For questions related to approximations
0
votes
0answers
14 views
Calculate the tendency of a set of samples
I develop an application in which I constantly get samples of heart pulse.
I defined an interval of $t$ seconds.
In each $t$ seconds I have $n$ samples.
In every interval, I want to calculate the ...
0
votes
0answers
30 views
FF neural network with single sigmoid output (calculation of probability)
first of all I'm sorry for my not very skilled English, but I will do my best to explain my problem.
I'm trying to create a feedforward neural network with one hidden layer (with probably arctan ...
4
votes
2answers
52 views
Does this ODE have an exact or well-established approximate analytical solution?
The equation looks like this:
$$\frac{\mathrm{d}y}{\mathrm{d}t} = A + B\sin\omega t - C y^n,$$
where $A$, $B$, $C$ are positive constants, and $n\ge1$ is an integer. Actually I am mainly concerned ...
0
votes
1answer
21 views
Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$
I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$
...
1
vote
0answers
25 views
${{\left( 1-p \right)}^{n}}{{\text{ }}_{1}}{{F}_{1}}\left( n+1,n+a+1,\lambda T \right)\approx ?$
I am trying to approximate
$$\frac{p\left( n+a,\bar\lambda T \right){{\left( 1-p \right)}^{n}}_{1}{{F}_{1}}\left( n+1,n+a+1,\lambda T \right)}{p\left( n+a,\bar\lambda T \right)}={{\left( 1-p ...
2
votes
1answer
26 views
Approximate radius of a group of n packed circles
I am looking for a formula to estimate the radius of a circle which would hold n number of circles with some radius r. I understand this is part of the packing problem which does not have a definite ...
1
vote
1answer
70 views
Newton-Raphson in $\mathbb{R}^n$
Let $U\in\mathbb{R}^n$ be open and $f:U\to\mathbb{R}^n$ be a $\mathcal{C}^1$ map. $\exists p\in U$ such that $\;f(p) = 0$ and $Df_{|p}$ is invertible. Define $\phi(x)= x-(Df_{|x})^{-1}f(x)$, show ...
1
vote
1answer
43 views
Weighted sum of ratio of partial sum of binomial coefficients
I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$,
$$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose ...
0
votes
0answers
32 views
Complete normed vector space
I came across this question and was not sure how to proceed. Is the normed vector space of all polynomial functions on [0,1] complete with respect to the infinity norm? Thank you
0
votes
0answers
43 views
Simpson's Rule derived from Trapezoidal Rule
I was just wondering if I could have some assistance in regards to the Trapezoidal Rule and Simpson's Rule.
I have a question where it asks to generalize the Trapezoidal Rule to the case of ...
2
votes
1answer
27 views
Approximating a log-power function
I can't figure out how the following approximation has been done, I would appreciate any guidance:
...
1
vote
1answer
27 views
A question about an expoential function
I got an exponential function as follows
$\displaystyle 1-\frac{1}{x}+\frac{e^{-x}}{x}$
Does anyone know how to approximate such a function in a simpler term? Many thanks!
2
votes
1answer
46 views
Choosing degree of Chebyshev approximation
Chebyshev approximation approximates a function by fitting a weighted sum of Chebyshev polynomials to it. It requires evaluating a function at $N$ sample points to form the weighting coefficients. ...
1
vote
1answer
44 views
Inaccuracy in numerical calculation of arclength of part of an ellipse
I am trying to numerically calculate the arclength of part of an ellipse according to:
$$ L = \int_0^{\phi_s}\sqrt{r^2+\left(\frac{dr}{d\phi}\right)^2} d\phi $$
where $r$ is defined as:
$$ ...
2
votes
3answers
41 views
Series evaluated to $m$ terms, approximating the error
Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms?
$$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$
...
4
votes
3answers
54 views
Taylor Series for $e^x$ where $x = 1$, estimating the Error
I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by ...
1
vote
0answers
34 views
Small Inhomogeneity of Differential Equation
Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$:
$$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=0$$
Let ...
0
votes
2answers
39 views
Continuation of smooth functions on the bounded domain
Given a bounded domain $\Omega\subset \mathbb{R}^n$ and a smooth function $f$ with bounded derivatives on $\Omega$, is it possible to extend $f$ to $\tilde{f} : \mathbb{R}^n \to \mathbb{R}$ such that ...
