0
votes
0answers
25 views

How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
0
votes
1answer
33 views

Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
2
votes
0answers
35 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
0
votes
0answers
29 views

Approximate function algorithm using a polynomial and Boor splines

I have a defined function and a set of points with equal distance between them. The problem is that I have to approximate the graphic of that function using a polynomial function of 3rd degree and a ...
0
votes
0answers
29 views

Proving Chebyshev's Theorem (polynomials)

A polynomial $P_n(x)$ of degree $n$ is a polynomial of best approximation to the function $f \in C[a,b]$ if and only if there are at least $n+2$ Chebyshev alternant points on the closed interval ...
0
votes
0answers
36 views

If there exists a polynomial of best approximation of degree n, there also exists a polynomial of best approximation of degree n+1.

First I'd like to say that although this question was asked before (here) and is from the same text, the answer used methods that were not introduced in the text. Let $P_n(x)$ be a polynomial of ...
1
vote
2answers
269 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
1
vote
1answer
65 views

Approximating Trig Functions with Polynomials

I was thinking about the graphs of different trig functions and noticed that most of them are of a similar shape to some different types of polynomials. For example: Higher degree polynomials create ...
2
votes
1answer
129 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.
1
vote
1answer
56 views

Determining the surface with given polynomial borders

Let's say we'd like to guess the shape (I'm not sure the word 'approximate' is appropriate here) of some surface when we are given its borders via third order polynomials (i.e. we are given their ...
0
votes
0answers
21 views

Approximating a piecewise algebraic polynomial

Let $f \in L_{\infty}[-\pi, \pi] $ with $|f|_\infty \leq 1$ be a piecewise algebraic polynomial $$f(t)=\sum_{k=1}^{M}P_k(t) \chi_{(t_k,t_{k+1}]}(t) , -\pi<t_1<...<t_{M+1}<\pi $$ So that ...
0
votes
1answer
55 views

Intuitive proof of interpolation polynomial existence

Problem: Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n ...
4
votes
1answer
79 views

Approximating $e^{inx}$ by polynomials

Show that every function $e^{inx}$ can be uniformly approximated on $[-\pi,\pi]$ by polynomials in $x$. Using the power series expansion, ...
0
votes
0answers
96 views

Approximations other than taylor series and pade approximation

I have a function which has the following form: $$ f(x)=K_1 \coth (Q_1 Q_2 \sqrt{x})^2 + \frac{1}{x}\left[K_2 + K_3 \coth(Q_1 Q_3\sqrt{x})\sqrt{x}\right]$$ and I want to find $x$ for $f(x)=1$. I'm ...
0
votes
0answers
37 views

What are the coeff. of this polynomial?

My matlab generates this answer for the problem: p = -20.2090 17.3368 272.9057 -0.7528 Is it correct? It seems I've gotten different results in another ...
4
votes
1answer
87 views

$f$ is approximated uniformly on $R$ by $p_n(x)$, then $f$ is a polynomial

Suppose $p_n(x)$ is a sequence of polynomials which converge to a function $f$, uniformly on $\mathbb{R}$. Show that $f$ is a polynomial. If there were a uniform bound $M$ on the degree of $p_n(x)$, ...
0
votes
1answer
62 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 ...
1
vote
1answer
280 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 ...
1
vote
1answer
244 views

What is a fitting parameter in least-squares?

I was doing some least-squares homework when I saw this term "fitting parameters". I was asked to implement the leas-squares fit using polynomials of $p-th$ degree to a generic dataset. This is done. ...
2
votes
1answer
80 views

Approximate N order polynomial as a weighted sum of lower order polynomials

I want to represent a polynomial such as $x^5$ with a sum of weighted polynomials so that $$x^5 - (ax^4 + bx^3 + cx^2 + dx + e) = \epsilon$$ My aim is to pick these weights $(a,b,c,d)$ assuming that ...
0
votes
1answer
242 views

Least-squares approximation polynomial

Consider the function $\displaystyle f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$ in the interval $I = [-1,1]$. Set $\beta = 0$. How do I get the expression for the least-squares polynomial, say $\tilde ...
1
vote
0answers
63 views

How to approximate $\sqrt[3]{x}$ when $x$ is rational number

One wants to approximate the real value of $\sqrt[3]{x}$ when $x$ is rational number. One want to approximate to two decimal digits. Is there any way to do this quickly?
2
votes
3answers
146 views

Precision with Taylor Expansions

when you take a 1st order taylor expansion of a function, so: $$f(a) + f'(a)(x-a)$$ does that mean that if the result is only accurate to one decimal place? so for a value a.bcd, d would be the ...
1
vote
0answers
120 views

Extending Hermite polynomial interpolation

Working with the definition of Hermite polynomials $x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then $$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} ...
2
votes
1answer
186 views

General bound on a polynomial's root with the largest norm

Is there a general bound on a polynomial's root with the largest norm? When Rouche's theorem is used, it still seems that the polynomial's root with the largest norm still needs to be found if we ...
2
votes
1answer
77 views

Density of polynomials having “additive-separation of their variables”?

Let $K$ be a compact subset of $\mathbb{R}^2$. Let $P$ be a polynomial in the variables $x$ and $y$. Given $\epsilon>0$, can we find two polynomials $P_1=P_1(x)$ and $P_2=P_2(y)$ such that $$ ...
2
votes
2answers
252 views

$\pi$ polynomials whose real zeros approximate $\pi$

Let's have the following polynomials $x^4+105x^2-1134=0$,$x^6+126^x4+10395x^2-115830=0$, $3x^8+550x^6+45045x^4+3378375x^2-38288250=0$ The positive real zeros of these equations are good ...
3
votes
1answer
182 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 ...
0
votes
2answers
131 views

Fitting a function to a polynomial

I have a black box called $F(t)$ 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 of $F(t)$ from the black box as output. It ...
2
votes
1answer
224 views

Polynomial approximation of $\chi^2$ distribution pdf

The $\chi^2$ distribution PDF is $$f_{\chi^2}(x;k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} \mathrm{e}^{-x/2} \mathbf{1}_{x \geq 0}$$ I am trying to find a polynomial approximation to this density ...
1
vote
2answers
268 views

Triple Recursion Relation Coefficients

I am reading Atkinson's "An Introduction to Numerical Analysis" and I am trying to understand how a certain equation was reached. It is called the "Triple Recursion Relation" for an orthogonal family ...
1
vote
2answers
92 views

Low degree approximation of the polynomial extension of the logical-or function

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$). Consider ...
3
votes
2answers
494 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 ...
1
vote
0answers
117 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 ...