For questions related to approximations
2
votes
2answers
81 views
Using differentials to approximate a function
So I have a homework problem that I cannot figure out. I am supposed to approximate the value of $\sqrt{(4.98)^2-(3.03)^2}$ using differentials.
What I have so far is $$f(x,y)=\sqrt{x^2-y^2}$$
...
4
votes
1answer
46 views
How to calculate (or approximate) “trimmed” (a+b)^n?
$a^n + C_n^{1}a^{n-1}b + ... C_n^{n-1}a^{1}b^{n-1}+b^n = (a+b)^n$
But how to calculate (maybe approximately)
$a^n + C_n^{1}a^{n-1}b + ... C_n^{i}a^{n-i}b^{i} = ?$
For info, the underlying problem ...
3
votes
1answer
113 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
1answer
141 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 ...
2
votes
1answer
27 views
Approximate radius of a group of n packed circles
I am looking for a formula to estimate the radius of a circle which would hold n number of circles with some radius r. I understand this is part of the packing problem which does not have a definite ...
2
votes
1answer
39 views
What is the meaning of “mean-field”?
In lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this ...
1
vote
1answer
43 views
Weighted sum of ratio of partial sum of binomial coefficients
I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$,
$$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose ...
1
vote
1answer
103 views
Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes
I have to numerically calculate the following definite integral
$$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$
for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
1
vote
1answer
52 views
Approximate function from sample data
Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function. I don't have function definition. It's described as a fuzzy inference system. I have the inference system and can manipulate sample data for each ...
1
vote
1answer
184 views
How to get NURBS control points from an array of points that should be part of its solution from controll points we are searching for?
We are talking about Non-uniform rational B-spline. We have some simple 3 dimensional array like
{1,1,1}
{1,2,3}
{1,3,3}
{2,4,5}
{2,5,6}
{4,4,4}
Which are ...
0
votes
1answer
66 views
How to show that a measurable function on $R^d$ can be approximated by step functions?
In Stein's Book: Real Analysis, Theorem 4.3 says that any measurable function $f$ on $R^d$ can be approximated by step functions. But the proof provided there only show that when $f=\chi_E$, with ...
0
votes
1answer
63 views
B-Spline Interpolation/Approximation
I've got a couple of probably very simple questions, yet some googling didn't bring up what I was looking for.
First what I want to do: I have a grid, and the gridpoints are function values. I want to ...
0
votes
1answer
52 views
Need some help understanding notation for composite gauss quadrature formula
Reading through some notes on 2-point gauss quadrature, I came across the following general formula. I'm currently doing an assignment with 3-point quadrature, and have gotten to a similar formula, ...
0
votes
1answer
49 views
Efficient method of approximating a distribution with Gaussian
Given a univariate uni-modal density function $f(x)$ (very hard to compute its cumulative distribution function (CDF) $F(x)$, not to mention its inverse CDF $F^{-1}(x)$),
how to find the best ...
0
votes
1answer
69 views
Integral approximation.
Can you help me to show that
$$\int^{\pi/2}_{0}{d\theta\over (1-m^2\cos^2\theta)^2} \approx {(2-m^2)\pi\over4(1-m^2)^{3/2}}$$
to first order, such that $0 \lt m \lt 1$
0
votes
1answer
62 views
Question about piecewise linear paths
I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it.
Consider a finite family of hyperplanes in a finite-dimension real ...
0
votes
1answer
68 views
Quadratic approximation of a cost function with a Taylor expansion
See also http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch12.pdf, page 92.
Given an instantaneous cost ...
0
votes
1answer
57 views
Expansions of Hermite functions
I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients.
Thank you for your ...
0
votes
0answers
67 views
Covering number of a set of matrices
Suppose that we are given $n$ vectors $x_1,x_2,\ldots,x_n$ in $\mathbb{R}^d$ that are in general position. Now consider the set $\mathcal{X}=\lbrace\sum_{i=1}^n\gamma_i x_i x_i^\mathrm{T} \mid ...
