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Questions tagged [approximation]

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).

4
votes
3answers
280 views

I've discovered a strange approximation for exponents, why does it work?

I was working with some code involving exponents in an environment where exponents can only be calculated if the base of the exponent is an integer. I needed a good fast way to approximate this ...
0
votes
0answers
4 views

Approximation for fractional function for given boundary conditions

I try to find an approximation for the following expression $$\tan\phi(x)=\frac{\mathrm{si}(x)\sin(kx)-\mathrm{si}(x-\tau)\sin(k(x-\tau))}{\mathrm{si}(x)\cos(kx)-\mathrm{si}(x-\tau)\cos(k(x-\tau))} \...
0
votes
0answers
18 views

How can I understand Newton's root finding algorithm from the equation?

How can we prove Newton's method works correctly? Here, let me describe the equation by $f(x)=\frac{1}{x}-a$ for some $a$. The root of $f(x)$ is $\frac{1}{a}$. Its derivative is $f'(x)=-\frac{1}...
1
vote
2answers
59 views

Approximate $\log(7)+\cos(1)$ with an error of less than $10^{-4}$

Evaluate $\log(7)+\cos(1)$ with an error of less than $10^{-4}$ Obviously the aim is to use Taylor's expansion with Lagrange's remainder, but where to center it? I was thinking in $e^2$, which seems ...
2
votes
1answer
23 views

why the proximal operator is well defined?

Let $h$ be a convex function. We define the proximal operator as $$prox_h(x)=argmin_uh(u)+\frac{1}{2}\|u-x\|^2$$ why is this operator well defined? We must see that exist that minimum and is unique. ...
0
votes
0answers
32 views

Casting a shadow from a spotlight on a sphere

There is a hollow hemisphere $x^2+y^2+z^2=4$ $(z<0)$ in the coordinate space, and in the $xy$-plane, there is a square whose center is the origin and whose length is $2$ on one side. When you ...
0
votes
0answers
32 views

“Intersection” of multiple circles with inaccuracies

I am trying to calculate the position of a point (POI) through GPS distance measurements, that I get from a "blackbox" system. I don't know where the point is (for testing purposes I can of course ...
1
vote
1answer
48 views

What is a simple upper bound for $\exp\left(-\frac{1}{2}(x-(2\log(1/\delta)^{1/2}))^2\right)$ given $x \ge0$ and $\delta \in (0, 1)$?

Question For $x \ge 0$ and small $\delta \in (0, 1)$, what is a "simple" good upper bound for $$u(x,\delta) := \exp\left(-\frac{1}{2}(x-(2\log(1/\delta)^{1/2}))^2\right), $$ that doesn't involve $x$ ...
0
votes
0answers
8 views

Approximation of a $\min$ function to evaluate 3-dimensional integral

I have this $\min$ function, and looking for a linear approximation (or any other approximation if linear doesn't exist) $$f(x_2) = \min\left(x_2,\,y_1\left[z-\frac{x_2}{y_2}\right]\right)$$ where $...
-1
votes
2answers
43 views

Approximation of Sin(2) within 10^-4

I am having some trouble with this problem. So far, I am thinking to use a similar approach to finding the approximation of sin(x) per guidance of my textbook. This would get me something along the ...
0
votes
1answer
27 views

Simple sig fig definition

The number $\tilde{p}$ is said to approximate $p$ to $t$ significant figures if $t$ is the largest non-negative integer for which $$\left|\frac{p-\tilde{p}}{p}\right| \le 5 \times 10^{-t}$$ Source ...
2
votes
1answer
46 views

Showing the general solution reduces to $x(t)\approx \frac{F}{2mw^2_0\theta}\left(\sin(w_0t)-\sin(wt)\right)$

Consider the equation for a periodically forced, damped oscillator: $$m\ddot{x}+r_0\dot{x}+k_0x=F\sin(wt)$$ where $m, r_0, k_0$ are positive constants. Suppose now that $r_0=0$ and let $w_0=\sqrt{\...
2
votes
1answer
48 views

Approximating $f(x,y)$ as $g(x/y)$

Specific problem So here is my problem. I have a function $$ f(x,y\,|\,\alpha) = \frac{x(\alpha y^2 - 1)}{y(\alpha x^2 - 1)} $$ were x,y and $\alpha$ correspond to some physical parameters and thus I ...
0
votes
1answer
29 views

If one rounds digits one by one starting from the end, then is the rounding same as when “cut-offing” around required the precision?

