For questions related to approximations
3
votes
2answers
3k views
Relation between Simpson's Rule, Trapezoid Rule and Midpoint Rule
I am studying numerical approximation and verifying $S_{2n} = \frac{1}{3}\left(T_n +2 M_n\right)$. ($S_n$ refers to Simpson's Rule approximation, $T_n$ refers to Trapezoid Rule approximation and $M_n$ ...
3
votes
2answers
227 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
2answers
316 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 ...
3
votes
1answer
215 views
Can the trigonometric functions be expressed, explained, or proven in terms of arithmetic?
I'm trying to wrap my head around sine, cosine, and tangent. I'm aware that they're commonly defined in high schools as ratios of the various parts of triangles set in the unit circle, but that's not ...
3
votes
3answers
659 views
Approximating log of factorial
I'm wondering if people had a recommendation for approximating $\log(n!)$. I've been using Stirlings formula,
$ (n + \frac{1}{2})\log(n) - n + \frac{1}{2}\log(2\pi) $
but it is not so great for ...
3
votes
1answer
497 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:
$$ \ln \binom{N}{r} = \ln \frac{N!}{(N-r)! r!} \approx (N-r) ...
3
votes
2answers
107 views
Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$
I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer.
I am interested to find the bounds on the value it can take or an ...
3
votes
4answers
94 views
Improving Newton's iteration where the derivative is near zero?
I'm implementing a root-solver for finding x coordinates of a function f(x), after I have an y-coordinate. The function is periodic, roughly sinusoidal with constant amplitude but non-linearly ...
3
votes
1answer
69 views
Approximating the logarithm of a Laplace transform
Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$.
Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e.
...
3
votes
1answer
57 views
Disc method of approximating volume
There are a couple things I'm unclear on regarding the disc method of approximating shape volume. Given $x=y^2$ and $x=4$, determine the approximate volume by revolving the shape around the line ...
3
votes
2answers
158 views
Multidimensional Interpolation within a polygon
Apologies in advance if I get terminologies wrong (not sure if "multidimensional interpolation" is the right term), I'm not really that great at maths, but here goes:
Suppose we have two 2D points, ...
3
votes
3answers
385 views
How to evaluate probability estimators with only external information?
Here's a problem that I have pondered over many times without ever coming to a satisfactory solution:
Let's say that we have a series of random events: V(i) for I = 1 to n. Each of these events will ...
3
votes
1answer
144 views
Laplace's method with unknown exponent.
Given the integral:
$$I = \int_0^a{e^{-\lambda g(x)}f(x)dx}$$
Where $g(x)$ and $f(x$) are both low order positive polynomials, and $\lambda \gg 1$, Laplace's method is commonly used to approximate ...
3
votes
1answer
87 views
A problem on $C^k$
Can anyone help me solve the following problem ?
Let $a,k >0$, $a$ is real and $k$ is integer. Consider the set $S$ of all function $f\in C^k([0,a])$ such that
1) $f(0)=0$ and $f(a)=1$
2) ...
3
votes
1answer
603 views
Approximation of a bounded measurable function with step functions?
I'm having trouble judging whether this statement is correct:
For an arbitrary bounded measurable function $f$ defined on $[0,1]$, $\exists{}\ $a sequence of step functions $\{\phi_n\}$, such that ...
3
votes
1answer
718 views
Projection of Gaussian in Spherical Coordinates
Consider a point with spherical coordinates $\vec{r}_0=(r_0, \theta_0, 0)$. The spherical gaussian distribution centered at $\vec{r}_0$ is $f(\vec{r})=Ne^{|\vec{r}-\vec{r}_0|^2/A}$, where $N$ is the ...
3
votes
1answer
183 views
Differentiability of Moreau-Yosida approximation.
I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...
3
votes
1answer
131 views
Calculate $E[X]$ using polynomial approximation of CDF
I have a black box called $F(t)$ ($~$($P~(X\le t)~$, $X$ is random variable) with me where I don't have any information on the exact expression of $F(t)$. But if I supply a $t\ge 0$ I will get a value ...
3
votes
2answers
314 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
2answers
394 views
Finding the real roots of a polynomial
Recent posts on polynomials have got me thinking.
I want to find the real roots of a polynomial with real coefficients in one real variable $x$. I know I can use a Sturm Sequence to find the number ...
3
votes
1answer
64 views
Linearization of an implicitly defined function.
Problem:
Given the equation: $xz^{2}+y^{2}z^{5}=19$
Also given: (3,4,1) is a solution to the equation. This point is not the only solution.
1) Find dz/dx and dz/dy (through implicit ...
3
votes
2answers
50 views
Discrete approximations of $\nabla^2{\bf v}$
I am writing a Navier Stokes solver. The vector field is represented as a grid with integer coordinates
I am looking at other people's computer code. I don't entirely understand the vector calculus, ...
3
votes
1answer
75 views
Brownian motion and hitting frequency
Suppose we have a Brownian motion $B_t$ with $B_0 = 0$ and $B_t - B_s \sim N(0,t-s)$. Every time $B_t$ hits $\pm h$, where $h$ is some "barrier" $>0$, I pay someone £1 and the brownian motion ...
3
votes
1answer
56 views
Iterative model fitting
I have a sequence of points $\{(x_k,y_k,z_k)\}$ and I need to fit some $2D$ model $P(x,y)$ that approximates $z$ in some sense.
The $z_k$$'s$ are noisy samples of some $2D$ function $z_k = f(x,y) + ...
