3
votes
2answers
85 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
0
votes
0answers
26 views

How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and ...
0
votes
0answers
2 views

Multidimentional Scaling with Pairwise distance “vectors”

Consider a random variable $z$ with a Gaussian distribution : $$ \mathbf{z} \sim \mathcal{N} ( \mathbf{m}, \mathbf{V} ) $$ Where $\mathbf{m}$ and $ \mathbf{V}$ are mean and variance parameters. ...
0
votes
0answers
38 views

Efficient approximation of derivatives of an integral

Suppose $ \phi(z) $ is the probit function (http://en.wikipedia.org/wiki/Probit). And $$ Z = \int \phi(\mathbf{w}^\top \mathbf{x}) \mathcal{N}(\mathbf{w}; \mathbf{\mu}, \mathbf{\Sigma}) d\mathbf{w} ...
0
votes
1answer
63 views

How do I use the linear approximation of a function given a value, a, and change in x?

My book gives a few definitions/formulas for obtaining linear approximation, but I'm having trouble understanding how to use them. Heres the question: a.) Use the Linear Approximation for f(x) = ...
1
vote
1answer
84 views

Differentiability Theorem Question

$f(x,y) = \begin{cases} \frac{1}{2} y \log(x^2+y^2), & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$ You may assume that this is a continuous function. Prove that f does not satisfy ...
1
vote
1answer
24 views

The value of $w$ also has a max error of $p\%$

Suppose $\frac{1}{w}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ where each variable $x,y,z$ can be measured with a max error of $p\%$ Prove that the calculated value of $w$ also has a max error of $p\%$ ...
0
votes
1answer
44 views

curve fitting with known first derivatives

I need to find a polynomial function $f(x)$ minimum order 2 that best satisfies the following it passes through points (x1,y1) and (x2,0) it is known that $f'(x_{1})=A$ and $f'(x_{2})=B$ with ...
4
votes
1answer
126 views

Find the order of the error for the approximation $f' '(x)$

Given $$f''(x) = \frac{ f(x+h) - 2f(x) + f(x-h)}{h^2}.$$ I realize that this is just an approximation - that it won't give the exact value of $f''(x)$ and therefore there is an error term. However, I ...
3
votes
1answer
653 views

Method for estimating the nth derivative?

When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
1
vote
2answers
281 views

Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ...
4
votes
1answer
142 views

Numerical differentiation issues

I've been using this to compute the first order derivative's value of a function $f$ in a given point: $$f'(x) = \frac{f(x+\epsilon) - f(x-\epsilon)}{2\epsilon}$$ For some $\epsilon = 0.0001$ or ...