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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1answer
224 views

Streaming algorithm for polynomial fitting data?

The specific problem I'm trying to solve is: $$h_k(x, n) = \left(\frac{\alpha}{n} + 1 - \alpha\right) \sum_{i=0}^{k} c_ix^i.$$ Given $k$ and a stream of tuples $(x, n, h_k(x, n))$ (where the $x$'s ...
2
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1answer
870 views

Approximating a large number of data points using (cubic) splines in l1/l2 norm.

I have a pretty large dataset ($x,y$) consisting of a few million points. There is a lot of noise in the data. I want to find a smooth but simple approximation/representation for this dataset, so that ...
1
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0answers
14 views

Fittig N-D periodic functions

Besides fourier series, are there any stable way to fit N-D nonlinear functions that demonstrate some degree of periodicity, sigmoid neural networks works poorly in this domain.
1
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1answer
98 views

still trying to figure out details of Newton's method

I understand mostly what going on when using Newton's method to approximate some value but there are some details I am still hazy on (like how you come up with the correct $f(x)$ for the update rule). ...
3
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1answer
180 views

Harlan J. Brothers's approximation to $ e $ ad infinitum?

Consider the series generated by Harlan J. Brothers's method for the number $e$ http://en.wikipedia.org/wiki/List_of_representations_of_e Can they be improved again and again or is there a limit so ...
5
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4answers
608 views

Lambert function approximation $W_0$ branch

I am looking for a simple, inexpensive and very accurate approximation of the Lambert function ($W_0$ branch) ($-1/e < x < 0$).
2
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1answer
43 views

experiencing methodological consternation in correctly applying Newton's method

In lecture, we were told that to find $\sqrt[3]{a}$, we use Newton's method as follows: $$ \begin{align} f(x) &= x^3 - a\\ f'(x) &= 3x^2\\ x_{n+1} &= x_n - \frac{f(x_n)}{f'(x_n)}\\ &= ...
2
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1answer
92 views

Interpretation of the variational principle for the Ritz approximation, solid Mechanics

Below $U$ and $V$ are recpectively the internal and external energy components of a given structural element: $$U+V=W$$ Expressing $U$ in terms of the strains $\varepsilon$ and the material ...
2
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1answer
77 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 $0<m&...
0
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1answer
67 views

Best way to approximate function

What is the best approximation for function like on attached image ? Function is increasing or decreasing from "spike" to "spike" Zoom to the first few members: All members:
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1answer
851 views

Bounding the modified Bessel function of the first kind

i'm looking for an upper bound for the modified Bessel function of the first kind of a +ive real argument. It seems that it satisfies the inequality : $$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$ ...
2
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2answers
238 views

Show that for a finitely differentiable function there exists a polynomial that both their $k$-th derivatives converge

We need to show that for all $f: [a,b] \rightarrow \Bbb R $ which is differentiable n times, and for all $\epsilon>0$ there exists a polynomial $p\colon[a,b] \rightarrow R$ s.t. $\forall n \...
5
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2answers
1k 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
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1answer
613 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
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0answers
341 views

Discrete approximation to a continuous probability density function

I want to approximate a continuous, finite probability density function, with a specified number $N$ of points, in the following way: If the pdf is 1-dimensional, defined over the section [0,1], then ...
4
votes
2answers
150 views

Ratio between trigonometric sums: $\sum_{n=1}^{44} \cos n^\circ/\sum_{n=1}^{44} \sin n^\circ$

What is the value of this trigonometric sum ratio: $$\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ} = \quad ?$$ The answer is given as $$\frac{\...
8
votes
3answers
486 views

the following inequality is true, but I can't prove it

The inequality $$\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)$$ holds for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find ...
3
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2answers
169 views

approximate $\int_{u=0}^b e^{-\frac{u^2 + ac}{au}}du.$

I'm trying to find an approximation (or exact solution if possible) for an integral of the form: $$\int_{u=0}^b e^{-\frac{u^2 + ac}{au}}du.$$ I was thinking of somehow applying a Gauss Hermite ...
2
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1answer
258 views

Approximate Nash Equilibrium

I am sort of confused by the notion of approximate Nash equilibrium. I will try to express my confusion in the following exercise. Problem. Is it true that for every two player game where every ...
2
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1answer
50 views

determining sign of function containing logarithm.

