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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).

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1answer
13 views

what too find the parameters T1?

we have sequence $$ t_n= [\frac{1}{n} Log[1 + n]-1-\frac{1}{6} Log[ (8 (\frac{1}{n})^3 + 4 (\frac{1}{n})^2 + \frac{1}{n} + \frac{1}{30} -\frac{1}{( \frac{240}{11 \frac{1}{n}} + \frac{9480}{847} ...
-2
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0answers
25 views

how this the result of this limit got [on hold]

if $$t_n =\log n!-\frac12\log 2\pi- \left(\frac12+n\right)\log n+n-\frac1k\log\left(1+\frac k{12n}\right)$$ we have $$t_n- t_{n+1}= (1/144) 1/n^3 + O(1/n^4)$$ then the rate of convergence of the ...
0
votes
1answer
19 views

What are good strategies for stable numerical approximations of special functions?

I am trying to write a scientific calculator for a very small microcontroller with no floating point unit. If the standard c math libraries were included the compiled code would be too large to fit on ...
0
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0answers
34 views

How to formulate this optimization problem to solve it efficiently?

Let $D\in \mathbb{R}^{N\times 1}$ be a column vector. It defines the demands from $N$ departments. Let $S\in\mathbb{R}^{2^N\times N}$ is a matrix. It can be defined as a supply matrix. It is ...
4
votes
2answers
37 views

P value for a z-score of 4.9? Or am I doing this wrong?

My question is as follows: A fair die is rolled $120$ times. Find the probability that $5$ is on the top: a. between $30$ and $40$ times, b. between $18$ and $50$ times, c. more than $70$ times. ...
0
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0answers
14 views

How to improve derivative approximation errors along the boundary using radial basis functions

I am using radial basis functions to approximate the derivatives of a function. The test function I am using is: $g=y\cos(x)+x\sin(y)$ on the interval from 0 to $2\pi$ in both x and y directions. The ...
0
votes
1answer
27 views

Approximate Trig Functions without the use of Taylor Series

I am familiar with how a trig function, i.e. $\sin(x)$, can be approximated by a MacLauren series; \begin{align} \sin(x_0) &\approx \sin(0) + \cos(0) x_0 - \frac{1}{2}\sin(0) x_0^2 - \frac{1}{3!}\...
1
vote
4answers
42 views

Proving approximation of $\text{erf}$ with Taylor expansion

I am asked to show that $$\text{erf}(x) \approx 1 - \frac{1}{\sqrt{\pi}}\frac{1}{x}e^{-x^2}$$ in a computational project. Numerically it is really easy to show that this approximation makes sense. ...
1
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0answers
16 views

Sampling extreme points from Minkowski Sum

I recently stumbled upon the following subproblem: we are given zonotopes $P_1, \dots, P_m$ in $\mathcal{V}$ representation (i.e. we are given the extreme points of each $P_i$). Denote the Minkowski ...
3
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2answers
44 views

What exactly does it mean to say “$dv=4\pi r^2\,dr$ can be thought of as the spherical volume element between $r$ and $r+dr$”?

In my textbooks and lectures (I'm a second year physics student) I often come across statements such as $$``dV=4\pi r^2dr\text{ can be thought of as the spherical volume element between }r\text{ and }...
0
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2answers
36 views

Find the approximate value of π

We re given a ruler, a compass and square sheets of side length $a$. Using these we need to find the approximate value of π. I tried to arrange the sheets similar to a graph paper. Then we can draw a ...
0
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0answers
30 views

Ordinary Differential equation Successive Approximations

Ordinary Differential equation Successive Approximations Taking $\phi(0,\lambda)$ as constant function, where it is equal to 0 which I consider as a first approximation, gives for the second ...
0
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0answers
23 views

Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem Solving a non-linear system of equations. Number of variable is same as number of equations. When I fix a set variables (say $\vec{y}$) and keep another set free (say $\vec{x}$), the system ...
0
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0answers
38 views

Finding parameters of noisy rounded sine wave

I have $n$ groups each with $m$ samples generated by the following formula $$s_{ij}=\lfloor 8 + 5 \times (a \times \sin(2 \times \pi \times b \times (c+x(j-1)) + d) + z_i) \rceil$$ $s$ is value of ...
0
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1answer
48 views

The Normal approximation to the Binomial (I cant find where im going wrong)

Here is the main question: Police estimate that 80% of drivers now wear their seatbelts. They set up a safety roadblock, stopping cars to check for seatbelt use If they stop 20 cars during the first ...
2
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0answers
45 views

An happy coincidence for the approximate solution of $x \tan(x)=k $?

Thinking more about this question where I proposed some approximate solution of the first positive root of equation $\color{blue}{x\tan(x)=k}$ for any $k >0$, I notice that, for the $[3,4]$ Padé ...
0
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0answers
31 views

Which functions can be well-approximated everywhere by algebraic curves?

Some analytic functions, like, $$ f(x) = \frac{1}{1+x^2}, $$ have no everywhere-convergent Taylor series. However, this function corresponds exactly to the roots of a certian algebraic curve, $$ (1+x^...
1
vote
2answers
29 views

What is the expansion of $\log(N+x) = \log(N) + [\dots\text{blank}\dots] $? ($N \in \mathbb{R}+$ and $0 \leq x \leq 1)$.

