3
votes
1answer
28 views

Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
1
vote
0answers
22 views

What is the derivative of $\frac{f^{(3)}(\xi(x))}{6}$ at $x=x_0$

The error of interpolating polynomial is $$ E_n(x)=\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x)) $$ The derivative of $E_n(x)$ is $$ ...
0
votes
0answers
64 views

Using Lagrange polynomial to obtain the Second Derivative Midpoint formula

The Second Derivative Midpoint/Central Formula is $$ f^{\prime\prime}(x_0)=\frac{f(x_0-h)-2f(x_0)+f(x_0+h)}{h^2}-\frac{h^2}{12}f^{(4)}(\xi) $$ I tried to get this formula using Lagrange polynomial. ...
2
votes
1answer
33 views

Can any function on naturals be interpolated to a smooth function on reals?

Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be an arbitrary function from naturals to naturals. Is it always possible to find a function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that for any $n ...
0
votes
1answer
26 views

properties of third order derivative

I have a question in my assignment (about interpolation) which has the condition that the third-order derivative is continuous at $x_2$ and $x_{N-1}$. That is, $S'''(x_2)=S'''(x_{N-2})$. The question ...
0
votes
2answers
35 views

Polynomial interpolation using derivatives at some points

Given $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y_5)$, we can interpolate a polynomial of degree 4 using Lagrange method. But, when we are given $(x_1, y_1), (x_2, y_2), (x_3, y_3), ...
1
vote
2answers
51 views

How to find the second derivative?

I use this article from Wikipedia to build it in my program. How to find the second derivative in $(x_i, y_i)$ point of this cubic interpolation, if I know other $(x_j, y_j)$ points?
0
votes
2answers
50 views

How can I find the gradient of this function? $f(r,t)=r^3\cos(t).$

$$f(r,t)=r^3\cos(t).$$ Is it not like this: $<3r^2,-\sin(t)>$
1
vote
0answers
59 views

Differentiation by interpolation.

I am asked to show that the formula: $$ f'(x)\sim \sum_{i=0}^n A_i f(x_i) $$ which is derived from differentiating the interpolation polynomial is similar to that derived from checking/evaluating the ...
0
votes
1answer
185 views

How do I interpolate the derivative of a catmull-rom spline?

I am creating an implementation of a cubic hermite spline in Python. One feature I would like to add is a method to compute the slope (IE the derivative) for a given T value. Currently, I can do it ...
1
vote
2answers
333 views

Piecewise interpolation with derivatives that is also twice differentiable

This question regards the issue of interpolation of one dimension real functions. If one has a finite set of function values and its corresponding derivatives, one could find unique continuous ...
0
votes
1answer
195 views

finding derivative at intermediate point of known data set

I have a function $y = f(x)$, $ x \in [0,1] $ and $ y \in [0,1]$ Set of values $(x_i,y_i)$ are known for n points. I need to find derivative at point $x_{\zeta}$ such that $y(x_{\zeta}) = 0.5$ Now ...
1
vote
1answer
165 views

3D Numerical differentiation with spline approximation

I have three 3D matrices X, Y, and Z that define a matrix V of the same size over some region. The matrices are regularly spaced. I'm trying to compute the gradient of V. I have read that ...
3
votes
1answer
395 views

Determining partial derivatives and cross products for bicubic interpolation using function values only?

I'm trying to implement a bicubic interpolation algorithm. In order to calculate the interpolated values, I need to calculate sixteen coefficients used in the calculation process - and that's where ...
0
votes
1answer
269 views

Derivative of a function defined by the divided difference of another function.

Given a function $f$ of class $C$ $^{n+2}$ in an interval $[a,b]$ and $x_{0}=a<x_1<x_2 ... <x_n = b$ a subdivision of $[a,b]$ into $n+1$ points. Given another function $g$ defined in the ...
0
votes
1answer
93 views

Fixing end derivatives up to the second order when interpolating points

I would like to interpolate a set of points in the real plane $(x_i,y_i), \ 1\leq i \leq n$ with specified end derivatives up to the second order. That is finding $f \in ...