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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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45 views

Relationship between $n$ and $P_{n}$

We often hear it said that either there is no relation between the natural, or counting numbers, $n$, and their counterparts the primes, $P_{n}$, or that if there is, it is so recondite as to be ...
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0answers
23 views

Question related to sigmoid function

I have been going through the Deep Learning course from http://neuralnetworksanddeeplearning.com/chap1.html I have come across this approximation in the function $w_jx+b$ $\Delta output \approx \...
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1answer
37 views

Approximating a Banach space “vector- valued function” by “simple functions”

I'm trying to prove the following claim: Let $T$ be a compact (metric) space, and let $\mathcal{X}$ be a Banach space over $\mathbb{K}$. Let $f : T \longrightarrow \mathcal{X}$ be a continuous ...
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0answers
21 views

Approximating square root for long expressions

I'm currently working on a problem which asks me to calculate the potential energy of a three spring system arranged in an equilateral triangle constrained to move in the x-y plane. As a consequence ...
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2answers
27 views

Small angle approximation for $\frac{1+\sin\theta}{5+3\tan\theta-4\cos\theta}$

Please could somebody explain how the expression involving $\theta$ that $$\frac{1+\sin\theta}{5+3\tan\theta-4\cos\theta}$$ approximates to for small values of $\theta$ is $1-2\theta+4\theta^2$?
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1answer
55 views

Approximation of $\prod _{k=p+1}^{\infty } \cos \left(\frac{p \,\pi}{2 k}\right)$

After this post, I started wondering about possible approximations of the infinite product $$A_p=\prod _{k=p+1}^{\infty } \cos \left(\frac{p \,\pi}{2 k}\right)\tag 1$$ where $p$ is an integer. As far ...
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2answers
55 views

Smooth approximation of $f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$ [on hold]

I'd like to find a smooth function to approximate $$f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$$ This function should be differentiable everywhere. Thanks.
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0answers
9 views

Proper way to write a theorem with approximation result.

I have a lemma that shows that we can approximate $a$ with $b$ if $n \gg 1$ and I use the result to prove a theorem. Which one of the following ways is better to write this lemma? Provided that $n \...
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1answer
35 views

By creating a Maclaurin series up to an including $x^4$ for $\ln(\cos x)$ shows that $\ln2 \approx \frac{\pi^2}{16}\left( 1+\frac{\pi^2}{96}\right)$

By creating a Maclaurin series up to an including $x^4$ for $\ln(\cos x)$ shows that $$\ln2 \approx \frac{\pi^2}{16}\left( 1+\frac{\pi^2}{96}\right)$$ So creating a Maclaurin series using the general ...
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1answer
29 views

Find a composite solution to the following problem

I'm trying to solve the following exercise: Find a composite solution to the following problem: $$ \epsilon y'' + y(y' + 3) = 0 \text{ for }0<x<1, \text{ where }y(0) = 1, \,y(1) = 1 $$ where $\...
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1answer
31 views

Asymptotic differential equation for large $t$. Can I just drop fast decaying terms?

I am looking at the differential equation $x'(t)=x(t) (1+s(t))$. and I want to make a statement about the behaviour of its solution $x(t)$ for large $t$, knowing that for large $t$ $s(t)=\frac{1}{2}...
4
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3answers
65 views

Famous fractions: Can any “special” numbers be approximated by simple ratios like $3.14\ldots$ as $22/7$?

The ratio $22/7$ dates back to antiquity as an approximation of $3.14\ldots$. I'm wondering whether there are any other "famous" numbers with a similar situation. That is, something like $e$ or $\phi$ ...
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1answer
40 views

Spline interpolation - why cube with 2nd derivative?

Background For spline interpolation, it looks the degree 3 cubic spline is accepted as the better way and in my understanding it requires 1st and 2nd derivatives at the joints to be the same. ...
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0answers
40 views

What is the best way to compare the numerical solution to the exact solution?

I have 2000 approximate solutions (2000 arrays of points (x,y)). In the following figure, I have randomly shown three approximate solutions versus the exact solution. By increasing the case number, ...
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1answer
36 views

Optimization over a binary (or discrete) variable

I have the equation $y=Kx$ where, for example, $x=\begin{pmatrix} x_1 \\ \vdots \\ x_{50} \end{pmatrix}$, where $x_i=0$ or $1$ $K \in M _{1000,50}(\mathbb R)$ a given constant matrix and $$y=\begin{...
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2answers
46 views

tanh implementations for FPGA neural nets

In trying to put a neural network on my FPGA, I am running into the problem of describing my activation function. On my computer, I typically use the well-known tanh, so I am inclined to stick to my ...
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2answers
49 views

Easy example of $Ax =b$ floating point arithmetic.

