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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5
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2answers
1k views

How to best approximate higher-degree polynomial in space of lower-degree polynomials?

My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)? Orginially, as the title of the post ...
4
votes
2answers
696 views

Method of dominant balance and perturbation

Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$ I can't perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant ...
1
vote
2answers
1k views

Approximation in $L^2$ by piecewise constant functions

Dear all, I'd like to know if there is any general result on the approximation of $L^2$ functions by piecewise constant functions. More specifically, I'd like to know if the following approximability ...
15
votes
2answers
214 views

A Mathematical Coincidence, or more?

According to the paper "Ten Problems in Experimental Mathematics", $$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \quad = \quad \frac{\pi}{8}\color{blue}{-7.407 \times ...
11
votes
1answer
768 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
10
votes
4answers
14k views

Find formula from values

Is there any "algorithm" or steps to follow to get a formula from a table of values. Example: Using this values: ...
7
votes
3answers
646 views

How is it that this shape can converge to what looks like a triangle but has a different perimeter?

I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered ...
7
votes
4answers
9k views

How to simplify or calculate a formula with very big factorials

I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to ...
6
votes
1answer
441 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
5
votes
3answers
945 views

How to find an approximation to $1 - \left( \frac{13999}{14000}\right )^{14000}$?

I want to find an approximation to the expression $$ 1 - \left( \frac{13999}{14000}\right )^{14000} $$ I tried by taking logarithm $$ \ln P = \ln\left(1 - ...
5
votes
0answers
278 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
5
votes
1answer
253 views

Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)

I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm. I know that Maple has a PSLQ implementation, but ...
4
votes
1answer
96 views

Is there a section of mathematics that studies near-integer equations.

When I saw: $$e^\pi-\pi \approx 20$$ I thought it was pretty cool. And : $$\pi^3 \approx 31$$ So now the thought comes to me is what positive integer value of $n$ will make the expression: ...
4
votes
3answers
129 views

Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
4
votes
4answers
210 views

How to compute a lot of digits of $\sqrt{2}$ manually and quickly?

After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in ...
4
votes
2answers
183 views

Evaluate $\sum_0^\infty \frac{1}{n^n}$

Courtesy of this xkcd comic I now know that $$ \sum_{n=1}^\infty \frac{1}{n^n} \approx \ln^e(3) $$ Echoing the views of the comic itself, if I ever find myself taking $\ln^e(x)$ then something has ...
4
votes
2answers
2k views

Approximations for the partial sums of exponential series

Though the question here (Partial sums of exponential series - Stack Exchange) is similar, it is more specialized and I rather need a general approximation for an arbitrary partial sum. Essentially, ...
3
votes
1answer
246 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 ...
2
votes
1answer
188 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
2
votes
1answer
154 views

Chebyshev coefficients from a polynomial

Is there an efficient algorithm for finding the coefficients in a Chebyshev basis of a polynomial? That is, given the set of $a_k$ such that: $p_n(x) = \sum_{k=0}^N a_k x^k $ Find the set of $c_k$ ...
1
vote
2answers
2k views

Interpolation, Extrapolation and Approximations rigorously

A foreign book mentioned that "when the Lagrange's interpolation formula fails (for example with large sample due to Runge's phenomenon), you should use approximation methods such as ...
0
votes
1answer
270 views

Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$?

The approximation $$\pi\approx\frac{22}{7}=3+\frac{1}{7}$$ suggests that the closest integer to $\frac{1}{\left(\pi-3\right)}$ is $7$. However, $$ \frac{1}{\left(\pi-3\right)^2}\approx49.879 $$ is ...
0
votes
2answers
497 views

Continuation of smooth functions on the bounded domain

Given a bounded domain $\Omega\subset \mathbb{R}^n$ and a smooth function $f$ with bounded derivatives on $\Omega$, is it possible to extend $f$ to $\tilde{f} : \mathbb{R}^n \to \mathbb{R}$ such that ...
26
votes
3answers
1k views

The right “weigh” to do integrals

Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of ...
12
votes
1answer
238 views

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
9
votes
2answers
790 views

Complex Zeros of $z^2e^z-z$

Can anyone give me a hint on showing (in a relatively elegant way, as I know the answer from WolframAlpha), that the complex valued function $z^2e^z-z$ has at most 2 roots with norm less than 2? ...
8
votes
2answers
844 views

Sum of inverse of Fibonacci numbers

If $F(n)$ is the nth Fibonacci number, How can I prove that: $$\sum_{i=1}^{\infty} \frac{1}{F(i)}\approx 3.36\, .$$
7
votes
3answers
54k views

Approximation symbol: Is π ≈ 3.14.. equal to π ≒ 3.14..?

