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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2
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1answer
511 views

Uniform approximation of continuous functions by polynomials in two variables

Consider a subset $K\subset \mathbb C^2$ consisting of pairs $(z,\bar z)$ such that $|z|=1$. Is there an easy way to see that continuous functions on $K$ can be uniformly approximated by polynomials ...
11
votes
2answers
789 views

Application de Stone-Weierstrass

Bonjour, J'ai rencontré le problème suivant dans le livre "Real and Functional Analysis" de Lang, au chapitre 3. J'explique d'abord le contexte, puis j'en viendrai à la question précise. Il faut ...
0
votes
1answer
192 views

Restoring the function by its graph

I need a function that will produce a graph similar to the one below. This function is odd, symmetrical relatively to origin in III quarter. A is an asymptote (the top part is similar to ...
2
votes
4answers
971 views

Approximating $\pi$ using Monte Carlo integration

I need to estimate $\pi$ using the following integration: $$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$ using monte carlo Any help would be greatly appreciated, please note that I'm a student trying to ...
1
vote
1answer
326 views

Complex-Analytic theorem similar to Runge's theorem

I'm trying to prove a result similar to Runge's theorem and Mergelyan's theorem (link at the bottom of the previous link), but without the condition of analyticity. The problem is as follows: Let γ : ...
6
votes
4answers
501 views

An approximation of an integral

Is there any good way to approximate following integral? $$\int_0^{0.5}\frac{x^2}{\sqrt{2\pi}\sigma}\cdot \exp\left(-\frac{(x^2-\mu)^2}{2\sigma^2}\right)\mathrm dx$$ $\mu$ is between $0$ and $0.25$, ...
3
votes
3answers
709 views

How to evaluate probability estimators with only external information?

Here's a problem that I have pondered over many times without ever coming to a satisfactory solution: Let's say that we have a series of random events: V(i) for I = 1 to n. Each of these events will ...
0
votes
1answer
146 views

Taylor Series. Reusing an approximation of a function

I have this function, $e^{-x}$ bounded between 0 and 1500 and I have an approximation by Taylor Series of the same function bounded between 0 and 0.5. I would like to express my function $e^{-x}$ ...
2
votes
1answer
882 views

Numerical integration - Gauss quadrature rule

How do we numerically integrate a rapidly decaying exponential function? A simple Gauss quadrature which is based on approximating the function by polynomial, I think will not work, since rapidly ...
0
votes
1answer
100 views

Development of a specific hardware architecture for a particular algorithm. Modelling fuctions by Taylor sSeries.

I'm trying to develop a architecture hardware to make a implementation of an algorithm that can be descompose in terms of sums, multiplications, subtractions and exponential functions. I'm trying to ...
1
vote
1answer
282 views

How to get NURBS control points from an array of points that should be part of its solution from controll points we are searching for?

We are talking about Non-uniform rational B-spline. We have some simple 3 dimensional array like {1,1,1} {1,2,3} {1,3,3} {2,4,5} {2,5,6} {4,4,4} Which are ...
1
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0answers
1k views

What are the advantages / disadvantages of using splines, smoothed splines, and gaussian process emulators?

I am interested in learning (and implementing) an alternative to polynomial interpolation. However, I am having trouble finding a good description of how these methods work, how they relate, and how ...
48
votes
4answers
4k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
0
votes
1answer
62 views

Estimating the equation of a line from its rounded values

So, I'm playing a game where there is a certain percent value that changes over time. I look at it occasionally, it seems to be approximately linear in time. However, the change is slow enough that ...
8
votes
1answer
349 views

Estimates for Stirling's formula remainder

It is proved in Advanced Calculus by Angus Taylor, § 20.8, that $$\log n!=\log \left( \left( \frac{n}{e}\right) ^{n}\sqrt{2\pi n}\right) +r_{n},$$ where $$r_{n}=\sum_{k=1}^{\infty }S_{k}$$ with ...
1
vote
0answers
233 views

Multivariate B-Spline Derivatives

To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension. What I'd like to know is how do you ...
3
votes
1answer
348 views

Quadratic splines, minimizing integral

Hi I have a problem with quadratic splines, I am supposed to find $ S_1 $ and $S_2$ that interpolates the following points $S(-1)=0$, $S(0)=1$, $S(1)=2$, and at the same time we want to find $S$ such ...
5
votes
3answers
5k 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: ...
2
votes
1answer
187 views

How to decompose a result of a multiplication?

I've got a multiplicative-with-noise model $F(x,y)=S(x)*R(y)*D(x,y)+N$, where $S(x)$ and $R(y)$ are unknown functions, $D(x,y)$ is a distance function, that is, a function that depends only on ...
2
votes
2answers
488 views

Is this integration approximation method known/used?

