2
votes
0answers
38 views

(newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
1
vote
0answers
25 views

Simpson's rule error rate for N-dimension

I'm doing a project that involves numerical method, but I'm not too familiar on calculus. I'm using Simpson's rule to integrate n-dimension gaussian, I was able to get the integration result for ...
0
votes
2answers
33 views

Numerical Analysis: Given a function and successive derivatives at one point, what's the value of the function at another point?

Example of an exercise I'm trying to solve: Find the value of $f ( 4)$ given that $f (6 )=350 , f ' (6 )=87 , f'' (6 )=30 , f ''' (6 )=4$ and all other higher derivatives of $f (x) at x=6$ are zero. ...
4
votes
1answer
20 views

Numerical computation of the $n^{\mathrm {th}}$ derivative of a multivariate function

From a multivariate function $f$, depending on $n\geq 1$ variables, which can be computed numerically, but which does not admit simple analytic expression, I would like to approximate numerically the ...
2
votes
1answer
57 views

Numerically calculate the second “left hand” derivative

The Problem I have a series of measurements for which I have to calculate the first and second derivative numerically in a "live" fashion, i.e. using only previous data. This is easy for the first ...
2
votes
1answer
51 views

Knowing if the real derivative exists

The numerical derivative is valid only if the real derivative exists. Is it possible to know if the real derivative exists without using symbolic derivative, and using computer operations?
2
votes
0answers
146 views

Runge-Kutta method accuracy

I got Runge-Kutta method here and I solve this system using it. So here's Runge-Kutta stuff $k_1 = f(t_n, y_n)$ $k_2 = f(t_n + h/2, y_n + hk_1/2) $ $k_3 = f(t_n+h, y_n - hk_1 + 2hk_2)$ $y_{n+1} ...
0
votes
1answer
29 views

Methods of computing the derivative of vector norms

I am very new to norms. Except the basic definitions and properties of the norm, I don't know too much about it. Now, I am very interested in computing the derivative of the norms. So, I am wondering ...
0
votes
1answer
37 views

numeric differentiate: show that the relative mistake can be at 100%

i have $f(x) = x+1$, a physical size, and the values $\tilde{f}(x_i)$ are measured at equally spaced points $$x_i=ih, \qquad 0 \leq i\leq 10^3, \qquad h=10^{-3},$$ with a maximum relative mistake of ...
1
vote
2answers
57 views

Develop second-order method for approximating f'(x)

I am stuck on the following question: Develop a second-order method for approximating $f'(x)$ that uses the data $f(x-h), f(x)$, and $f(x+3h)$ only. Any hints/tips?? Thanks!
1
vote
2answers
57 views

Finding A,B,C s.t $f'(a)+O(h^2)=\frac{Af(a)+Bf(a+2h)+Cf(a+3h)}{h}$

Find constants A,B,C s.t for differtiable three times function f, $f'(a)+O(h^2)=\frac{Af(a)+Bf(a+2h)+Cf(a+3h)}{h}$ I know that $f'(a)+O(h^2)=\frac{f(a+h)-f(a-h)}{2h}$ so I need to solve ...
0
votes
1answer
25 views

Second Order forward finite difference scheme

Show that $d^2u/dx^2(x_i)=[(-u_{i+3})+(4u_{i+2})-(5u_{i+1})+2u_i]/h^2 +O(h^2)$ provided all terms in the expression are well defined is a second order finite difference scheme for second order ...
0
votes
0answers
93 views

Numerical differentiation with non-uniform step sizes?

Most of text I've read so far about numerical differentiation with higher-order methods has a fixed step size, including the one described on Wikipedia. I have a set of data from a simulation from ...
1
vote
1answer
101 views

Picard method to solve a diferential equation

I have to obtain by Picard method the solution to this problem. $$x'=x+t, x(0)=x_0$$ doing $$x_j=x_0 + \int_{0}^{t} f(s,x_{j-1}(s))ds$$ i have obtain ...
2
votes
0answers
51 views

To calculate a derivative of a set of points, is it more correct to interpolate finite differences or to derivate the interpolation?