0
votes
0answers
22 views
Product of Standard uniform & CLT
Suppose that $U_{i},\dots,U_{n}$ are iid $U(0,1)$. Use the central limit theorem to find an approximation for:
$$P(U_{1}\times U_{2} \times\dots\times U_{25}\leq 6\times 10^{-6} )$$
Answer:
Using ...
1
vote
0answers
34 views
Smooth Approximation of $L^p$ function
Given a bounded domain $\Omega \subset \mathbb{R}^n$, is it possible to approximate every $L^p(\Omega)$ function (where $1\leq p < \infty$) by smooth functions $\mathit{C}^{\infty}$ ?
1
vote
0answers
17 views
Is there a way to estimate the range of fitting coefficients from only the data?
Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain
$f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$
where ...
0
votes
1answer
70 views
Confused by the answer of this Riemann sum approximation question.
$$\begin{array}{ | c | c | } \hline
n & \sum \limits_{k=1}^{n} \left(\frac{1}{x_k}\right)\left(\frac{1}{n}\right) \\ \hline
100 & 5.19 \\ \hline
200 & 5.88 \\ \hline
300 & 6.28 ...
2
votes
1answer
78 views
Application of Runge's theorem
Runge's theorem states:
Let $K$ be a compact subset of $\mathbb C$ and let $S\subset \overline{\mathbb C}\setminus K$, such that $S$ contains at least one
point in each connected component of ...
1
vote
1answer
35 views
Properties of Lebesgue functions
If $f\in \mathcal {L}$ then there exists a sequence $\{f_k\}$ of step functions s.t. $\lim_{k\to\infty} f_k(x)=f(x)$ for almost all $x$ and $$\lim_{k\to\infty} \int|f(x)-f_k(x)|\,dx=0.$$ If I have ...
2
votes
1answer
42 views
How to use undefined value in Composite Simpson's Rule
I have to use the Composite Simpson's Rule to approximate the integral $\int_0^1 t^2\cdot sin(\frac{1}{t}) dt$. I've used the Composite Simpson's Rule, but when I work through the algorithm, one step ...
1
vote
2answers
73 views
Weierstrass Approximation Theorem for continuous functions on open interval
I am studying for my introductory real analysis final exam, and here is a problem I am somewhat stuck on. It is Question 2, in page 3 of the following past exam (no answer key unfortunately!):
...
0
votes
0answers
57 views
How to find integer solutions of an equation using approximation methods?
If I have a function called $f(x)$ that have several roots, integers and not integers. How can I find just the integer ones by approximations methods?
A simple example would be ...
2
votes
1answer
22 views
Handling matrix of differential operator when using the Ritz method for an extremum problem
The internal energy (or strain energy) of a typical structural member under a displacement field $\{u\}$ can be represented as:
$$
U = \frac{1}{2}\int_{V}[u]^T[B]^T[F]^T[B][u]dV
$$
where $V$ is the ...
0
votes
0answers
13 views
Can a measurable function in $R^d$ be approximated by step function? [duplicate]
Consider the question here: How to show that a measurable function on $R^d$ can be approximated by step function?
I have work on it for a while, now I can show that (by essentially apply Egorov's ...
2
votes
0answers
37 views
Useful approximation of the pdf
Good day to everyone.
In my research work I came out with a function, which looks like this (it is the pdf of some random variable):
$$f(x,\rho,\psi)=\frac{2}{\pi }+\sqrt{\frac{2}{\pi }} ...
0
votes
1answer
66 views
How to show that a measurable function on $R^d$ can be approximated by step functions?
In Stein's Book: Real Analysis, Theorem 4.3 says that any measurable function $f$ on $R^d$ can be approximated by step functions. But the proof provided there only show that when $f=\chi_E$, with ...
0
votes
0answers
35 views
Approximation of $\tan$, $\tanh$, $\sin$, $\cos$, …
Is there a 'simple' way of approximate all these math functions?
I'm interested in $\tanh$, $\cos$, $\sin$, $\arccos$ and much more :)
Im searching for a way to implement these functions by my own ...
2
votes
0answers
21 views
$\epsilon$-net of a $n$-dimensional $\ell_2$-ball
Let $B$ be an $\ell_2$-ball of radius $r$ in $\mathbb{R}^n$.
