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
50 views

How to find a sequence by its limit?

Is there any way to construct non-trivial sequence by its limit? Something like $\begin{cases} a_1=2 & \\ a_{n+1}=\dfrac1{2}\left(a_n+\dfrac2{a_n}\right) \end{cases}$for $\sqrt2$. I'm especially ...
3
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2answers
117 views

A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?

Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why, $$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$ $$\sum_{n=0}^\infty P(n) q_4^{n+1}\approx\...
0
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1answer
110 views

How do I know that this is the density of the Chebyshev Points?

By knowing that a discrete distribution of points go asymptotically to the density: $\displaystyle p(x)= \frac{1}{\pi \sqrt{1-x²}}$ in $[-1, 1]$ I am able to conclude that interpolating at those ...
1
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1answer
161 views

Minimizing the $L^2$ error when approximating with trigonometric polynomials

I want to find approximations ${\rm g}_{n}\left(x\right) \in T_{n}$ of $\,\,{\rm f}\left(x\right)$, so that the error $$ \left\vert\left\vert\,{\rm f} - {\rm g}_{n}\,\right\vert\right\vert^{2} = \int_{...
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3answers
316 views

Series Expansion of $\arcsin\left(\frac{a}{a+x}\right)$

Can anyone think of a good approximation to: $$ \arcsin\left(\frac{a}{a+x}\right)\ $$ accurate at $x = 0$? The Taylor series is not available...perhaps some other kind of method?
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1answer
207 views

Regularized distance function on Riemannian manifolds

Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let $\epsilon&...
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1answer
47 views

Affine approximation of countinuous functions?

Is it true that any real continuous function can be approximated by a piecewise affine function? If true, can you suggest a link or something related to the question? Thanks
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2answers
84 views

Approximating $e^{-x}$

So we all know that the Taylor series for $e^x$ is $1 + x + \frac{x^2}{2} + \frac{x^3}{6}+\ldots$ Similarly for $e^{-x}$ it comes comes out to be $1 - x + \frac{x^2}{2} - \frac{x^3}{6}+\ldots$ Now for ...
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6answers
4k views

How to calculate square root or cube root? [duplicate]

I was reading Richard Feynman biography when I read that one time he was able to calculate the cube root of large number in his brain by just using simple facts of everyday life. So my question is ...
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0answers
21 views

Can $u\in W^{1,2}_0$, $\Delta u\in L^2$, $u\geq 0$ be approximated by a sequence smooth function $u_k\geq 0$

Assume that $\Omega\in R^3$ is a bounded Lipschitz domain. $u\in W^{1,2}_{0}$, $\Delta u\in L^2$, $u\geq 0$. Is it possible to approximate u by a sequence of nonnegative smooth functions $u_k$, $u_k|_{...
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1answer
409 views

Approximate a complex measurable function pointwisely almost everywhere by polynomials

This is Exercise 13.12 in Rudin's Real and Complex Analysis: Let $f$ be a complex-valued measurable function defined in $\mathbb{C}$. Then there is a sequence of polynomials $P_n$ such that $\lim_{...
3
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2answers
311 views

Sums of central binomial coefficients

Are there closed forms for $$\sum^n_{i=0} \binom{2i}{i}$$ and $$\sum^n_{i=0} \binom{2i}{i}^2$$? Also, how can these sums be approximated?
4
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1answer
184 views

Is it possible to approximate all angles with certain pythagorean triples?

With sticks $a,b$ and $c$ of length $3,4$ and $5$, you able to draw a right (tri)angle. But are also able to construct an angle $\cos\alpha=\frac35, \alpha=\arccos(\frac35)=$$53.13010...^°$. Is it ...
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1answer
35 views

Showing that $\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0$ but the matching series does not converge

I want to show that: $$\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0 $$ And also $${\displaystyle \sum_{n=1}^{\infty}{n \choose \left\lceil \frac{n}{2}\right\...
0
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1answer
229 views

Variational calculus applied to the strain energy functional in solid mechanics

The question is basically about when to apply the variational operator... Given the general functional representing the strain energy of a solid under a given stress state $\sigma$ and strain state $\...
0
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1answer
46 views

normal as approximation to binomial

Among 784 checks, 479 had amounts with leading digits of 5, but checks issued in the normal course of honest transactions were expected to have 7.9% of the checks with amounts having leading digits of ...
0
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1answer
191 views

