Tagged Questions
0
votes
0answers
45 views
Simpson's rule characteristics
I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that:
...
0
votes
1answer
45 views
What is meant by the norm of this object?
I want to know how to interprete this norm
$|\nabla\nabla f|$, where f is a vector?
Is it the Hessian? I think its a tensor? But how does it look like? I mean the components....
0
votes
0answers
17 views
Numerical methods for ODE
(excuse my english in advance)
I have an ODE $$\dot{\alpha}(t,x)=V(t,\alpha (t,x),\lambda)$$
$$\alpha(0,x)=x$$ where the field $V$ is $C^{\infty}$ with respect to $t$,$x$ and a vector of parameters ...
-2
votes
1answer
57 views
Numerical Analysis & Optimization
Write a function that computes accurate values of $$f(x) = (x+1)^{1/3} −x^{1/3}$$ for $x>0$.
0
votes
0answers
92 views
Using Secant Method to find number of roots
I have a discrete function,
$ y=F(N) $
where $N$ is positive integer, and I don't know number of roots of this function.
Can I use Secant Method to find number of roots?
For example, If this function ...
0
votes
1answer
156 views
Chebyshev rational approximations to $\cos x$
How can we construct all the Chebyshev rational approximations of degree $3$ for $f(x) = \cos(x)$.
So, I note that we first get the Taylor series of $\cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} ...
1
vote
1answer
528 views
Orthogonal polynomials and Gram Schmidt
How can we use the Gram Schmidt procedure to calculate $L_0,L_1, L_2, L_3$, where ${L_0(x), L_1(x), L_2(x), L_3(x)}$ is an orthogonal set of polynomials on $(0, \infty)$ w.r.t. the weight function ...
3
votes
2answers
92 views
a question on $O$- notation
I'm trying to understand $O$-notationbetter. I've found a really helpful answer : Big O notation, $1/(1-x)$ series
But I have trouble with quotient. Let me discuss about this example function:
...
-1
votes
3answers
103 views
number with finite binary representation and infinite decimal representation
One can easily find numbers with finite decimal representation with infinite binary representation. (Like $0.3$ and $0.01010101..$)
I assume there is an opposite case, meaning a number with finite ...
1
vote
1answer
82 views
Creating a 3D surface from 2D graphs
So I have two sets of equations:
$\mathcal{A}$ = \begin{equation}
\{ f(y_{0},x), \, f(y_{1},x) , \;... \;, f(y_{n},x) \}
\end{equation}
$\mathcal{B}$ = \begin{equation}
\{ g(y,x_{0}), ...
0
votes
1answer
483 views
Relationship between rate of convergence and order of convergence
What is the difference between rate of convergence and order of convergence? Have they any relationship to each other? For example could i have two sequences with the same rates of convergence but ...
0
votes
0answers
59 views
relative error relation
Let $x$ be a non-null quantity. Let $\hat{x}$ be its approximation. I am trying to find the relation between: $\frac{\left | x-\hat{x} \right |}{\left | x \right |}$ and $
\frac{\left | x-\hat{x} ...
0
votes
0answers
27 views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega ...
0
votes
0answers
34 views
Proof of convergence Finite Volume
Hi all can some one give me a good reference to the proof of convergence and error estimation of ELLIPTIC PDE in divergence form by a Cell centered Finite Volume Method..???
Arwin
2
votes
2answers
130 views
Stability analysis of this differential equation
Below is a problem I recited from an exam I took. I wasn't able to solve it on time. Could someone show me how to simplify it?
If
$$f(t,y(t)) = y'(t)$$
$$y'(t) = -\lambda y(t)$$ where $\lambda$ is a ...
0
votes
0answers
24 views
Optimal Expansion of a complex function
I have the following problem:
I want to find an optimal exponential expansion of a function
$$f: \Re \rightarrow C,$$
which (I suppose) could be expanded in series
...
1
vote
2answers
133 views
Numerical integration given a derivative of a function of two dependent variables
I want to solve the following equation of an integral valued function:
$Q = \int_{0}^{x_p}f(t_p,x)dx$
for some particular $x_p$ at a fixed time $t_p$, given some known $Q$ and an initial $f(0,x)$. ...
2
votes
0answers
163 views
calculating the amplitude of a cosine function
I want to be able to be able to get the amplitude of the following function:
$$||A||\cos(2 \omega t + a)+||B||\cos(3 \omega t +b)+||C||\cos(5 \omega t +c)$$
I am trying to find a way to get the ...
0
votes
2answers
167 views
Mapping a variable having very vast range to the interval (0,1)
I am trying to get a continuous mapping from (0,∞) to (0,1). What would be a good mapping?
Context:
I need to rank tuples based on values of two of their attributes ...
1
vote
0answers
141 views
Finding a force function from bodies in equilibrium
(This is an edited version of the original question, since I'm starting a bounty)
I'm trying to find a function $y$ from given data. Reverse optimization, so to speak.
Say we have two ...
1
vote
1answer
74 views
How to solve quadratic over square root of quartic equals constant?
