1
vote
0answers
29 views

Higher-order difference quotients

The Mean Value Theorem for Divided Differences says that if $f$ is $n$ times differentiable, and $x_0< x_1 < \dotsb < x_n$, then there is a point $\xi\in (x_0, x_n)$ such that $f[x_0, x_1, ...
0
votes
1answer
54 views

derivative B-spline with own knot set

Define the spline function of degree $q$ on the interval $[\xi_0,\xi_K]$ $$f(t)=\sum_{j=1}^{K+q}b_j B_j(t)$$ where $B_j$ are degree $q$ B-spline basis functions determined by the knots ...
1
vote
1answer
46 views

Why does secant method converge

Assume $f$ is continuous and twice differentiable on $[a,b]$ such that $f'(x)>0$ and $f''(x)>0$, $x \in [a,b]$. If $f(b)>0$ and $f(a)<0$ and I choose $x_0=a$,why are we gauraunteed ...
0
votes
0answers
39 views

Solving system of differential equations

I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us ...
0
votes
0answers
36 views

On an interesting boundary condition

So I am tackling an interesting boundary condition, where $B(Du)=0$, for $x\in\Omega$, where $B$ is the signed distance function to $\Omega^*$ (where $\Omega,\Omega^*$ are convex domains in $\Bbb ...
1
vote
2answers
37 views

Solution of $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$

I'm trying to prove that $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$,given that $A$ is symmetric positive-definite has only one minimum. I've found the derivative is $f'(x)=Ax-b$, and in order to find ...
3
votes
1answer
30 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 ...
0
votes
0answers
28 views

Implementation of Total Variation Regularization Algorithm (Lagged Diffusivity Algorithm)

I am trying to compute the derivative of an experimentally-measured quantity as a function of time. The data are fairly noisy, which causes problems. For instance, using finite differences (central ...
0
votes
2answers
43 views

Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
1
vote
0answers
23 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
2answers
86 views

Is there a general formula for estimating the step size h in numerical differentiation formulas?

Using three-point central-difference formula $$ f^{\prime}(x_0)\approx \frac{f(x_0+h)-f(x_0-h)}{2h} $$ and for $f(x)=\exp(x)$ at $x_0=0$ we have $$ \begin{array}{c, l, r} h & f^{\prime}(0) ...
0
votes
1answer
31 views

How to verify the gradient of a symbolic function using numerical gradient?

I have a function $f$, which takes as inputs a three arrays and returns an array. I have written a symbolic function $g$ to calculate the gradient of this function and I want to verify that it ...
0
votes
0answers
80 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. ...
1
vote
1answer
19 views

Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
6
votes
3answers
160 views

optimal way to approximate second derivative

Suppose there is a function $f: \mathbb R\to \mathbb R$ and that we only know $f(0),f(h),f'(h),f(2h)$ for some $h>0$. and we can't know the value of $f$ with $100$% accuracy at any other point. ...
2
votes
0answers
68 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
47 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
38 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
29 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
114 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
55 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
181 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
30 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
70 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
59 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
30 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 ...
1
vote
1answer
117 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
53 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
46 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
99 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
64 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
345 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
68 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
47 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
64 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
207 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
481 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
230 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 ...
4
votes
2answers
738 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
86 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
334 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
293 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
456 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
61 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
810 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
352 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
145 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
139 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 ...