I would like to know the sign of the following term in general. I tried approximately and it was negative. Is there any $m_0$ such that for all $n>m>m_0$, the following function is positive or ...
2
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1answer
329 views

Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t)$

I try to solve this problem all day, but can't reach any progress in it. Can you give me some hints? Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t) = \frac{1}{\sqrt[4]t}$ ...
1
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1answer
59 views

Approximation of beam

Assume that there is a simply supported beam subjected to concentrated moments $M_0$ at each end. The governing equation is $$EI\frac{d^2y}{dx^2}-M(x)=0$$ with the boundary conditions $y(0)=0$ and $y(...
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0answers
38 views

a question in projection

Let $V=L^2(\Omega)$, and $$k=\{v \in V ~s.t ~||v||_{L^2(\Omega)}\leq 1 \}$$ I need to find projection for any $u \in V$ on $k$. Please help me.I do not have any idea about this problem. I have many ...
0
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1answer
111 views

Approximating sums of powers of integers: why does $\sum_{i=1}^n i^r \div \frac{n^{r+1}}{r+1} \to 1$ as $n \to \infty$?

I know there are exact formulas for sums of integer powers of integers ($\sum_{i=1}^n i^r$ with $r \in \mathbb{N}$), but I was interested in approximating them. One way occured to me through a ...
2
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0answers
351 views

numerical approximation to logarithm

we know that $$ \ln(x)=\frac{1}{x} \int_{0}^{1}dt \frac{1}{1+xt} $$ then given a cuadrature formula inside $(0,1)$ is that true $$ \ln(x)= \frac{1}{x}\sum_{i}\frac{w_{i}}{1+xt_{i}} $$ wht other ...
5
votes
5answers
396 views

Bertrand's postulate in another point of view

I was just wondering why can't we use prime number's theorem to prove Bertrand's postulate.We know that if we show that for all natural numbers $n>2, \pi(2n)-\pi(n)>0$ we are done. Why can't it ...
1
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2answers
602 views

Fitting a set of points in a plane to a smooth curve obtained by joining a half-line and an arc of a circle

I have a set of points in the plane and I want to find a curve that best fits these points (e.g., in a least squares manner, or using some other convenient "measure"). I want that the curve be either: ...
6
votes
2answers
333 views

Good upper bound for $\int_0^1 (1 + 2x)/\sqrt{x + x^3}$

I am trying to obtain an upper and lower estimate for the integral $$I = \int_0^1 \overbrace{\frac{1}{\sqrt{x^2+1}} \left( \frac{1}{\sqrt{x}} + 2\sqrt{x}\right)}^{\Large f(x)}\,\mathrm{d}x,$$ and an ...
4
votes
1answer
1k 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 ...
1
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1answer
643 views

Find the 2nd-degree polynomial that approximates with the method of the least squares the:$f(x)=\frac{1}{10}x^2-2x+10$

It is known that a rectangular set of polynomials $\phi_k(x), k=0,1,\cdots,n$ for each $x\in[a,b]$ as to a weight function $w(x)$ can be constructed with the use of the following recursive type (Gram-...
1
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0answers
145 views

How to estimate linear operator?

Suppose the complex column vector $\mathbf{x}$ is linearly transformed by the complex matrix $\mathbf{T}$ into $\mathbf{y}$: \begin{align} \mathbf{y} = \mathbf{T}{x} \end{align} Assuming $\mathbf{T}$...
0
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1answer
114 views

Signal approximation using linear combination of functions

How I can approximate the signal $x(t)=0.001\,t^3 \exp(-0.1t)$ in the interval $[0,100]$ using a linear combination of the following functions: $f_1(t)=A_1$ $f_2(t)=A_2\cos(0.05t)$ $f_3(t)=A_3\cos(...
2
votes
1answer
133 views

Understanding weighted linear least square problem

I am having difficulty in understanding about weighted linear least squares. Could anybody explain me instead of minimizing the residual sum of squares why we need to minimize the weighted sum of ...
4
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2answers
69 views

Does this ODE have an exact or well-established approximate analytical solution?