If one rounds digits one by one starting from the end, then is the rounding same as when "cut-offing" around required the precision? That is does (for $1/10^3$): $0.84562...4356 \rightarrow 0.8456 \...
2
votes
1answer
79 views

How to approximate the solution of $y=a \arctan(x/a)-\arctan (x)$

After this question, related to the Prandtl–Meyer function $$\nu(M) = \int \frac{\sqrt{M^2-1}}{1+\frac{\gamma -1}{2}M^2}\frac{\,dM}{M}= \sqrt{\frac{\gamma + 1}{\gamma -1}} \tan^{-1}\left( \sqrt{\...
0
votes
0answers
23 views

Best Approximation of a piecewise polynomial [closed]

I am stuck trying to figure out this question If we are given the below piece wise function $ g(x) = \begin{cases} -1 & for -1 \leq x < 0\\ ...
2
votes
1answer
62 views

Is there an analytic approximation to this integral

The following integral does not have an analytical form.. however, would it be possible to find an analytical approximation? $$\int_b^c e^{-a\sqrt{x}} e^{-a \sqrt{x-b}} dx $$ with $c > b$ and $a&...
4
votes
0answers
60 views

Approximations to series of Ramanujan-type

Recently I have been playing around with series of the form $$\sum_{k=1}^{\infty}\frac{k^{s}}{e^{kz}-1} = \sum_{k=1}^{\infty}\sigma_{s}(k)e^{-kz}$$ for $s \in \mathbb{Z}$ and where $\sigma_s(k)$ ...
0
votes
0answers
19 views

Simple lower bound on Gaussian CDF evaluated at sum: $G(s + t)$ in terms of $G(s)$, with $s, t \ge 0$ and $s \le 1$

Let $G: s \mapsto \int_{-\infty}^s g(s)ds$ be the CDF of the standard Gaussian (with $g(s) := (2\pi)^{-1/2}\exp(-s^2/2)$ the density) and $s \le 0 \le t$. Question what is a simple lower bound for ...
0
votes
1answer
28 views

Can we combine convolution and higher powers for locally maximising a function?

Can we somehow find local maximum function value (for strictly positive functions) using a convolution? My idea is based on the result that $$ \lim_{p\to \infty}\left[\frac{1}{N}\sum_{k=1}^N {(a_k)} ^...
0
votes
0answers
12 views

Bound the residual between function and Taylor approximation

Taylor approximation of function f(•) can be written as follows: $$f(x+h)=F_{x,p}(x+h) + o(||h||^{p})$$ Assume that f is convex and p times differentiable on $dom / f$. Denote by $L_p$ uniform bound ...
1
vote
0answers
17 views

Calculating the marginal expectation of a joint pdf on the unit sphere

Consider a random matrix $\mathbf{X} \in \mathbb{C}^{T\times M}$ with independent columns $\mathbf{x}_1,\dots,\mathbf{x}_M$ distributed on the unit sphere $\mathcal{S} = \{\mathbf{x} \in \mathbb{c}^T: ...
1
vote
1answer
42 views

Approximating pi rate of convergence

I have been reading about a method for approximating $\pi$ using two uniform distributions and the ratio of points that lie within the circle compared to the square formed by the two uniform ...
7
votes
4answers
400 views

Justify: if $x\gt 0$, $\;\lim_{n\to\infty} \sqrt{n}\cdot{\overbrace{\sin\sin\cdots\sin}^{n\space\text{sines}}(x)}=\sqrt{3}$

I believe that I have managed to show that (if $x\gt 0$) $$\lim_{n\to\infty} \sqrt{n}\cdot{\overbrace{\sin\sin\cdots\sin}^{n\space\text{sines}}(x)}=\sqrt{3}$$ I did this by defining a sequence as $a_0=...
1
vote
1answer
36 views

Monte-Carlo approximation with small samples

Let me suppose I have one function of $y$ given $x$ : $f(y\mid x)$ and $N$ samples of $x$ : $\{x_i\}_{i=1}^N$. Here, I’d like to create a distribution over the space of $y$ based on this function $f$ ...
0
votes
3answers
41 views

Is $|\ln(1+t) - t| \leq t^2$ for $|t| \leq \frac{1}{2}$ obvious?

