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

1
vote
0answers
38 views

a question in projection

Let $V=L^2(\Omega)$, and $$k=\{v \in V ~s.t ~||v||_{L^2(\Omega)}\leq 1 \}$$ I need to find projection for any $u \in V$ on $k$. Please help me.I do not have any idea about this problem. I have many ...
0
votes
1answer
111 views

Approximating sums of powers of integers: why does $\sum_{i=1}^n i^r \div \frac{n^{r+1}}{r+1} \to 1$ as $n \to \infty$?

I know there are exact formulas for sums of integer powers of integers ($\sum_{i=1}^n i^r$ with $r \in \mathbb{N}$), but I was interested in approximating them. One way occured to me through a ...
2
votes
0answers
345 views

numerical approximation to logarithm

we know that $$ \ln(x)=\frac{1}{x} \int_{0}^{1}dt \frac{1}{1+xt} $$ then given a cuadrature formula inside $(0,1)$ is that true $$ \ln(x)= \frac{1}{x}\sum_{i}\frac{w_{i}}{1+xt_{i}} $$ wht other ...
5
votes
5answers
396 views

Bertrand's postulate in another point of view

I was just wondering why can't we use prime number's theorem to prove Bertrand's postulate.We know that if we show that for all natural numbers $n>2, \pi(2n)-\pi(n)>0$ we are done. Why can't it ...
1
vote
2answers
598 views

Fitting a set of points in a plane to a smooth curve obtained by joining a half-line and an arc of a circle

I have a set of points in the plane and I want to find a curve that best fits these points (e.g., in a least squares manner, or using some other convenient "measure"). I want that the curve be either: ...
6
votes
2answers
333 views

Good upper bound for $\int_0^1 (1 + 2x)/\sqrt{x + x^3}$

I am trying to obtain an upper and lower estimate for the integral $$I = \int_0^1 \overbrace{\frac{1}{\sqrt{x^2+1}} \left( \frac{1}{\sqrt{x}} + 2\sqrt{x}\right)}^{\Large f(x)}\,\mathrm{d}x,$$ and an ...
4
votes
1answer
1k views

Linearization of an implicitly defined function.

Problem: Given the equation: $xz^{2}+y^{2}z^{5}=19$ Also given: (3,4,1) is a solution to the equation. This point is not the only solution. 1) Find dz/dx and dz/dy (through implicit ...
1
vote
1answer
643 views

Find the 2nd-degree polynomial that approximates with the method of the least squares the:$f(x)=\frac{1}{10}x^2-2x+10$

It is known that a rectangular set of polynomials $\phi_k(x), k=0,1,\cdots,n$ for each $x\in[a,b]$ as to a weight function $w(x)$ can be constructed with the use of the following recursive type (Gram-...
1
vote
0answers
145 views

How to estimate linear operator?

Suppose the complex column vector $\mathbf{x}$ is linearly transformed by the complex matrix $\mathbf{T}$ into $\mathbf{y}$: \begin{align} \mathbf{y} = \mathbf{T}{x} \end{align} Assuming $\mathbf{T}$...
0
votes
1answer
114 views

Signal approximation using linear combination of functions

How I can approximate the signal $x(t)=0.001\,t^3 \exp(-0.1t)$ in the interval $[0,100]$ using a linear combination of the following functions: $f_1(t)=A_1$ $f_2(t)=A_2\cos(0.05t)$ $f_3(t)=A_3\cos(...
2
votes
1answer
133 views

Understanding weighted linear least square problem

I am having difficulty in understanding about weighted linear least squares. Could anybody explain me instead of minimizing the residual sum of squares why we need to minimize the weighted sum of ...
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 \right)}^...
2
votes
1answer
88 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 that ...
2
votes
0answers
140 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
113 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
469 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 Simpson'...
2
votes
1answer
290 views

Approximating a log-power function

I can't figure out how the following approximation has been done, I would appreciate any guidance: $$y=-60+10\log_{10}\left[\frac{\left(\frac{99}{100}\right)^m}{\frac{1}{11}\left(\frac{1}{3}\right)^m+\...
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
780 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
297 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: $$ r=\frac{...
2
votes
3answers
67 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
57 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
507 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
46 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
156 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
31 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
776 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
884 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
56 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
230 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!): http://...
0
votes
0answers
526 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 $\sin(\frac{2x\pi}{3})$...
2
votes
1answer
185 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
141 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 }} e^{-\frac{\...
0
votes
2answers
677 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 $m(E)...
2
votes
0answers
101 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
326 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
427 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
67 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
99 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
84 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 \\ j=n_2+2n_3+\ldots+(k-1)n_k\end{smallmatrix}}\left(\frac{-t^2}{n}\right)\frac{1}{\prod_{i=...
0
votes
3answers
890 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
480 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
289 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 ...