The equation looks like this: $$\frac{\mathrm{d}y}{\mathrm{d}t} = A + B\sin\omega t - C y^n,$$ where $A$, $B$, $C$ are positive constants, and $n\ge1$ is an integer. Actually I am mainly concerned ...
0
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1answer
42 views

Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$

I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$ ...
1
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0answers
37 views

${{\left( 1-p \right)}^{n}}{{\text{ }}_{1}}{{F}_{1}}\left( n+1,n+a+1,\lambda T \right)\approx ?$

I am trying to approximate $$\frac{p\left( n+a,\bar\lambda T \right){{\left( 1-p \right)}^{n}}_{1}{{F}_{1}}\left( n+1,n+a+1,\lambda T \right)}{p\left( n+a,\bar\lambda T \right)}={{\left( 1-p \right)}^...
2
votes
1answer
90 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 ...
0
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1answer
100 views

Newton-Raphson in $\mathbb{R}^n$

Let $U\in\mathbb{R}^n$ be open and $f:U\to\mathbb{R}^n$ be a $\mathcal{C}^1$ map. $\exists p\in U$ such that $\;f(p) = 0$ and $Df_{|p}$ is invertible. Define $\phi(x)= x-(Df_{|x})^{-1}f(x)$, show that ...
2
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0answers
140 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 ...
0
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0answers
114 views

Complete normed vector space

I came across this question and was not sure how to proceed. Is the normed vector space of all polynomial functions on [0,1] complete with respect to the infinity norm? Thank you
1
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0answers
470 views

Simpson's Rule derived from Trapezoidal Rule

I was just wondering if I could have some assistance in regards to the Trapezoidal Rule and Simpson's Rule. I have a question where it asks to generalize the Trapezoidal Rule to the case of Simpson'...
2
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1answer
292 views

Approximating a log-power function

I can't figure out how the following approximation has been done, I would appreciate any guidance: $$y=-60+10\log_{10}\left[\frac{\left(\frac{99}{100}\right)^m}{\frac{1}{11}\left(\frac{1}{3}\right)^m+\...
1
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1answer
35 views

A question about an expoential function

I got an exponential function as follows $\displaystyle 1-\frac{1}{x}+\frac{e^{-x}}{x}$ Does anyone know how to approximate such a function in a simpler term? Many thanks!
2
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1answer
784 views

Choosing degree of Chebyshev approximation

Chebyshev approximation approximates a function by fitting a weighted sum of Chebyshev polynomials to it. It requires evaluating a function at $N$ sample points to form the weighting coefficients. ...
1
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1answer
300 views

Inaccuracy in numerical calculation of arclength of part of an ellipse

I am trying to numerically calculate the arclength of part of an ellipse according to: $$ L = \int_0^{\phi_s}\sqrt{r^2+\left(\frac{dr}{d\phi}\right)^2} d\phi $$ where $r$ is defined as: $$ r=\frac{...
2
votes
3answers
67 views

Series evaluated to $m$ terms, approximating the error

Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
5
votes
3answers
1k views

Taylor Series for $e^x$ where $x = 1$, estimating the Error

I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by ...
1
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0answers
57 views

Small Inhomogeneity of Differential Equation

Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$: $$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=0$$ Let $...
0
votes
2answers
510 views

Continuation of smooth functions on the bounded domain

Given a bounded domain $\Omega\subset \mathbb{R}^n$ and a smooth function $f$ with bounded derivatives on $\Omega$, is it possible to extend $f$ to $\tilde{f} : \mathbb{R}^n \to \mathbb{R}$ such that ...
0
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0answers
46 views

Product of Standard uniform & CLT

Suppose that $U_{i},\dots,U_{n}$ are iid $U(0,1)$. Use the central limit theorem to find an approximation for: $$P(U_{1}\times U_{2} \times\dots\times U_{25}\leq 6\times 10^{-6} )$$ Answer: Using ...