All questions with tag [math: approximation]


Estimate the area restricted by $f(x) = \log(x+1)/x \, , f(x-1), \, y=0$, and $y=a$.

I need to estimate the area between the functions $f(x) = \log(x+1)/x \, , f(x-1), \, y=0$, and $y=a$. where $a>1$. Now I have tried quite a few ways to do this, but nothing comes to mind. I tried writing out the taylor series, I tried changing this around. Nothing really gave a decent approximation. A decent in my mind would be anyth...
2022-07-25 20:44:17

$C^1$ approximation of a continuous curve.

Suppose I have two points $\alpha,\beta \in \Bbb{R}^n$. Define $$ X=\{\gamma \in C^1([0,1] , \Bbb{R}^n),\ \gamma(0)=\alpha,\gamma(1)=\beta ,0 <|\gamma'|<K\}$$ parametrized curves joining $\alpha$ and $\beta$. If I have a sequence $(\gamma_n) \subset X$ such that the paths $\gamma_n$ are all contained in a compact set $K \subset \B...
2022-07-25 20:42:49

Help with Chebyshev Economization of $\exp(x)$?

This might be a foolish inquiry, so I ask forgiveness beforehand if it is. This is a really usual instance of Chebyshev Economization, yet I still do not recognize just how the coefficients are located. I intend to approximate $\exp(x)$ over the interval $[-1, 1]$. $$\exp(x)=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$$ I will specify $$P_5(x)=1...
2022-07-25 19:32:37

Steiner Tree Approximation

My question is about a subtlety regarding the $2$-approximation for the Metric Steiner Tree problem. The classical Metric Steiner tree problem: Given a metric space on $n$ points and a subset $S$ of the points, find a tree that spans $S$ with minimal total cost. The classical $(2-\frac{1}{n-1})$-approximation: Take a minimal spanning tree on ...
2022-07-25 16:22:53

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. $\mathcal{L}_X(s)=\int_0^\infty e^{-s t} d F(t)$ Can one conclude immediately that, as $s \to 0$, $\log \mathcal{L}_X(s) \approx -s \mathbb E X + o(s^2) $ ? If n...
2022-07-25 16:22:10

asymptotic limit of $\int_0^{\infty}\left(1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}\right)^qdt$

Help me please with the adhering to indispensable. I've asked this inquiry prior to, yet it ends up that it was wrong inquiry. I need to get an asymptotic restriction (with $k$ mosts likely to $\infty$, and also $q$ dealt with) of it. For $q\ge 2, t\ge 0, k \in Z$ $$ \int_0^{\infty}\left(1-\frac{t^2}...
2022-07-25 10:55:44

Describe growth of $\epsilon n$

For all $\epsilon$ we have that $f(n)\le \epsilon n$ where n is an all-natural number. What can we claim concerning the development of $f(n)$? Plainly $f(n)=O(n)$, can we claim anything sharper?
2022-07-25 10:48:41

asymptotic limit at the integral

I would love to get an asymptotic restriction at the adhering to indispensable: for $p\ge 2, n \in N$, $t \ge 0$ $$ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t $$ I assume replacement $t=\frac 12 \sqrt{(n+1)!}y$ need to function. Yet after the substittution, I do not recognize what to do. Thanks for your...
2022-07-25 10:37:16

Algorithms for finding closed form approximations for integrals (with no closed form solutions)

It is popular that several integrals have no closed form remedies, generally what you would certainly do is address them numerically. My inquiry is if there are algorithms that offer you excellent closed form estimates rather.
2022-07-24 09:40:58

Newton's Method, and approximating parameters for Bézier curves.

I've been wanting, for quite a while now, to polish up some source code I wrote for approximating arbitrary Bézier curves to given series of points. I managed to accomplish quite a bit, but I hit a road bump today while trying to write documentation for it. I'm using Philip J. Schneider's "An Algorithm for Automatically Fitting Digitized Cu...
2022-07-24 09:19:51

Approximate solution for an exponential equation

Trying to solve this question: Probability of ball ownership I got at an expression for the solution, P: $$\frac{P}{M} = (1 - \frac{1}{M+N})^{N(1-\frac{P}{M})+M}$$ Where M, N are parameters. The problem is, I got P both in the left side and in the exponent in the right side, and I have no idea how to simplify it. My goal is to have a simple appr...
2022-07-24 06:31:09

Approximating the logarithm of sum

I would like to approximate $$ \ln(\sum_{k=0}^n(n-2k)^p) $$ Here $p\geq 2$
2022-07-24 05:26:43

Extending Hermite polynomial interpolation

Working with the definition of Hermite polynomials $x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then $$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} [f'(x_j)\hat{H}_{n,j}(x)],$$ where $H_{n,j}(x) = [1-2(x-x_{j})L'_{n,j}(x)]L_{n,j}^2(x)$, $\hat{H}_{n,j}(x)=(x-x_{j})L_{n,j}^2(x)$ and $L_{n,j}(x)$ is...
2022-07-24 05:25:02

Approximating log of factorial

I'm asking yourself if individuals had a referral for estimating $\log(n!)$. I've been making use of Stirlings formula, $ (n + \frac{1}{2})\log(n) - n + \frac{1}{2}\log(2\pi) $ yet it is not so wonderful for smaller sized values of $n$. Exists something much better? Otherwise, should I compute the specific value and also button to the approxim...
2022-07-24 05:22:04

Proof about binomial coefficient

I today see a approximated formula, when $n \ll u $: $$\log {u \choose n} \approx n \Big(\log \frac{u}{n} + 1.44\Big)$$ I would love to recognize just how to confirm it.
2022-07-22 18:17:45

Elementary bound of binomial coefficient

I'm functioning my means via an Erdős paper from the sixties and also several of the primary bounds he asserts appear to be simply past my reach. The expression looks horrible yet possibly there is a brilliant simplification? In the adhering to expression, $n$ and also $s$ declare integers and also $c$ is an approximate actual constant, why is i...
2022-07-22 15:37:26

Calculation of a 'double' sum

Let $n \in N$ and also $q\geq 2$. I am attempting to compute the adhering to amount: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any aid will certainly be valued. Thanks.
2022-07-20 17:51:45

Approximation of a bounded measurable function with step functions?

I'm having problem evaluating whether this declaration is correct: For an approximate bounded quantifiable function $f$ specified on $[0,1]$, $\exists{}\ $a series of action features $\{\phi_n\}$, such that $\{\phi_n\}$ merges to $f$ pointwisely a.e. on $[0,1]$. By the Simple Approximation Theorem, this holds true if we are permitted to make...
2022-07-20 17:43:53

How many significant figures are needed in base 2?

$x \in \mathbb{R}$ $2^{500}<x<2^{501} $ How several substantial numbers are required in base 2, to recognize in high approximation whether $2^x$ is integer?
2022-07-20 17:41:36

Numerically estimate $a^b$

Possible Duplicate: How can I calculate non-integer exponents? What is the most efficient way to estimate $a^b$ ($a > 0$) numerically? My goal is not to use built-in math functions (like $e^x$ or $\log x$), i.e. only using +-*/. I successfully accomplished this task by estimating $e^x$ and $\log x$ (below), because $a^b=e^{b\log a}$. ...
2022-07-20 17:24:01