In a textbook of probablity and statistics there is a use of approximation by Taylor series that $\ln(1 + t) \doteq t$. It is stated that that the accuracy of the approximation is due to $|\ln(1+t) - ...
0
votes
1answer
28 views

A Better Approximation at Lower Input Values

I'm doing some research on an algorithm and to give an upper bound I wanted to simplify the term, call it $T$, below. \begin{equation} T(i)=\sum_{k=1}^{i-1}\sec\Big(\big(\frac{3}{4}\big)^{k-1}\big(...
0
votes
1answer
50 views

Solving this differential equation $H\nabla^2C - \frac{D}{L}\frac{\partial C}{\partial z}=0$

I have a differential equation: $$H\nabla^2C - \frac{D}{L}\frac{\partial C}{\partial z}=0$$ If I substitute $j(x,t) = j_0\exp(ikx+\omega t)$ into the above equation I get $$H\frac{\partial^2C}{\...
0
votes
3answers
55 views

Find analytic solution: for what $n_0$, $\forall n > n_0$ : $n \cdot p ^{n-1} \leq \delta$ holds?

Suppose $p$ is a fixed number in $(0, 1)$ and $\delta$ is a small positive number s.t. $ 0 < \delta < p$. What is $n_0$ such that for any $n > n_0$, the following holds: $$ n \cdot p ^{n-1}...
0
votes
1answer
33 views

Approximate scalar dot product with a vector's sum

I have two vectors $u$ and $v$ of size $n$. The $u$ vector is a linear increasing function. For the $v$ vector the individual elements are not known, only its sum. Is it possible to approximate the $u ...
-1
votes
1answer
78 views

cosine approximation [closed]

For cosine approximation , how to arrive at (9.11) from (9.10) ? Note: I do not think cosine angle addition formula helps here. ...
0
votes
0answers
29 views

Compare $\|h\|^k$ and $\sum_{i_1, \ldots, i_k} h_{i_1}\cdots h_{i_k}$ when $h \ll 0$

I have a question because when I search something on mathstackexchange, I've read something that I'm not sure it's true, and I would like to know if I'm right or I'm wrong. Actually, we consider $h \...
6
votes
1answer
57 views

Prove that: Probability of connectivity of a random graph is increasing with the size of the graph

In a random graph $G(n, p)$, the exact probability of the graph being connected can be written as: $$ f(n) = 1-\sum\limits_{i=1}^{n-1}f(i){n-1 \choose i-1}(1-p)^{i(n-i)} $$ This probability is ...
1
vote
1answer
64 views

Integral calculus from the modern viewpoint [closed]

This is a soft question. What is the purpose of teaching techniques of integration at the college level? More specifically, in the sense of putting integration into practice, what value does ...
1
vote
2answers
63 views

Little-o meaning in equation

I am completely new to little-o notation, I came across it in a lecture about algorithmically approaching a function minimum: $$ f(x^{n+1}) = f(x^n) + \nabla f(x^n ) \cdot (x^{n+1} − x^n) + o( \| x^{n+...
4
votes
1answer
88 views

$n$-sphere enclosing the Birkhoff polytope

I am not a mathematician by training, so please feel free to correct my logic or descriptions when necessary: Let $P$ denote a $\textit{permutation matrix}$: $$ \begin{equation} P := \{X \in \{0,1\}^{...
2
votes
0answers
152 views

EXP(x) approximation in old 1980's computer ROM

The presentation : Old 1980’s ROM (Apple 2e, Commodore 64, ...) uses a Taylor’s series-like to evaluate the exponential function EXP(x) : ...
0
votes
0answers
31 views

