the value f ' (a), where f is differentiable at t = a.
Let p be the polynomial of degree n that interpolates f at t0, t1, ... , tn, where the interpolation nodes t1 are clustered near t = a. Then p provides us with an approximation for f, and we can estimate the value f ' (a) by evaluating the derivative of p at t = a.
f ' (a) = p ' (a)
As with a numerical integration formula, it can be shown that the value p ' (a) can be expressed as p ' (a) = A0 p(t0) + A1 p(t1) + ... + An p(tn)
the weights Ai are determined by the system of equations:
q0 ' (a) = A0 q0 (t0) + A1 q0 (t1) + ... + An q0 (tn)
.
.
.
qn ' (a) = A0 qn (t0) + A1 qn (t1) + ... +An qn (tn)
where q0(t) = 1, q1(t) = t,...., qn(t) = t^n
"
Does that tell you anything? If not I could possibly type out one of the examples they give.
There are many different ways, and different answers, to "estimate" anything. I would suggest fitting a quadratic equation to the curve, then differentiating that at x= -1. If then , , and . Of course, . If this is a problem specifically asking you to do the numerical differentiation with "linear algebra" then I would suspect there is a formula in the text book close to this problem! Of course, you can use linear algebra to solve for a, b, and c.