0
votes
0answers
40 views
Approximation when $|a(t)|\ll b$
If $|a(t)|\ll b$ is it alright to take $a\left({a\cdot \dot{a} \over b^2}\right)$ as $0$?
Would the following argument make sense?
I know that we can take $\left({a\cdot a \over b^2}\right)$ as $0$ ...
0
votes
0answers
58 views
proof of one inequality with sums
Please help me to prove the following inequality:
Fix $k, m \in Z_+$ and for $j \in Z_+$ set
\begin{align*}
a_j^{(1)}=a_j=\sum_{i=0}^{\min\{j,k\}}\frac{1}{i!6^i}\frac{(-1)^{j-i}}{(2(j-i)+1)!}
...
0
votes
0answers
36 views
Minimizing a function of two variables in an asymptotic limit
I am trying to find an analytic solution to the following function in the limit of small, positive $p_x$, $p_z$ and large, positive $m,n$ where $mp_x$, $mp_z$, $np_x$, and $np_z$ are $O(1)$:
...
0
votes
0answers
54 views
Orthogonal basis for waveform expansion
I have many signals where each signal has a different waveform f(x). One example of such a waveform could be this f(x) sampled at 11 x positions:
I am looking for a basis, Bi, for a series ...
0
votes
0answers
47 views
A question applying Ritz Galerkin.
Let S = (0,1)^2 contained in R^2. If we see the boundary value problem
-(Triangle)u = (pi)^2 cos(pi*x1) in S, del(subscript n) u = 0 on the boundary, we have to
1) Provide a functional J, where ...
0
votes
0answers
31 views
Steiner Tree Approximation
My question is about a subtlety regarding the $2$-approximation for the Metric Steiner Tree problem.
The classical Metric Steiner tree problem:
Given a metric space on $n$ points and a subset $S$ ...
0
votes
0answers
106 views
Calculation of sum
I am wondering if it is possible to calculate or approximate the following sum
$$
\sum_{k=0}^l\frac{(l-2k)^p(2l+k(k-1))l^{k-1}}{(k+3)(k+2)}
$$here $p \geq 2$.
Thank you.
0
votes
0answers
127 views
Comparing norms of a vector
Let $a$ be a vector in $\mathbb R^m$, such that $\sum_{i=1}^{m}a_i=0.$
I would like to compare $\sqrt{2m(2m−1)}\|a\|_{\infty}$ and $\sqrt{2m}\|a\|_2$, in the case when the vector $a$ satisfies the ...
0
votes
0answers
68 views
Approximating a function with a sine function: transform into constant amplitude?
I have a smooth function, it is stationary. So I tried approximating my function with regression by fitting a sine function that changes period, phase & frequency every observation to get the ...
0
votes
0answers
92 views
How do I create a shape from a square corners' values?
I'm working on a 3D algorithm, so my problem applies to cubes, not squares. But for convenience, I'll stick to 2D.
Each corner of a square can contain up to 100 units, depending of the values at each ...
0
votes
0answers
90 views
An asymptotic approximation for exponential integrals.
For a function $f(x)$ I have seen this identity being used at times,
$\lim _ {N \rightarrow \infty} \int _ {-\infty} ^{\infty} \sqrt{N} e^{-Nf(x)} = \sum _{\frac{df}{dx}\vert_{x_i}=0} e^{-Nf(x_i)}$
...
0
votes
0answers
66 views
How trustworthy is this kind of approximation?
I need to speed up an algorithm that takes as input the number of chips of several players in a tournament. Their chips can be seen as a normally distributed curve.
Since my algorithm can handle a ...
0
votes
0answers
121 views
linearization or linear approximation question
I have done this question but when I draw the graph of Lo(x) I dont get a linear function...I get some weird looking graph kinda U shaped. its above f(x) so I think I have the wrong answer..
the ...
0
votes
0answers
195 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 ...
