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).

learn more… | top users | synonyms

1
vote
3answers
1k views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
1
vote
1answer
159 views

Find fast exact value for numbers in the form $\sum_{k=min}^{Max}\frac{1}{k}$

I know I could start multiplying by all denominators and try to get the exact value that way but is there some smarter way or shortcut? Let's take simple example: $\displaystyle ...
3
votes
2answers
146 views

Numerical Approximation Involving Trig

I have a graphics problem that reduces to this: (Computer equation) alpha = arctan(X / ((Y / (Z * cos(alpha) - k)) * Z * cos(alpha))) (LaTeX) $$\alpha = ...
4
votes
2answers
594 views

Approximation vs. Interpolation

Sorry if this is a silly question, i'm just getting back into math after a long time away. My question is regarding approximation and interpolation. In which cases is it appropriate for one technique ...
1
vote
0answers
160 views

Terminology: Galerkin approximation

Dear all, I'm preparing a paper in which I'm trying to prove that my numerical approximation (Galerkin) of some mechanical problem indeed provides an approximation of the solution. In order to do so, ...
1
vote
0answers
122 views

Efficient way to recompute weights when shifting range of Legendre polynomial bases

I am storing a 2D (Cartesian) density function as a 2D patch with known X/Y limits and a set of 11 coefficients of the third order 2D Legendre polynomial basis functions over that patch. This works ...
3
votes
1answer
764 views

Stirling's Approximation and Binomial Random Variable

I am trying to follow equation (1.13) in MacKay's Information Theory textbook (http://www.inference.phy.cam.ac.uk/itprnn/book.pdf). It is: $$ \ln \binom{N}{r} = \ln \frac{N!}{(N-r)! r!} \approx (N-r) ...
3
votes
2answers
285 views

Multidimensional Interpolation within a polygon

Apologies in advance if I get terminologies wrong (not sure if "multidimensional interpolation" is the right term), I'm not really that great at maths, but here goes: Suppose we have two 2D points, ...
2
votes
2answers
125 views

Calculate other tangents which are related

I am using a small microcontroller, which has limited processing resources. I need to calculate the three tangents: ...
5
votes
3answers
449 views

$\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$

I should be able to answer this myself, but feel insecure anyway. I want to know, whether a function f(n) is bounded if n goes to infinity (and if it's bounded, the limit). Heuristically it appears ...
0
votes
1answer
192 views

How would you determine the measurement error in the following example?

I'm at a bit of a loss as to how to determine the error in measurement in a project I'm working on. The project involves taking a picture of an object, and then using the image to determine the width ...
19
votes
2answers
564 views

Maximum of Polynomials in the Unit Circle

Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by $$ ...
1
vote
2answers
946 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 ...
1
vote
2answers
457 views

best real approximation to complex numbers

I have a system of equations and its answers are complex, but I want real numbers. Is there any way to find the best real approximation to a complex number?
1
vote
1answer
177 views

bound of Erlang distribution

Is there any known polynomial bound of the Erlang distribution? I'd like to say that, given $k$ and $\lambda$ with probability p the r.v. is going to be less than some value x.
0
votes
2answers
6k views

With a few data points can a generate a close equation to meet them?

I have 1x = -40, 2x = -41 , 3x = -54 and getting a few more. Can I generate a equation for a graph that gets close to this? I was trying to get Wolfram Alpha to ...
7
votes
4answers
3k views

Is there an analytic approximation to the minimum function?

I am looking for an analytic function that approximates the minimum function. i.e., $|f(x_1,x_2) - \min(x_1,x_2)| < \zeta$ for some $\zeta$ that may be related to $|x_1 - x_2|$. Or may be a series ...
5
votes
2answers
3k views

Approximations Involving Exponential Functions

I am reading a text and I am curious to know how certain approximations were reached. The first function approximations is: $$ 1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) ...
1
vote
1answer
407 views

Having the eigenvalues how to find the eigenvectors?

This is a more practical question. In Pari/GP I have difficulties to use mateigen for some real matrices because of extreme long computation time and frequently "missing eigenspace" due to too low ...
1
vote
0answers
203 views

Gauss interpolation in 3D and friends

I was looking for approaches on how to adequately interpolate the values for a continuous 3D function for which I have the exact values in a 3D grid of equidistant points. I found that linear ...
6
votes
3answers
490 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 ...
3
votes
2answers
325 views

Asymptotic number of unlabeled graphs

A rather tight lower bound $c(n)$ of the number of unlabeled digraphs of order $n$ (loops allowed) seems to be $$c(n) = 2^{n^2}/n!$$ because there are $2^{n^2}$ labeled graphs, almost all of them ...
3
votes
1answer
610 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. ...
0
votes
1answer
937 views

SVD for a complex matrix and approximation to a real matrix

Suppose M is a 20 by 3 complex matrix, and I'd like to SVD. For example, in Matlab, I can do easily with: [U, S, V] = svd(M); where U, S, and V are complex ...
2
votes
0answers
207 views

When is it valid to convert a function inside a probability integral to the indicator function?

I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid. We want to model the lifetime $T$ of a website ...
9
votes
2answers
610 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? ...
63
votes
9answers
4k views

Find the average of $\sin^{100} (x)$ in 5 minutes?

I read this quote attributed to VI Arnold. "Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he ...
4
votes
3answers
522 views

How to find a Newton-like approximation for that function?

I want to find the complex fixpoint $t=b^t $ for real bases $b> \eta = \exp(\exp(-1))$. added remark: I'm aware that there is a solution using branches of the Lambert-W-function, but I've no ...
1
vote
0answers
448 views

Approximate linear density function for a normal distribution

I'm working on implementing Order Preserving Encryption for Numeric Data, and part of the algorithm includes approximating density of the distribution as a linear density function $f(p) = qp+r$ where ...
1
vote
2answers
1k views

How to find closest exponential approximation?

I have a bunch of data, and I'd like to find the closest exponential aproximation I can to fit the points. I'm guessing there's a (relatively) straightforward way to do this. For example, if I have ...
16
votes
5answers
1k views

How best to explain the $\sqrt{2\pi n}$ term in Stirling's?

I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation $$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq ...
2
votes
1answer
517 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
797 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
976 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
506 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
712 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
147 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
897 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
101 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
vote
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 ...
50
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
350 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
235 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
351 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 ...