For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

learn more… | top users | synonyms

4
votes
2answers
69 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
42 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
37 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
78 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 ...
0
votes
1answer
100 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 ...
2
votes
0answers
132 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
111 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
1
vote
0answers
430 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
269 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
35 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
701 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
283 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
64 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? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
5
votes
3answers
1k 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
54 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
463 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
44 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
147 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
30 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
647 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 ...
4
votes
1answer
743 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
55 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 ...
1
vote
1answer
55 views

Significant digits/rounding

I have a bunch of criteria to evaluate for a product, and each is scored on a scale from 0 to 5. Each criterion has a weight associated with it. If I find the weighted score of a criterion, it is ...
2
votes
1answer
215 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
1k 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
510 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
171 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 ...
3
votes
0answers
139 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
2answers
614 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 ...
2
votes
0answers
95 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
269 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
388 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
65 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) ...
1
vote
2answers
97 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
83 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
814 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?
2
votes
0answers
432 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
143 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
268 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
1k 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
7k 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
30 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 ...
2
votes
2answers
49 views

Formula for the pseudofrequency using approximations

We know that $w_1=\frac{\sqrt{4w_{0}^2-b^2}}{2}$ and $(1+u)^{\alpha}=1+\alpha u + \mathrm{O}\left(u^2\right)$ I need to show that ...
1
vote
0answers
173 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 ...
1
vote
1answer
310 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 ...
3
votes
0answers
82 views

How to solve a distance problem inside of a picture?

sorry for my bad english. I have the following problem: In the picture you can see 4 different positions. Every position is known to me (longitude, latitude with screen-x and screen-y). Now i want ...
1
vote
1answer
61 views

approximation of law sines from spherical case to planar case

we know for plane triangle cosine rule is $\cos C=\frac{a^+b^2-c^2}{2ab}$ and on spherical triangle is $ \cos C=\frac{\cos c - \cos a \cos b} {\sin a\sin b}$ suppose $a,b,c<\epsilon$ which are ...
0
votes
2answers
137 views

Aproximate calculation in decimals

I am trying to refresh on precision of calculations. If we have the decimal fractions: $.234673$, $.322135$, $.114342$, $.563217$ each known to be correct to six figures why are each of the decimals ...
0
votes
1answer
386 views

Need some help understanding notation for composite gauss quadrature formula

Reading through some notes on 2-point gauss quadrature, I came across the following general formula. I'm currently doing an assignment with 3-point quadrature, and have gotten to a similar formula, ...
3
votes
0answers
65 views

Root calculation by hand (division-like algorithm)

I remember from my highschool days a division-like algorithm for calculating square, even cubic roots. I know the continued fraction method, some series and Newton's method. I have checked similar ...