I'm working on a math problem which might be solvable if I can re-express $\log(N+x)$ as $\log(N) +$ 'something. The problem I am having with the Taylor series expansion about $x=0$ is that it ...
0
votes
1answer
27 views

How to expend $\log_a(\log_ax)$ for $a\in(0;1) \land x\in(0;1)$?

Here are some logarithm rules : $\log_ay=\frac{\ln y}{\ln a}$ $\log_a(A\cdot B)=\log_aA+\log_aB$ $\frac{1}{\ln a}=\log_ae$ Hence: $$\log_a(\log_ax)=\log_a \left(\frac{1}{\ln a}\cdot \ln x\right) = \...
2
votes
3answers
46 views

Why is $\sqrt{1 + x^2}$ approximately equal to $1 + \frac{x^2}{2}$?

I saw this in Shankar’s Physics book and couldn’t make out the reasoning behind it. I would assume the dx and derivative have nothing to do with it. https://i.imgur.com/iyTxPwW.jpg
0
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1answer
20 views

Unbiased Estimation of Sum of Reciprocals over a Symmetric Distribution by Taylor Expansion

Random variable $X$ follows a symmetric and unkown distribution. $\lbrace x_n \rbrace$ are a large (~$10^6$) sample drawn from $X$ Expectation $a = E[X]$ is known. Consider the taylor expansion of $f(...
0
votes
1answer
18 views

Approximate Average of Variance of Many Bernoulli Distributions

$B_{1}, ... , B_{n}$ are many (~$10^6$) random variables following Bernoulli distributions. $p_{i} = P(B_{i} = 1)$ Define $p_{avg} = \frac{1}{N}\sum_{n} p_{i}$ Can I say: $\frac{1}{N}\sum_{n}{p_{i}(...
1
vote
0answers
20 views

Error term, second-order Taylor approximation, integral

Let $L:\mathbb{R}^2 \rightarrow \mathbb{R}$ be the Laplacian of the 2D Gaussian, $$L(\boldsymbol{x}) = \frac{1}{2\pi\sigma^4}\left( \frac{\left\|\boldsymbol{x}\right\|^2}{\sigma^2}-2\right)\exp\left(-...
1
vote
2answers
29 views

Name and derivation of the approximation $\frac{1+x}{1+y} - 1 \approx x-y$?

I am wondering if there is a name and way to derive the following approximation: $$\frac{1+x}{1+y} - 1 \approx x-y$$ I'm essentially interested in how to refer to this.
0
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0answers
12 views

Approximation with inequality constraints

Suppose $\mathbf x = [x_1\; x_2\; \cdots\; x_n]$ is a discrete approximation of a function at $n$ points. I want to get another approximation of this function at $n/2$ even points, say $\mathbf y = [x'...
0
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0answers
24 views

Another mini-max approximation

Let $n\ge 1$ be an integer. $\mathcal P_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$. My question is : Is it true that $\inf_{x_0,x_1,...,x_n\in[a,b], x_0&...
2
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1answer
36 views

Taylor approximation of cos function

I have the following problem: Knowing the linear approximation of the Taylor approximation of the form(1): $$ f(x_{0} + \Delta x) \approx f(x_{0}) + f'(x_{0}) \Delta x $$ I have to determine the ...
0
votes
1answer
41 views

Approximating functions with rational functions

If I have an analytic function of a complex variable, I can write a Taylor series and in some cases can truncate the high powers to obtain a good approximation over some part of the function's domain. ...
0
votes
1answer
29 views

How to find a numerical approximation of this sum.

I need to find a good approximation of $S_n$ for big n's. $S_n = \sum_{i=0}^n \frac{n!}{i!(n-i)!}{2^{-n}}\log_2\frac {n!}{i!(n-i)}.$ I've computed this sum using Python on $1\leqslant n\leqslant500$ ...
2
votes
1answer
44 views

Problems with Taylor Series to Approximate Square Roots

I am currently looking into comparing different methods to compute the value of square roots, and I've decided to compare two relatively well known methods - the famous Babylonian Method and the ...
0
votes
1answer
17 views

Metrics for the similarity of two sets of data

I am trying to model a certain (discrete) behavior measured from source A, and the literature in the field have a model for a source A'. The behavior itself for sources A and A' are pretty similar ...
1
vote
0answers
43 views

Approximating $\log(1+\exp(z))$ when $z$ is complex

There exist beautiful numerical approximation for calculation of the function $$f(z) = \log(1+\exp(z)).$$ In case if $z$ is real, the following can be used $$f(z) = \begin{cases} z & z \gg 1 \\...
1
vote
1answer
61 views

Approximation of an integral involving x and 1-x

I am looking for an approximation of the integral $F(k,R)=\displaystyle\int_0^1\frac{\mathrm{d}x}{(Rx)^{-2}+(1-x)^{-k}}$, that is valid to within 1% over the range $2<k<10$ and $R>1$. Is ...
2
votes
0answers
43 views

Is there a name for this iterative formula for finding near-miss approximations for the square root of X?

