0
votes
2answers
28 views

Evaluate the following function using as many significant figures as required to get a final result of 4 digits accuracy

I need to evaluate $$ f_5(0.2) = 5! \left[ e^{0.2} - \left( 1 + (0.2) +\frac{(0.2)^2}{2!}+\frac{(0.2)^3}{3!} + \frac{(0.2)^4}{4!} +\frac{(0.2)^5}{5!} \right) \right] $$ using as many as required ...
0
votes
0answers
12 views

Decomposing a matrix into a finite set of elements

I'm looking to approximate real, symmetric matrix $\mathbf{A}$ of size $N$ with a unique restriction. First, it is known that the matrix has zero mean $\left < A_{ij} \right>=0$. Next, it is ...
10
votes
1answer
162 views

Function for which trapezoidal rule outperforms midpoint rule for every $n$

Is there a continuous elementary function $f:[0,1]\to [0,\infty)$ such that for every $n$ the trapezoidal approximation to $\int_{0}^{1}f(x)\,dx$ with $n$ trapezoids is strictly better than the ...
-1
votes
0answers
20 views

How to do a linear approximation with several parts

if i have partial data set of $ \langle x,y\rangle\in \mathbb{R}$ for a given function $f$, and i want to approximate it by $n$ partial linear functions how would i calculate those linear functions, ...
0
votes
1answer
37 views

Find a polynomial using minimax approximation

Find a polynomial with the maximum 1. degree which best approximates the $f(x)=e^x$ function in terms of minimax approximation in $[0,1]$.
0
votes
0answers
5 views

Are there other names for multilayer perceptrons or multidimensional interpolants based on Kolmogorov's approximation work?

Are there other names for multilayer perceptrons that are used outside of the neural net community? At its core, multilayer perceptrons form a multidimensional interpolant of the form $$ ...
0
votes
0answers
39 views

Archimedes' Apprxomation of Square Roots

Supposing a square root $\sqrt{X}$, let $x$ be the approximation of $\sqrt{X}$, then we get these 2 formulas to estimate $\sqrt{X}$: $x_{n+1}=\frac{x_n+\frac{X}{x_n}}{2}$ and ...
1
vote
0answers
26 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
0
votes
1answer
51 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
1
vote
1answer
65 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
2
votes
2answers
72 views

$(1-x)^y ≈ e^{-xy}$

Here is an approximation I often see in biology articles but don't really understand: $$(1-x)^y ≈ e^{-xy}$$ I think this $e^{-xy}$ closely approximates $(1-x)^y$ whenever $x$ is small. Can you help ...
3
votes
1answer
25 views

Show that a subset $V \subseteq C[a,b]$ is a Haar subspace

Let $C[a,b]$ be the set of continuous functions on $[a,b]$, then a linear subspace $V \subseteq C[a,b]$ of finite dimension $n+1$ is called an Haar subspace iff one of the following equivalent ...
1
vote
2answers
52 views

Cebysev (Tschebyscheff-) Approximation

I have to find the best Tschebyscheff-Approximation of $x(t) = \sin(t)$ on the Interval $[0,d]$ by a line where $0 < d < 2\pi$. But I have no idea how to perform such an approximation, do you ...
1
vote
1answer
26 views

Complete orthonormal system

Let $H$ a linear space with inner product. An orthonormal system $\{e_1, e_2, \dots \}$ is called complete in $H$ if $x=0$ is the only element that satisfies the relations: $$(x, e_n)=0, \ \ \ ...
1
vote
0answers
29 views

Solve Van der Pol equation by Padé approximation

I want to solve the Van der Pol equation: $$f''+ \mu \, (f^2-1)f'+f=0, \quad f = f(t),$$ by Padé approximation. I know the solution should be the combination of $\sin{t}$ and multiplied by $\mu$, ...
0
votes
1answer
32 views

Discrete Function Approximation Error - Which type? (Applied math, signals)

I have two functions, one derived via software, and we can call it the exact function, $f_{exact}$. The other is a result I got through hardware, and we can call it the approximation, $f_{approx}$. ...
1
vote
0answers
9 views

Simplifcation of the Function Required

I want to simplify the following expression $$ \sum_{x=0}^{D+m} \frac {x!} {(n+1)_x} (\frac {\theta_{eff}} {\mu})^x {D+m \choose x} $$ The parameters $D, m, n, \theta_{eff}, \mu, $ are constants. ...
2
votes
2answers
53 views

Approximation of difference of harmonic numbers

Harmonic number $H_n$ is equal to $$H_n = \sum_{i=1}^n \frac{1}{i}$$ Asymptotic expansion of harmonic humber is $$(1) H_n = \ln n + \gamma + \frac{1}{2n} - O\left(\frac{1}{n^2}\right)$$. Very popular ...
0
votes
0answers
60 views