Minimal/Small convex partition of a set of points

Given a set of $n$ point tuples $(x_1, y_1), \ldots, (x_n, y_n)$ with $x_i, y_i \in \mathbb{R}^m$. I am interested in a partition of a hypercube the points $x_1, \ldots, x_n, y_1, \ldots, y_n$ are ...
1
vote
2answers
60 views

How to solve equations like $\alpha \sin x -\beta\sin 2x +\gamma=0 $

Can I solve this equation without Newton-Raphson method? I have $\alpha=47.02$ $\beta=112.5$ and $\gamma=50$. When I have to use Newton-Rapson to solve trigonometric equations ? I will greatly ...
0
votes
1answer
50 views

Approximation by smooth functions by changing values at arbitrarily small interval

Assume $f\in C[0,1]$ is smooth (i.e. infinitely many times differentiable) on $(0,\frac 12)$ and $(\frac 12,1)$. Let $\epsilon>0$ be arbitrarily small. Can we approximate $f$ in supremum norm by ...
1
vote
2answers
34 views

Measures for homogeneity of a finite sequence - approximating an image with lines.

I am trying to approximate a grayscale image with lines, using a greedy algorithm. For our purposes, the line is a finite sequence, with each number being between $0$ and $1$, where $0$ is pure white ...
3
votes
1answer
53 views

Approximating Integral over Sphere

I would like help approximating the surface integral $$\int_{S^{n-1}_{\ge 0}}\frac{1}{\hat n\cdot p}dS$$ where $\hat n$ is the unit normal to the sphere at the given point, $p\in\mathbb{R}^n$ is a ...
0
votes
1answer
40 views

Longest path between 2 given vertices in undirected unweighted graph

Consider undirected unweighted graph. My problem is to find the longest simple path between two given vertices or its approximation. I was thinking of solution like this - Find the shortest simple ...
-1
votes
0answers
15 views

matrix non-linear transform approximation

I have a square matrix denoted as $A$ and an element-wise square operator $\Delta$, such that $\Delta(A)=(a_{ij})^2$ for all $i,j, (a_{ij})$ is the ith row and jth column element of A. Is there exists ...
1
vote
1answer
20 views

Approximation to a parametrized integral

Consider the following integral: $$ I_1(r)=\int_{-r}^{r} \int_{-\sqrt{r-x^2}}^{\sqrt{r-x^2}} \frac{1} {(x^2+y^2) \log^2\big(\frac{2}{\sqrt{x^2+y^2}}\big)}\,dy\,dx.$$ I am not able to evaluate ...
1
vote
4answers
98 views

Iterative calculation of $e^x$

Is there an iterative approximation method for calculating $e^x$, which only use basic operations (add, subtract, multiply, division), and which is capable of using an initial guess? So, I have an ...
0
votes
0answers
13 views

Integrals similar to Taylor shifts but more smooth & less localized?

I am sometimes slightly annoyed by the extreme focus of local-only behavior for functions which a function's Taylor expansions around some real point shows us. Does there exist any theorem involving ...
0
votes
2answers
55 views

proof of $\mathrm{arg}(1+\frac{z}{n})=\mathrm{Im}(\frac{z}{n})+o(\frac{1}{n})$

I'm looking for the proof of the following statement:$$\mathrm{arg}(1+\frac{z}{n})=\mathrm{Im}(\frac{z}{n})+o(\frac{1}{n}).$$ I found it on page 108 of Arnold's『Ordinary Differential Equations(1973)』,...
3
votes
0answers
58 views

Approximating a Convex Piecewise Linear Function by Convex Polynomial

I have a convex piecewise linear function defined on some interval $x \in [a,b]$ $$ f(x) = \max(a_0 x + b_0, ..., a_n x + b_n) $$ where $n$ is large, too large to calculate quickly when used as a ...
0
votes
1answer
70 views

How to approximate integral of $e^{x \cos \phi}$ when $x \gg 1$?

Let $I(x)$ be $I(x) = \int _{-\pi} ^\pi e^{x \cos \phi}d\phi$. When $x \gg 1$, how to get major terms of (how to approximate) $I(x)$? There must be the solution but I don't have any idea. In the ...