I'm approximating an integral with only exponentials. i.e., it is equal to $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{c_j e^{i\cdot d_j \cdot t}}}{\displaystyle\sum_{k=a}^b{r_k ...
2
votes
1answer
162 views

Using derivatives to evaluate an integral of exponentials

Reading a book on fractional calculus reminded me that I'd like to know more on the following method/idea. Given an integral: $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{e^{i\cdot ...
3
votes
2answers
178 views

Next term in $(1+a/n)^n \rightarrow \exp (a)$

Working on the generalized birthday problem, where you draw with replacement from $\{1,2,3, \ldots,d\}$ and look for the number of draws $n$ for which you have greater than $1/2$ chance of a match I ...
1
vote
0answers
292 views

How effective is this alternative to integration?

I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following: $f(x) = ...
2
votes
2answers
265 views

Sequence of smooth functions approximating a 2d cylinder step function

Let $f(x,y)=1$ if $(x,y)$ is in the unit disk and $f=0$ otherwise. I would like to approximate $f$ by a sequence of smooth functions. The functions need to be evaluated quickly so the results I'm ...
6
votes
2answers
1k views

I want to graph an equilateral triangle on graph paper

I want to graph an equilateral triangle. It would be ideal if I had a set of three points: $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ with $x_i, y_i \in \mathbb{Z}$ as the vertices. However, this is ...
1
vote
1answer
257 views

How to fit a 3-D parametric equation to datapoints

Consider that I have $3$ parametric equations as function of time and describe the motion of a body in space: $x = f(t)$ $y = g(t)$ $z = h(t)$ These curves are pretty simple and can be modeled ...
6
votes
4answers
4k 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 ...
4
votes
1answer
473 views

Large Deviation Properties of a function of a geometric random variable

Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is $p_s = (1 - x)^{s - 1} x$, Consider the following function of ...
8
votes
3answers
715 views

Why are some mathematical constants irrational by their continued fraction while others aren't?

Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore ...
4
votes
1answer
661 views

Approximating $\pi$ using Monte Carlo integration

I'm trying to approximate $\pi$ using Monte Carlo integration; I am approximating the integral $$\int\limits_0^1\!\frac{4}{1+x^2}\;\mathrm{d}x=\pi$$ This is working fine, and so is estimating the ...
10
votes
3answers
417 views

Solving randomized recurrence relation

I'm looking at the random sequence $x_n$ with $x_0=x_1=1$ and \begin{equation} x_{n+1}=2x_n\pm x_{n-1} \end{equation} where we choose the $\pm$ sign independently with equal probability. Now ...
6
votes
0answers
404 views

Using Padé approximants for the quaternion exponential

On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
25
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 ...
6
votes
3answers
1k views

How to approximate/connect two continuous cubic Bézier curves with/to a single one?

I subdivide a cubic Bézier curve at a given t value using de Casteljau’s algorithm, which yields two cubic Bézier curves. Afterwards I “scale” the second curve (proportionally). I’d like to reconnect ...
10
votes
5answers
572 views

Approximation theorems

The Weierstrass' approximation theorem for continuous functions on a compact space by using polynomials is well-known. As far as I know, there are some variants of this theorem, e.g. Stone-Weierstrass ...
7
votes
2answers
790 views

Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?

The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some properties of the gamma function: $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$ $\Gamma (1)=\Gamma ...
3
votes
2answers
431 views

Math behind a “fling”? (i.e. on a mobile touch device)

I'm working on a game which relies on "flinging" an object. That is, click and hold on the object, and then drag and release it, and it continues on the path you were dragging it. Of course, the most ...
2
votes
2answers
380 views

Approximating a cosine

Let $\theta_{kl}$ be an angle such that $\cos\theta_{kl}=\frac{1}{2}(\cos(\frac{2\pi k}{n})+\cos(\frac{2\pi l}{n}))$. Given that definition, if I introduce a new variable $t$ is the following a ...
2
votes
3answers
187 views

Determine speed of the object at the current time by the non-uniform time sample

Here is a time sample: $Q = \{(t_i, x_i) | 0 \leq x_i \leq x_{i+1}, 1 \leq i \leq n\}$ and rules: (1) $T_1 \leq t_{i+1} - t_i < T_2$ where $T_1, T_2 > 0$ (2) $x_i$ comes with error: $x_i = ...
70
votes
3answers
3k views

Is there an integral that proves $\pi > 333/106$?

The following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves ...
4
votes
4answers
3k views

Why is $22/7$ a better approximation for $\pi$ than $3.14$?

This seems counterintuitive, but $22/7$ is closer to $\pi$ than $3.14=314/100$ which has a significantly greater denominator. Why is $22/7$ a better approximation for $\pi$ than $3.14$? This has ...
4
votes
2answers
9k 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 ...
6
votes
3answers
348 views

Are there variations on least-squares approximations?

In least-squares approximations the normal equations act to project a vector existing in N-dimensional space onto a lower dimensional space, where our problem actually lies, thus providing the "best" ...
16
votes
6answers
9k views

Simple numerical methods for calculating the digits of $\pi$

Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that ...