Approximating piecewise linear function

I'm trying to derive an analytic approximation to the following piecewise linear function: $$ f(x) = \left\{ \begin{eqnarray} \frac{x}{x_s} & & \text{if} & x < x_s \\ ...
2
votes
1answer
38 views

Error formula for linearization

Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
2
votes
0answers
30 views

Time Discretization

I wonder why we work with constant discretization in Time Discretization of numerical approximation for numerical scheme if we take not necessarily constant Discretization Is the numerical scheme ...
2
votes
1answer
138 views

Stone-Weierstrass theorem with $p(1/x)$

I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial.
0
votes
1answer
25 views

How to find sequence from set of values

I have set of probability values arranged in ascending order, p1<p2<p3<...<pM. Now I want to assign set of numbers in the same manner in which probabilities are increasing. It means I ...
0
votes
1answer
27 views

Affine approximation of countinuous functions?

Is it true that any real continuous function can be approximated by a piecewise affine function? If true, can you suggest a link or something related to the question? Thanks
1
vote
2answers
84 views

Pade Approximations convergence acceleration

Why Pade Approximatoins accelerate the convergence of series? Generally speaking, what is there an advantage in the sence of convergence acceleration using rational interpolation? Thanks much in ...
1
vote
0answers
29 views

Convergence acceleration using rational approximation

How Pade Approximations accelerate the convergence of series??? Here I don't mean only the Pade Approximations, just in general how do the rational approximations contribute to series convergence ...
0
votes
0answers
46 views

Coefficient $a_k$ of generating function

Given a generating function $F(z)$, am I right to say that the coefficient $a_k$ of $[z^n]$ is computed by $\frac{F^{(k)}(0)}{k!}$ $(1)$. Since we have the definition of $F(z)$ is: $F(z) = \sum_{i ...
1
vote
0answers
28 views

Approximation of series

Can anyone provide an approximation for the series: $$\sum_{i=0}^{\infty}\frac{\left(1-\Gamma\left(i+1,\frac{b^{2}}{N_{01}}\right)\right)v^{-i-1}}{\mu_{1}N_{01}^{i}}\\\times ...
6
votes
1answer
142 views

Simple approximation to a sum involving Stirling numbers?

I have also posted this question at http://mathoverflow.net/questions/141552/simple-approximation-to-a-sum-involving-stirling-numbers#141552. I have an exact answer to a problem, which is the ...
1
vote
1answer
57 views

Approximating a function

I'm sorry if this question in not well formed... I would like to perform a computation of the following function: f() = -2*X1 -1*X2 +0*X3 + 1*X4 +2*X5 (The ...
0
votes
1answer
46 views

curve fitting with known first derivatives

I need to find a polynomial function $f(x)$ minimum order 2 that best satisfies the following it passes through points (x1,y1) and (x2,0) it is known that $f'(x_{1})=A$ and $f'(x_{2})=B$ with ...
2
votes
1answer
66 views

Given the first n derivatives of a function at two points, is it possible to approximate the function between these points?

That is, the function is on an interval $f:[a,b]\rightarrow\mathbb{R}$ and smooth; and at the boundaries of the interval $(a,b) \in\mathbb{R}$, all $f^{(m)}(a)$ and $f^{(m)}(b)$ are known for ...
2
votes
2answers
426 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 ...
1
vote
1answer
313 views

Meaning of $\alpha$ in Laguerre polynomials

I found that generalized Laguerre polynomials are: $$ L_n^{\alpha} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!}.$$ However, I wonder what is the meaning of $\alpha$ in this ...
1
vote
0answers
22 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...
11
votes
1answer
321 views

Approximation of Semicontinuous Functions

Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$. Does there exist a ...
1
vote
1answer
98 views

understanding the least squares criterion

I was given 20 data points and asked to choose the most suitable lowest degree polynomial to fit them using the least-squares criterion. I looked it up, but what i found seems far too complex or just ...
0
votes
0answers
130 views

When does distribution bootstrap mean converge to distribution sample mean?

Let $\bar{X}_n$ denote the sample mean of n iid random variables. Let $\bar{X}^*_n$ be the bootstrap sample mean. Does $\left|\mathbb{P}\left(n^{1/2}\bar{X}_n\leq ...
0
votes
1answer
70 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 ...
4
votes
1answer
208 views

Integral with Bessel function

Let $n$ be half an odd integer, say $n=k+1/2, k \in \mathbb{N}$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral: $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot ...
1
vote
0answers
73 views

Approximation in the Plane of Constant Curvature

In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve. Now, The ...
10
votes
5answers
567 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 ...