This could be a trivial question, but what is exactly the difference of between these two expressions? Am I correct to state the both interchangeably whenever I need to express the approximation of ...
4
votes
4answers
91 views

Generalization of piece-wise linear functions over a metric space

Suppose we want to construct a function $f$ from a compact metric space $(X,\rho)$ to a Euclidean space $\mathbb{R}^n$ that is Lipschitz continuous with a constant $L$: $$ \forall x,y \in X . ...
4
votes
3answers
1k views

How to calculate $\exp(-x)$ using Taylor series

We know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...
4
votes
1answer
105 views

$f$ is approximated uniformly on $R$ by $p_n(x)$, then $f$ is a polynomial

Suppose $p_n(x)$ is a sequence of polynomials which converge to a function $f$, uniformly on $\mathbb{R}$. Show that $f$ is a polynomial. If there were a uniform bound $M$ on the degree of $p_n(x)$, ...
4
votes
1answer
234 views

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow ...
3
votes
1answer
349 views

What is the proof of the rules of significant figures?

I wanted to know how do we know that the rules that we follow when doing arithmetic with significant figures are correct? Like why when adding or subtracting we keep the same number of decimal places ...
3
votes
2answers
818 views

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
3
votes
1answer
693 views

Approximation for Lambert W function near zero

I am looking for a good approximation for the $W_0$ branch of the Lambert $W$ function. I am looking for values $0 < x < e$ only, so I expect something simpler than the general Taylor expansion. ...
2
votes
1answer
53 views

How do you Taylor expand the log likelihood function of the Poisson distribution?

This question is an extension to this previous question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a ...
2
votes
1answer
812 views

Approximating a large number of data points using (cubic) splines in l1/l2 norm.

I have a pretty large dataset ($x,y$) consisting of a few million points. There is a lot of noise in the data. I want to find a smooth but simple approximation/representation for this dataset, so that ...
2
votes
1answer
815 views

Bounding the modified Bessel function of the first kind

i'm looking for an upper bound for the modified Bessel function of the first kind of a +ive real argument. It seems that it satisfies the inequality : $$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$ ...
1
vote
2answers
200 views

Can every continuous function on complex domain be approximated by polynomials pointwise?

Do you know any theorem that will help me with this question: Let $f$ be any continuous function on complex plane. Show that there is a sequence $(P_n)$ of polynomials such that $P_n$ converges ...
1
vote
2answers
734 views

Product of Gamma functions II

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right) \end{align} and can it be shown that \begin{align} \prod_{k=1}^{20} ...
1
vote
1answer
2k views

Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s and q.

The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 ...
1
vote
1answer
2k views

How to approximate an integral using the Composite Trapezoid Rule

I'm trying to estimate the value of the following integral on the interval $[0,1]$ $$ I = \int_0^1 \frac{1}{1+x} dx $$ So, using the composite trapezoid rule (and with $n=4$, ie I'm only using the ...
0
votes
1answer
36 views

When is right to kill $r^l$ and/or $r^{(-l-1)}$?

When we solve the Laplace equation in spherical polar coordinate, we get the radial part whose solution is: $$R=Ar^l+Br^{-(l+1)}$$ Now, some solutions keep this two terms, but when we derive the ...
0
votes
3answers
878 views

Approximating the integral $\int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$

How would I go about evaluating $I = \int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$ to four decimal accuracy?
6
votes
5answers
1k views

What exactly is “approximation”?

There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
5
votes
1answer
96 views

Inverse of $\frac{1-e^{-x}}{x}$ on $(0,1)$

I am trying to invert (or to estimate the inverse of) $$y=\frac{1-e^{-x}}{x}$$ for $y\in(0,1)$. The function 'looks' monotonically decreasing between $x=0$ and $x=\infty$, but I have not been able to ...
5
votes
0answers
73 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
5
votes
5answers
3k views

Numerically Efficient Approximation of cos(s)

I have an application where I need to run $\cos(s)$ (and $\operatorname{sinc}(s) = \sin(s)/s$) a large number of times and is measured to be a bottleneck in my application. I don't need every last ...
4
votes
1answer
673 views

Newton-Raphson for reciprocal square root

I have a question about using Newton-Raphson to refine a guess of the reciprocal square root function. The reciprocal square root of $a$ is the number $x$ which satisfies the following equation: ...
4
votes
2answers
113 views

Approximation of DE

It depends on my previous question. Closed form solution of DE I don't want to deal with Airy functions. How can I approximate this DE in continous domain $[0,1]$? $$y''(x)+(x+1)y(x)=0\quad\text{ ...