Intuitive proof of interpolation polynomial existence

Problem: Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n ]$...
7
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1answer
456 views

Approximate Holder continuous functions by smooth functions

Let $g \in C^{\alpha} (B_1)$ be given. Can we find a sequence $(f_n) \subset C^{\infty} (B_1)$ such that $f_n \rightarrow g$ in $C^{\alpha}(\overline{B_1})$? If so, how can it be done? I have tried ...
2
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1answer
53 views

Approximate $f(x) = \frac{1}{x}$ using $e^x, \sin(x)$ and $\Gamma(x)$

My task is to approximate $f(x) = \frac{1}{x}$ using a linear combination of $e^x, \sin(x)$ and $\Gamma(x)$ in the interval [1, 2] with a step width of 0.01. How is this possible?
2
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0answers
51 views

Is there a known algorithm for approximating all the real and imaginary zeros of any well behaved equation of a single variable?

Does there currently exist a general algorithm (or set of algorithms used together) that will approximate all the zeros of any well behaved non-differential equation of a single variable which has a ...
6
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1answer
804 views

On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is ...
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1answer
76 views

Understanding an Approximation

I am reading the paper A Group-theoretic Approach to Fast Matrix Multiplication and there is an approximation in the paper I don't fully understand. In the proof of Theorem 3.3. it is stated that $$ ...
2
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0answers
63 views

Approximating a function using its integral

Question: Let $f:\Bbb R \to \Bbb R \in C^{1}, \forall \delta>0:$ $$F_\delta = \frac 1{2\delta}\int^{x+\delta}_{x-\delta} f(t) \, d(t)$$ in $[a,b]$ prove that $\forall \varepsilon>0 \exists \...
2
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1answer
146 views

show that any continuous function can be approximated uniformly

I do not know where to start because i have not dealt with a question like this before. I feel that i have to use the Stone-Weierstrass theorem, but im not sure how to use it.
3
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1answer
177 views

$n \approxeq k + 2^{2^k}(k+1)$. How can one get the value of $k(n)$ from this equation?

I am trying to find approximation for this sum. Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$ Doing following way. Let $a_k(n) = \frac{n\choose k}{2^{2^k}}$. I tried to ...
2
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2answers
141 views

Pade Approximations convergence acceleration

Why Pade Approximatoins accelerate the convergence of series? Generally speaking, what is there an advantage in the sence of convergence acceleration using rational interpolation? Thanks much in ...
3
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1answer
249 views

Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$

I need to find the asymptotic approximation of this sum $$\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$$ Can you please share a link to theory or hint how it can be solved? Here is my attempt $n \...
4
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1answer
108 views

Approximating $e^{inx}$ by polynomials

Show that every function $e^{inx}$ can be uniformly approximated on $[-\pi,\pi]$ by polynomials in $x$. Using the power series expansion, $$e^{inx}=1+inx+\frac{(inx)^2}{2!}+\frac{(inx)^3}{3!}+\ldots$...
4
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1answer
119 views

A Neat Rotation Matrix Identity

Let $\mathbf{R}_i$ be $N$ rotation matrices that represent a rotation around axes $\mathbf{\omega}_i$ by an angle $|\mathbf{\omega}_i|$. Now say we know that the product of these matrices is unity, i....
3
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0answers
52 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
0
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1answer
140 views

How do I use the linear approximation of a function given a value, a, and change in x?

My book gives a few definitions/formulas for obtaining linear approximation, but I'm having trouble understanding how to use them. Heres the question: a.) Use the Linear Approximation for f(x) = ln($...
5
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1answer
115 views

When does $f(a),f(f(a)),f(f(f(a)))…$ produce better and better approximations to $x=f(x)$?