I want to find the cardinal of a (not too big) $\epsilon$-net of $B$, that is the cardinal of a finite set $V\subset B$ such that $\forall ...
4
votes
1answer
33 views
Find the order of the error for the approximation $f' '(x)$
Given $$f''(x) = \frac{ f(x+h) - 2f(x) + f(x-h)}{h^2}.$$
I realize that this is just an approximation - that it won't give the exact value of $f''(x)$ and therefore there is an error term. However, I ...
2
votes
4answers
90 views
Analytical approximation of an integral
I think there is no analytical solution for
$$
\int_{K}^{\infty} \frac{exp(-x)}{x} dx
$$
where $K > 0$. Instead, is there an analytical approximation?
2
votes
1answer
35 views
Approximation of a function
Let $f:[0, 1] \rightarrow [0, \infty]$ be a function of $x$, with a parameter $\theta > 0$, such that
$f$ is continuous
$f$ is strictly decreasing
$f(0) = \infty$
$f(1) = 0$
For example, $f(x) ...
0
votes
0answers
23 views
approximate measurable function by continuous ones [duplicate]
Say that I have a measure space $(X,\mu)$ and a measurable function $f$ which is non-negative and bounded from above. $\mu(X)<\infty$. Now, the approximation under my concern is almost everywhere ...
2
votes
2answers
73 views
Asymptotic expansion for $\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du$?
Is there an asymptotic expansion for the function:
\begin{equation}
g(x)=\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du,
\end{equation}
over the domain $x\in [0,\infty)$ in terms of ...
1
vote
0answers
53 views
Expansion in powers
Let $n=2k, k \in Z_+$. Let
$$P_k\left(\frac{t}{\sqrt n}\right)=n!\sum_{\begin{smallmatrix} n_1+\ldots+n_k=n \\ ...
0
votes
3answers
114 views
Approximating the integral $\int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$
How would I go about evaluating $I = \int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$ to four decimal accuracy?
1
vote
0answers
70 views
Approximations of the incomplete elliptic integral of the second kind
For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
3
votes
2answers
107 views
Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$
I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer.
I am interested to find the bounds on the value it can take or an ...
1
vote
1answer
41 views
Find taylor polynomial that approximates e^x with accuracy at least 1.
Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$.
I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
1
vote
1answer
27 views
What is a fitting parameter in least-squares?
I was doing some least-squares homework when I saw this term "fitting parameters".
I was asked to implement the leas-squares fit using polynomials of $p-th$ degree to a generic dataset. This is done.
...
1
vote
2answers
259 views
Power series for $\ln(1+x^2)$
In the problem I am asked to use a power series representation of $\ln(1+x)$ to approximate the integral from $0$ to $0.5$ of $\ln(1+x^2)$ to within 4 decimal places.
So far I have found a series for ...
0
votes
1answer
21 views
Approximation related to resonance
Can someone help me with this problem.
We have $$x(t)=N \sin (w_{0} t)+\frac{w_0}{w_1}e^{\frac{-t}{T}}\sin (w_{1}t)$$
and $w_1=(1+\frac{\delta_1}{N^2})w_0$ for some $|\delta_1|\leq 1$.
I need to ...
1
vote
2answers
31 views
Formula for the pseudofrequency using approximations
We know that $w_1=\frac{\sqrt{4w_{0}^2-b^2}}{2}$ and $(1+u)^{\alpha}=1+\alpha u + \mathrm{O}(u^2)$
I need to show that $\frac{w_1}{w_0}=1+\frac{\lambda}{N^2}+\mathrm{O}(\frac{1}{N^3})$ such that ...
0
votes
0answers
15 views
Approximate size of mesh
this is programming/mathematic question and i am not sure how to properly call this problem so excuse me if this is well-known problem already.
I have an unknown-sized mesh consisting of nodes. Every ...
1
vote
0answers
57 views
Approximate CDF of the sum of a gaussian and a truncated gaussian
I am looking for a quick-to-compute approximation of the CDF of $X+Y$, where $X \sim N(0,\sigma_1^2)$ and $Y$ is a truncated gaussian, more specifically, a gaussian with mean $0$, standard deviation ...
0
votes
1answer
59 views
B-Spline Interpolation/Approximation
I've got a couple of probably very simple questions, yet some googling didn't bring up what I was looking for.
First what I want to do: I have a grid, and the gridpoints are function values. I want to ...