Solve $Ax =b$ with two-digit floating-point arithmetic. We have $$ A= \begin{pmatrix} 1 & 1\\ 1 & 0,99\\ \end{pmatrix} $$ and $$ b = \begin{pmatrix} -1 \\ 1 \\ \...
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1answer
48 views

finding an approximation for $\cos(1)$ (Explanation of a solution)

So I want an approximation $C$ for $\cos(1)$ by using the taylor expansion of the Cosinus-function. How many terms of the taylor expansion and how many decimal places do I need for $$ | C - \cos(1)| &...
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1answer
18 views

f, h cont diff. Find necessary and sufficient conditions for these functions to be first-order approximations of each other at the point (0,0).

Suppose that the functions $f: \mathbb R^2 \to \mathbb R$ and $h: \mathbb R^2 \to \mathbb R$ are continuously differentiable. Find necessary and sufficient conditions for these functions to be first-...
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1answer
19 views

question on expected value of exponential functions

When is the approximation $\mathbb{E}\left\{{e^{f(x)}}\right\} = e^{\mathbb{E}\left\{f(x)\right\}}$ tight? I mean for which function of f(x) and which conditions? there is any theorem on it?
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1answer
66 views

Does this system of equations have attractors and periodic solutions?

I want to solve the following exercise: Determine the critical points of the system \begin{align} \dot{x} &= x^2- y^3\\ \dot{y} &= 2x(x^2 - y) \end{align} Are there attractors in this system? ...
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2answers
22 views

Round off the number 8.03567 to four significant digits and compute the percentage error.

I rounded off the number 8.03567 to four significant digits and it'll be 8.036 but, the problem is how can I compute the percentage error because there's not enough data to compute.
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1answer
40 views

Efficient and accurate approximatiion of logarithm of binomial coefficients

I am searching for an efficient and accurate way to approximate the logarithm of binomial coefficients since I have to deal with extremely large numbers in C++. Using Stirling approximation, I am able ...
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0answers
12 views

How to linearly approximate the L-2 norm of a complex vector?

The complex vector is given by ${\bf x}=[x_1,x_2,\cdots,x_k]$. How to linearly approximate the $l_2$-norm of $\bf x$?
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1answer
21 views

Dealing with log functions

I have a function where all variables are linear except for the log function. In one equation I have $\log(x)$ and another equation I have $\log(1-x)$. How can I linearize $\log(x)$ and $\log(1-x)$?
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1answer
35 views

Approximating $\lceil|z|\rceil$ by complex polynomials

I want to approximate function $$z\mapsto \lceil|z|\rceil$$ by complex polynomials. I construct regions $$K_n=\bigcup_{i=0}^\infty K_n^i$$ where each $K_n^i$ is an annulus with a section removed and ...
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0answers
24 views

How could I obtain this approximation of the May-Wigner theorem?

I'm trying to understand the complete proof of the May-Wigner theorem. We have a real random $n\times n$ matrix $B$ with its non-zero elements $B_{ij}$ are chosen independiently from a fixed ...
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0answers
11 views

Unbounded Separable Basis Function

Suppose to have a function like $F(x, y)=\frac{1}{1-xy}$ with $0<=xy<1$. I would like to write this function like $F(x, y)=\frac{1}{1-xy}=\sum_j \beta_j K_j(x, y)= \sum_i \alpha_i f_i(x)g_i(y)$...
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2answers
32 views

Trouble understanding Taylor expansion approximation

I'm not really understanding this Taylor expansion approximation. It states that it uses the fact that $2|\epsilon_i| < 1$ and that it is a first order approximation. The $P_i$ is a probability, so ...
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1answer
17 views

Error Bound of composite trapezium rule

Given the function: $f(x) = \cos(2x) \exp\left(-x^2\right)$ I estimated $\int_{-2}^2 f(x) \ dx$ using the formula. I need to calculate the error bound using the formula: $$ R = −\frac{b−a}{12} \cdot ...
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0answers
30 views

Find the value of $n$ that minimizes a complicated integral

I need to approximate this function: $$f(x)=\left\{\begin{matrix} \sqrt{21.49-(x+1.28)^2 }\ \ \ \text{if } x<-3.17\\ \sqrt{60.56-x^2 }-2.87\ \ \ \text{if } -3.17\leq x\leq 3.17\\ \sqrt{21.49-(x-...
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1answer
45 views

Approximation of the length of a number

Consider a positive integer number $2^{{10}^n}$ where $n\geq 5$. Question: How to prove the length of the number $2^{{10}^n}$ is equal to $30103{\underbrace{00\cdots0}_{n-5}}$. Example: Consider $n=...
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0answers
16 views

Normalise the root mean square error

I have $10$ people in a group and they undergone a surgery. I have the root mean square of each subject before and after the surgery. ...
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2answers
44 views

Proving difference between two functions is small when $x$ is small…

I'm really struggling to prove the following claim and I was wondering whether anyone could help me. Claim:$$ 0<|x| \leq 10^{-4} \implies \bigg{|} \frac{x}{e^x-1} - \frac{1}{\sum_{k=1}^3 \frac{x^...
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1answer
50 views

Can we approximate continuous functions arbitrarily well with polynomials? (beyond Weierstrass )