I have a series of points extracted from numerical simulations. I also recently discovered the amazing power of finite differences. Nevertheless, I was used to estimate my derivatives from the ...
0
votes
0answers
43 views

How to choose a numeric approach for derivates

I would like to find the derivate from some combined logistic and exponential functions that all describe the same data numerically. $$f'(t)=\frac{f(t+h)-f(t)}{h}$$ seems not the best choise for ...
0
votes
1answer
93 views

Condition or Proof: Minimizer of one function is maximizing another function

I have two real functions $f(X),g(X)$ where the argument $X$ is a real matrix. The solution $X^*$ for the problem of minimizing $f$ is ending up maximizing $g$ as well. I am looking for a way to prove ...
2
votes
0answers
55 views

automatization of numerical derivation

I would like to know if there is an automized or fast way to numerically derivate a large number of tab-delimited files (derived from the program kaleidagraph) and to automatically extract some key ...
7
votes
2answers
263 views

In what sense is the derivative the “best” linear approximation?

I am familiar with the definition of the Frechet derivative and it's uniqueness if it exists. I would however like to know, how the derivative is the "best" linear approximation. What does this mean ...
2
votes
1answer
66 views

Can we use (higher order) derivatives to help integration?

I'm wondering if it's possible to use derivatives to ease the evaluation of an integral. For instance, I know that to evaluate an integral with enough precision I need to evaluate it at $n$ points. ...
1
vote
1answer
46 views

Why no roundoff error when dividing by h in this approximation

Why the approximation $f'(x) = ( f(x+h) - f(x-h) ) / 2h$ produces no roundoff error if $h = 2^{-k}$, where $k$ is any integer. Thanks very much in advance.
1
vote
1answer
58 views

Discrete numerical derivative with respect to d/d(n*x)

How can I generate a stencil for a d/d(n*x) operator? I am writing a program that needs a method to calculate line derivatives in an image. If we want to calculate the simplest forward derivative ...
0
votes
1answer
181 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 ...
2
votes
2answers
394 views

Trapezoid rule error analysis

How can I prove that the max error of the trapezoid rule for the integral $\int_{a}^{b}{f(x)\, \mathrm{d}x} $ is: $$\Delta=-\frac{1}{12n^2}f''(c)(b-a)^3 \text{for } c \in (a,b) \ ?$$ I know that to ...
2
votes
2answers
211 views

Integral of $f(x) \exp(ikx)$ with finite bounds calculated using Fourier transform, and its derivative

I have an integral which I need to calculate numerically along the lines of $$ I(k)=\int_0^{L} \exp(i k x)f(x) dx $$ where $x$ and $L$ are real. $f(x)$ is not necessarily periodic and differentiable ...
3
votes
2answers
505 views

How to obtain (prove) 5-stencil formula for 2nd derivative?

My question seems pretty easy. Prove the correctness of the following approximation: $$f(x)''= \frac{-f(x-2h)+16f(x-h)-30f(x)+16f(x-h)-f(x+2h)}{12h^2}$$ I rendered myself deeply saddened upon ...
0
votes
0answers
83 views

Numerical derivative without knowing change in variable

Suppose that we have a multivariate function $f(a,b,c,a_1,b_1)$, where $a,b,c,a_1,b_1$ are all real numbers. $f = \frac{{c({a^2} + {b^2}) + {{({a^2} + {b^2})}^{1/2}}}}{{c{{({a^2} + ...
1
vote
2answers
278 views

How do I calculate numerically a tensor in polar coordinates?

You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid? a) fit a (spline) surface, ...
0
votes
1answer
223 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 ...
3
votes
1answer
418 views

Given a cubic function, and its quadratic derivative- can I recover the cubic from quadratic?

Background: I'm trying to learn how to work with cubic and quadratic bezier splines for various drawing libraries, and working through how to approximate a cubic spline with a quadratic spline. It's ...
0
votes
0answers
59 views

Percentage variation dependance of a function of two variables

The language is a sort of barrier in this case (even in my native language) so I'll try to make an example here to clarify the question. Given a function $f(a,b)$ I want to answer the question: to ...
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 ...
0
votes
2answers
135 views

Deriving the formula and its error

$$f''(x) \thickapprox\dfrac{1}{2h^2}[f(x+2h) - 2f(x) + f(x - 2h)]$$ I'm supposed to be deriving the above formula and establish an error formula in using them. This is one of a series of problems ...