We need to solve the following equation for l.
$$\frac{n_1 - \ell n_2 + \ell ^2 n_3}{\sqrt{d_1 - \ell d_2 + \ell ^2d_3 - \ell ^3d_4 + \ell ^4d_5}} -
\cos{a_0} = 0$$
We have already tried ...
1
vote
1answer
113 views
convergence / fixed point method
Any help with the following:
Problem:
Consider the fixed point problem: $x=f(x)$ and given: $x_{n+1}=\frac{n}{n+1}f\left ( x_{n} \right )$. If $x_{0}$ is a fixed point where $\left | f^{'}\left ( ...
0
votes
1answer
253 views
Could explain me how eigenvector helps with compute gradient and how make differentiate operation on decrete space like digital image?
Could you explain me how eigenvector helps with compute gradient and how make differentiate operation on descrete space like digital image?
I know that this question is so connected with computer ...
3
votes
2answers
94 views
Is there an algebraic solution to $e^{-x/a}+e^{-x/b}=1$ ($a\neq b$, $a,b$ constants)?
Is there an algebraic solution for the to find the intersection of the following two functions for values of $x\geq 0$:
$$f_1(x)=1-2e^{-x/a}=f_2(x)=-1+2e^{-x/b}$$
$a$ and $b$ are positive constants.
...
1
vote
0answers
54 views
Solving an equation in several variable of the form $a_1x_1x_2+a_2x_2=c$
Consider the following equation in two variables
$$a_1x_1x_2+a_2x_2=c$$
where $x_i$ are variables and the other constants can be any real numbers.
In three variables, this is
...
2
votes
2answers
81 views
Interpolating polynomials
So I have this question on a homework and I just can't seem to figure it out.
Let $f \in C^4 [0,1]$ and let $p$ be a polynomial of degree $\le 3$ such that $p(0) = f(0)$, $p(1) = f(1)$, $p'(0) ...
1
vote
0answers
52 views
solution for related function
If we have $f(x)=(2A(A+\delta)-1)x^{2}-2Ax+1$, then the values of $x$ are equal to $\frac{1}{A+\sqrt{1-A(2\delta+A)}}$ and $\frac{1}{A-\sqrt{1-A(2\delta+A)}}$. The question is how to find the ...
4
votes
1answer
225 views
Series expansion for iterated function
I would like to find the MacLaurin expansion of an iterated function. Finding the first few terms is not hard, but it doesn't take long before Mathematica runs out of memory using the straightforward ...
5
votes
2answers
385 views
Lagrange Inversion of power series for fractional exponents?
I understand how they obtained the inversion of sin(x) shown here, using the Lagrange Inversion Formula, and have even written a MATLAB script to solve the inversion when input and output exponents ...
0
votes
2answers
57 views
a numerical concluding about (a/(a+b)) and (c/(c+b))
Let $a,b,c$ be three integers greater than $0$, and assume there is a real number $t$ such that
$$
\frac{a}{a+b}=\frac{\left\lfloor t\right\rfloor}{\left\lfloor t\right\rfloor+1}.
$$
Is there a way to ...
1
vote
1answer
86 views
Help with the proof of a Gaussian type inequality and some numerical results
I am trying to figure out if I made a mistake in the following proof - in particular I have been trying to verify the inequality (*) below. I have attached my proof but I have been getting some error ...
0
votes
1answer
1k views
Prove Simpson's rule (including error) using the integral remainder
I have to prove Simpson's rule including the error with the help of the integral remainder. However, I have practically no idea how to start.
Let $f: [a,b] \rightarrow \mathbb{R}$ be continuously ...
6
votes
2answers
166 views
How to prove the error estimate of the Newton-iteration?
I'm trying to get familiar with the Newton-iteration over here but I got stuck at the proof of the error estimate.
Let $f: [a,b] \rightarrow \mathbb{R}$ be continuously differentiable twice, concave ...
3
votes
1answer
290 views
Approximating an integral representation of the Number Partition Problem
One can write out an integral whose solution gives the number of solutions to the NP-Complete Number Partition Problem and I'm wondering if anyone has an suggestions or ideas on who to solve or ...
2
votes
1answer
99 views
Efficient Sampling
I'm trying to sample a lot of points efficiently. I'm wondering if the following method is possible.
I sample points of a function (evaluate the function) mod $n$. I.e. I calculate f(element one), ...
6
votes
3answers
455 views
Numerically estimate the limit of a function
Is there an algorithm that will allow me to numerically compute the limit of a function f(x) in a principled way?
The most naive algorithm would be to continue to compute the function for larger ...
3
votes
1answer
996 views
Numerically Solving a Second Order Nonlinear ODE
Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically.
$$f''(R) + \frac{2}{R} f'(R)=\frac{0.7}{R} \left( \frac{1}{\sqrt{f(R)}} - \frac{0.3}{\sqrt{1-f(R)}} ...
5
votes
1answer
127 views
applications of the “soft maximum”
There is a little triviality that has been referred to as the "soft maximum" over on John Cook's Blog that I find to be fun, at the very least.
The idea is this: given a list of values, say ...