I noticed a pattern in near-miss approximations for $\sqrt 2$ and eventually figured out a general iterative formula for the square roots of whole numbers: $\frac {Nn + XDd} {Dn + Nd}$ Where $X$ = ...
1
vote
1answer
23 views

How to use power series to find the approximate value of this integral function with $p$ as a variable?

$$L(p)=\int_0^{2^{-\frac{1}{p}}}(1+(x^{-p}-1)^{1-p})^{\frac{1}{p}}dx, p\in \mathbb{R}, p\ge 1$$ Parametrize $x$ as $\cos^{\frac{2}{p}}(t)$, then we can find a equivalent form of the integral, namely ...
0
votes
1answer
16 views

Families of functions used to approximate local neighbourhoods of non-smooth functions?

For local function approximation perhaps the most famous example how to do this is probably to use a basis of monomial functions $$\{1,x,x^2,\cdots,x^k,\cdots,x^n\}$$ In the most local setting, this ...
21
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14answers
5k views

Calculating the square root of 2

Since $\sqrt{2}$ is irrational, is there a way to compute the first 20 digits of it? What I have done so far I started the first digit decimal of the $\sqrt{2}$ by calculating iteratively so that ...
1
vote
2answers
29 views

Approximation of exponential expression

I have given the equation $$au_x-ap+1+e^{-ap+b}=0,$$ where $p>0$ is the unknown. $u_x$ denotes the derivative of a given function, $a$ and $b$ are merely constants. I want to express $p$ ...
1
vote
1answer
32 views

The monotone of an integral function

$$L(p)=8\int_0^{2^{-\frac{1}{p}}}(1+(x^{-p}-1)^{1-p})^{\frac{1}{p}}dx, p\in \mathbb{R}, p\ge 1$$ May I ask if this $L$ function is increasing ot decreasing?
0
votes
1answer
16 views

Why do error models often contain “squares” of a value?

In many error minimization or approximation models, they often do operations on "sum of squares" of the calculated value. (E.G. residual sum of squares) What is the purpose of squaring the error? Is ...
1
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0answers
49 views

Common differentiable approximation of $f(x) = |x|$ and $f(x,y) = \sqrt{x^2+y^2}$

Is there a common approximation (like commonly used in applications) of the functions $$ f(x) = |x| $$ and it's two variable version $$ f(x,y) = \sqrt{x^2 + y^2} $$ I'm trying to come up with a ...
0
votes
1answer
20 views

Confidence interval - To use standard error or standard deviation?

We are given a few true-false questions. I would like to know if my reasoning is correct. 25 readings are made on the elevation of a mountain peak. These averaged out to 81,000 inches, and their SD ...
0
votes
0answers
39 views

Is there any way to eliminate the singular point to solve this integral by hand or by approximations?

$$\int_0^1 (1+(x^{-p}-1)^{1-p})^{\frac{1}{p}}dx, p\in \mathbb{R}, p\ge 1$$ I find that $x=0,1$ are singular points(discontinuous points of this integral), is there a way to eliminate those singular ...
1
vote
2answers
68 views

Binomial Coefficient Stirling's Approximation

I have the following expression, $$\frac{\binom{n}{j}}{\binom{n}{j-l}}$$ I have approximated this and wrote: $$\frac{\binom{n}{j}}{\binom{n}{j-l}}\approx \frac{\frac{n^j}{j!}}{\frac{n^{j-l}}{(j-l)!}}...
2
votes
1answer
54 views

What are the most elegant approximations of the sum of binomial terms, by using Stirling's approximation?

What are the most elegant approximations by using Stirling's approximation for the sum of binomial terms as below? $Sum(r) = \sum\limits_{k=r}^{n} \binom{n}{k} p^k (1-p)^{n-k}$ I tried to plug in ...
34
votes
1answer
2k views

I have found a formula for dividing numbers in easy steps

I found an easy method for division and it depends on some factors. I wanted to find an answer for $1000/101$ with easy steps. My starting point is here. I formulated this method by 2 hours of hard ...
6
votes
3answers
77 views

Why is $x^n\approx \left(n(x^{1/4096}-1)+1\right)^{4096}$?

There's an old-school pocket calculator trick to calculate $x^n$ on a pocket calculator, where both, $x$ and $n$ are real numbers. So, things like $\,0.751^{3.2131}$ can be calculated, which is ...
0
votes
1answer
18 views

Approximate a curve with limited number of samples

Preface: We have an analogue measurement device that changes its behaviour with temperature. We use an environmental chamber and cycle through a range of -20°C and 120°C to find out how the many ...
1
vote
2answers
64 views

Which expression best approximates the real number $0.138943$? [closed]

I am curious to know if there is a fraction or an expression which approximates this number: $0.138943$.
0
votes
1answer
27 views

Approximation of an Exponential Correlation Matrix with a Constant Correlation Matrix

I am given with a $N \times N$ matrix $$ \begin{align*} A = \begin{bmatrix} 1 & \rho_{h} & \ldots & \rho_{h}^{N - 1} \\ \rho_{h} & 1 ...