3
votes
1answer
72 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
1answer
66 views
Calculating the divisor, known to be small, of two Stirling approximations of the logarithmic Gamma function without overflows
Earlier, I asked a question on MathOverflow
regarding how one might analytically approximate a function of the form: $f(n) = \prod_{i=1}^{n-1} (1-ai)$ for $a \ge 0$, $(ai) < 1$, and $n > 10^5$ ...
3
votes
1answer
73 views
is the approximation of the sum true?
Someone commented under my question Calculation of the moments using Hypergeometric distribution that
$$
\sum_{k=0}^l\frac{{l \choose k}{2n-l \choose n-k}(2k-l)^q}{{2n\choose n}}\sim \sum_{k=0}^l ...
3
votes
1answer
1k views
What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline?
I am reading a book about computer graphics. It is confusing about the various splines and their algorithms. What is the difference between natural cubic spline, Hermite spline, Bézier spline and ...
3
votes
2answers
74 views
Approximation or bound of $\operatorname{Pr}(X<\operatorname{E}(X))$
$X$ is a continuous random variable (we can assume some statistic (e.g., mean and variance) are known, but the distribution is unknown). Consider a probability ...
3
votes
1answer
382 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. ...
3
votes
1answer
380 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 ...
3
votes
0answers
54 views
How to solve a distance problem inside of a picture?
sorry for my bad english. I have the following problem:
In the picture you can see 4 different positions. Every position is known to me (longitude, latitude with screen-x and screen-y).
Now i want ...
3
votes
0answers
48 views
Root calculation by hand (division-like algorithm)
I remember from my highschool days a division-like algorithm for calculating square, even cubic roots. I know the continued fraction method, some series and Newton's method. I have checked similar ...
3
votes
0answers
100 views
Runge-Kutta Error Analysis
Could anyone explain to me how to reduce the error propagated by using Runge-Kutta of order 4? Or can anyone give me a nice reference to it.
3
votes
0answers
37 views
Bound on Permutations [duplicate]
I am trying to prove the following inequality,
$$n^{(l)} = n(n-1)\cdots(n-l+1) \geq \frac{n^l}{e}\quad\text{ for }\quad 2 \leq l \leq \sqrt{n}\;.$$ So my approach is to observe that $n^{(l)} = ...
3
votes
0answers
70 views
Question about linearization
Given a data matrix $D\in\mathbb{R}^{N \times N}$
Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$,
if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
3
votes
1answer
120 views
Rayleigh-Ritz method for an extremum problem
I am trying to use the Rayleigh-Ritz method to calculate an approximate solution to the extremum problem with the following functional:
$$ L[y]=\int\int_D (u_x^2+u_y^2+u^2-2xyu)\,dx\,dy, $$
$D$ is the ...
3
votes
0answers
42 views
When can the commutator of two matrices be neglected in series expansions?
Under what conditions can the higher order commutators in the Baker–Campbell–Hausdorff formula be neglected when the commutators does not vanish exactly and there is no small parameter in the ...
3
votes
0answers
144 views
Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation
Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$.
How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
3
votes
1answer
144 views
Remainder term for Gauss-Laguerre quadrature
I need to calculate a quadrature rule with maximum degree of accuracy that looks like this:
$$
\int_0^\infty e^{-x}f(x)dx = \sum_{i=0}^n A_if(x_i) + R_n(f)
$$
where $n=2$.
For $R_n(f)$ I have this ...
3
votes
0answers
74 views
Numerically estimate $a^b$ [duplicate]
Possible Duplicate:
How can I calculate non-integer exponents?
What is the most efficient way to estimate $a^b$ ($a > 0$) numerically?
My goal is not to use built-in math functions (like ...
3
votes
0answers
59 views
Is that observation really a property of the log of coefficients of continued fractions (example: cf(log(3)/log(2))
I'm again looking at the problem of approximation of perfect powers of 2 to that of 3 (I assume $\small q_N = 2^S / 3^N \gt 1 $) and specifically using the continued fraction representation of $\small ...
3
votes
0answers
159 views
Approximating a system of differential equations as a Bézier curve
I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve.
Sorry if my ...
3
votes
0answers
151 views
Approximation of a real number as a linear combination of two reals with coprime integral coefficients
Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
2
votes
4answers
295 views
What is the correct value of $\pi$
I have seen that:
$\pi = 22/7$
$\pi = 3.14\ldots$
$\pi = 17 - \sqrt{192}$.
$22/7 \gt \pi$
$22/7 \lt\pi$
My brain storming doubt was, is A = B? Is B = C? Is C= A = B? how D and E are correct or ...
2
votes
3answers
253 views
How to calculate the area of bizarre shapes
I'm looking for an algorithm to calculate the area of various shapes (created out of basic shapes such as circles, rectangles, etc...). There are various possibilities such as the area of 2 circles, 1 ...
2
votes
2answers
3k 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 ...
2
votes
4answers
91 views
Analytical approximation of an integral
I think there is no analytical solution for
$$
\int_{K}^{\infty} \frac{exp(-x)}{x} dx
$$
where $K > 0$. Instead, is there an analytical approximation?
2
votes
4answers
627 views
Approximating $\pi$ using Monte Carlo integration
I need to estimate $\pi$ using the following integration:
$$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$
using monte carlo
Any help would be greatly appreciated, please note that I'm a student trying to ...
2
votes
2answers
360 views
Where is the error on this approximation to $\pi$
I came across this website (see also) where the author (a supposedly alternative mathematician) claims to have a better approximation to $\pi$.
$$\pi\approx 3.1547…$$
Can someone tell me where is ...