I tried to approximate the solution to $x=f(x)$ for some given $f$, by guessing $x=a$, then I observed that $x=f(a)$ was an even better approximation, and $x=f(f(a))$ and so on was even better, so why ...
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0answers
36 views

Convergence acceleration using rational approximation

How Pade Approximations accelerate the convergence of series??? Here I don't mean only the Pade Approximations, just in general how do the rational approximations contribute to series convergence ...
2
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1answer
440 views

Error analysis of exponential function

By definition: $$ e^x = \lim_{n \rightarrow \infty} ( 1 + \frac{x}{n} ) ^ n$$ I am interesting in calculating the error $$\left | e^x - \left( 1 + \frac{x}{n} \right) ^ n \right|$$ for some fixed $n \...
1
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2answers
480 views

Least squares problem linearization

Suppose we want to find the best coefficients $a$ and $b$ that fits the data we have according to a model of the form $$ y = a t e^{bt} \text{ or } y = a e^{bt} \text{ or } y = a \left( \frac{x}{b+x} \...
2
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2answers
2k views

How to find upper bound on absolute error with composite trapezoid rule

Obtain an upper bound on the absolute error when we compute $\displaystyle\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points. The formula I'm ...
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10answers
1k views

How can I find the value of $\ln( |x|)$ without using the calculator?

I want to know if there is a way to find for example $\ln(2)$, without using the calculator ? Thanks
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1answer
91 views

Can we get an analytical solution to this equation involving the Lambert W function?

Can we get an analytical solution to the variable $t$: $$H\left(1+W\left(A\exp\left(Bt\right)\right)\right)=1+W\left(X\exp\left(Yt+Z\right)\right)$$ $W(x)$ is the Lambert W function.$A$ $B$ $X$ $Y$ $Z$...
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2answers
133 views

How to approximate this series?

How to approximate this series, non-numerically? $ S_n = \sum_{n=1}^{50} \sqrt{n}$
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1answer
165 views

Which form of Euler-Maclaurin formula to use?

This question may be rather elementary, but I am sort of confused about various forms of the Euler-Maclaurin summation formula and their use. For instance, let us suppose that we want to approximate ...
12
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2answers
15k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = y_i+(a_1k_1+a_2k_2+...
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0answers
50 views

Coefficient $a_k$ of generating function

Given a generating function $F(z)$, am I right to say that the coefficient $a_k$ of $[z^n]$ is computed by $\frac{F^{(k)}(0)}{k!}$ $(1)$. Since we have the definition of $F(z)$ is: $F(z) = \sum_{i &...
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3answers
67 views

Estimating $\sum_{k=1}^N a_kb_k$ given the means $\bar a_k,\bar b_k$ and determining the error

I need to calculate the following expression: $$\sum_{k=1}^N a_k b_k$$ I know the average values of $a_k$ , defined as $\overline {a_k} = {\sum_{k=1}^N a_k \over N } $ and $b_k$ , defined as $\...
2
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1answer
35 views

Approximate sector between two lines?

I need to approximate a red figure. I know coordinates of three points (little transparent circles). I also know a count of segments I need to divide this figure. The angle may be from 0 to Pi and may ...
0
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1answer
200 views

Cubic splines on a grid

I trying to work out how to interpolate on a grid with cubic splines. Let the point at which I'm trying to interpolate be at {xp,yp}. At the moment I am fitting splines across the rows and then ...
0
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0answers
193 views

Approximations other than taylor series and pade approximation

I have a function which has the following form: $$ f(x)=K_1 \coth (Q_1 Q_2 \sqrt{x})^2 + \frac{1}{x}\left[K_2 + K_3 \coth(Q_1 Q_3\sqrt{x})\sqrt{x}\right]$$ and I want to find $x$ for $f(x)=1$. I'm ...
3
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1answer
422 views

Upperbound approximation to the sum of Euler's totient function

I am currently working on a solution to a problem related to the density of finite coprime sets. I believe that I have found a solution to this problem - though it can only be expressed in terms of ...
0
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1answer
262 views

Proximal functions

I am a little bit new to proximal functions and I am currently stuck with the following problems How would I derive the prox function for the regularizer (h(x) function) : $\alpha\sum_{k+} $ and for ...
1
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1answer
331 views

Proximal operator

What is a proximal operator and how would one derive it in general for a function? In particular, if I had a function: $ f(x) = x^TQx + b^Tx + c $ How would I get the proximal operator for this if Q ...
0
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1answer
1k views

Normal approximation to the log-normal distribution

Intuitively, it seems that a lognormal distribution with a tiny $\sigma/\mu$ ratio might look quite a bit like a normal distribution. Can this be formalized in any way (e.g., by stating upper bounds ...