Let $f:(0,1) \to \mathbb{R}$ be continuous, and let $\delta:(0,1) \to \mathbb{R}$ be continuous and positive. Does there always exist a polynomial $p(x)$ satisfying $|f(x)-p(x)| < \delta(x)$ for ...
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0answers
22 views

Constant factor greedy approximation for b-fat objects maximum disjoint set

An object O is called b-fat if the ratio of the radius of the smallest circumdisk CD, O ⊆ CD and the radius of the largest indisk ID, ID ⊆ O is atmost b. Given a set of b-fat objects in the plane, we ...
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3answers
91 views

Approximation of a square root of e with a precision of $10^{-3}$ using Taylor formula

So, I have to approximate a value of square root of e: $\sqrt{e}$ with a precision of $10^{-3}$. I have calculated the first and second derivative: So instead of $\sqrt{e}$ I need to approximate the ...
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1answer
25 views

Approximating best fit gradients using averages of ratios

If I have a function $y=f(x)\approx ax+b$ where $a,b$ are constants that correlate to a set of $n$ points, could I use the following as an approximation of the gradient? $$a\approx\frac1n\sum_{i=1}^n\...
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3answers
107 views

Linearizing the trigonometric functions or: Squaring the circle by Fourier transformation

It's an easy exercise to approximate the cosine and the sine function by a piecewise linear function on the unit interval $[0,1]$. Let $\tau = 2\pi$. Let $$\boxed{\cos_\bigcirc(x) = \cos(\tau x)\\\...
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0answers
38 views

Estimating the value of a random number

Consider variables $y$, $A$ and $B$ and the following relationship between them: $$y = A + B.$$ $A$ has a uniform distribution between $[0,K]$. Distribution of $B$ is unknown, but $B$ is a random ...
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2answers
51 views

Computing the error function for Euler's number

By the error function for the sum $$\sum_{i = 0}^\infty \frac{1}{i!},$$ I mean the function $$f : \mathbb{R}_{> 0} \rightarrow \mathbb{N}$$ defined as follows. For each $\varepsilon \in \mathbb{R}...
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0answers
8 views

Derive a lower bound of weighted sum

I have the following expression $$w_1 a_1 + w_2 a_2 + \cdots + w_n a_n,$$ where $a_i$ are non-negative constants in the range $[L,U]$ and $w_i\in [0,1]$ are the weights satisfying $$w_1 + w_2 + \...
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0answers
38 views

Root objects and the simplest possible analytic continuation of the Riemann zeta function.

The equation I am trying to solve is: $$\lim\limits_{k \rightarrow 3} \left( \sum\limits_{n=1}^{n=k} \frac{1}{n^s}+ \frac{1}{k^{s - 1} \cdot (s - 1)}\right)=0 \tag{1}$$ The simplest possible ...
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4answers
606 views

Approximating $\frac {\log(5/4)}{\log(3/2)}$ to a rational number

I'm making a phone game, and I need to approximate $\frac {\log(5/4)}{\log(3/2)}$ to a rational number $p/q$. I wish $p$ and $q$ small enough. For example, I don't want $p$, $q\approx 10^7$; it's way ...
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1answer
26 views

Approximation $\ln \frac{(h+f)!}{(h-g)!}$ using Stirling’s when $f+g=o(h)$

Stirling’s approximation can be extended to a very well known inequality - $$\sqrt{2\pi n}\left(\frac{n}{e}\right)^n \leq n! \leq e\sqrt{n}\left(\frac{n}{e}\right)^n$$ How can we use this to prove, ...
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1answer
36 views

Jacobian for a semi linear differential equation problem

How would I find the jacobian in this case? Normally the Jacobian is calculated using partial derivatives of F with respect to each of the variables, but since we are using the centered finite ...
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0answers
22 views

Approximating a function in the form $\frac{1}{1-\sum_n b_n \exp(-(x-x_n)^2/a_n^2)}$

I need a good analytical approximation for the function which has the form: $$f(x)=\frac{1}{1-\sum^N_n b_n \exp(-(x-x_n)^2/a_n^2)}$$ $$|b_n|<1$$ I need this function approximated as a sum of ...
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0answers
20 views

Understanding the Power Method to find eigenvalues

For $A=\sum_i\sigma_i u_iv_i^T$, let $B=A^TA=\sum_i\sigma_i^2 v_iv_i^T$. Then $$B^k=\sum_i\sigma_i^{2k}v_iv_i^T$$ If $\sigma_1>\sigma_2$, then $B^k\to \sigma_1^{2k}v_1 v_1^T$. I don't know why a ...
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1answer
44 views

Convergence behavior of a rational iterative procedure of the form $\frac{1}{2}(z+1/z)$.

I am trying to prove the following result. First, we have an auxiliary sequence satisfying $E_{n+1}=\dfrac{E_n^2}{(1+\sqrt{1-E_n^2})^2}$ with $E_0<1$. It is called the Landen transformation. ...
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0answers
37 views

Use the given data to find an approximation for $f(x)$.

I have the following question in the book: But I do not know how to answer it, the only piece of information that my professor said is that a function differentiable means that it can be approximated ...