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Tagged with legendre-polynomials numerical-methods
31
questions
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Investigating the numerical accuracy of a truncated Legendre polynomial expansion of an unknown function
I have an integral equation involving an unknown function $f(x)$, of the most basic form
$$
\int_{-1}^{1} e^{iω(t) x} f(x) \ dx = g(t)
$$
I am solving for an approximation of $f(x)$ by substituting in ...
- 15
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0
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49
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Interpolation and general Gaussian quadrature
I just finished a course on numerical mathematics, and have become quite interested in interpolation and how it ties into numerical integration. What suprised me while studiyng quadrature was the fact ...
- 358
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0
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Any comment to speed up the calculation of double-integral having Legendre polynomials?
I want to compute the following double integral at $t=t_{0}$ rapidly. I tried different methods, but all are time consuming for I,J,M >7.
Any comments to speed up the calculation???
$$\frac{1}{2}\...
- 29
1
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1
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267
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Uniqueness of the nodes for Gauss-Legendre quadrature
Gauss-Legendre quadrature approximates $\int_{~1}^{1}f(x)dx$ by $\sum_{i=1}^nw_if(x_i)$.
Wikipedia says that
This choice of quadrature weights $w_i$ and quadrature nodes $x_i$ is the unique choice ...
user1095295
2
votes
1
answer
1k
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Legendre Polynomials integral
I've been asked to calculate:
$$
\int_{0}^{1} P_{\ell}(x)dx,
$$
where $P_{\ell}(x)$ is a Legendre polynomial by using:
i)The generating function:
$$
\sum_{\ell=0}^{\infty}P_{\ell}(x)t^{\ell}=\frac{1}{\...
1
vote
1
answer
88
views
$\sum_{n=0}^{\infty}\frac{x^{n+1}}{n+1}P_n(x)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$ [duplicate]
Using the generating function of Legendre polynomials, show
\begin{equation}
\sum_{n=0}^{\infty}\frac{x^{n+1}}{n+1}P_n(x)=\frac{1}{2}\ln
\left(\frac{1+x}{1-x}\right)
\end{equation}
My attempt
I ...
- 175
2
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0
answers
327
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Legendre polynomials satisfying a recurrence relation.
Monic Legendre polynomials (which are orthogonal polynomials) on $[-1,1]$ are defined as follows:
$$p_{0}(x)=1, \ p_{1}(x)=x$$
and $p_{n}(x)$ is a monic polynomial of degree $n$ such that $$\int_{-1}^{...
- 53
2
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0
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169
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Question on the method of Gauss-Legendre
The Gauss-Legendre method seems to be one of the most important methods in solving differential equations numerically. I have a few questions related to it:
My teachers always assumed - without a ...
- 1,542
2
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2
answers
1k
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Why is the Gaussian-Legendre Quadrature so effective?
I understand how it works, how its derived, etc. The proof of it has been shown to me. That is to say, I know how Legendre polynomials are derived, I know they are orthogonal, I know we sample a ...
- 3,571
1
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2
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Tail recursive formulation of the Legendre polynomial relation
The recursive formula for Legendre polynomials is widely known:
$(n + 1) P_{n+1}(x)
= (2n + 1) x P_{n}(x) - n P_{n-1}(x).$
Let us rewrite the above as follows for convenience:
$P_{n}(x)
= \frac{2n - ...
2
votes
1
answer
623
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Why do the Legendre Polynomials have these coefficients?
I learned of the Legendre polynomials for the first time, in the context of finding an orthogonal basis for $\text{span} \{1, x, x^2, ... \}$.
According to Wolfram, the Legrndre Polynomials are
$$...
- 23.8k
1
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1
answer
202
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What non-zero function is w-orthogonal to all the polynomials of degree less than or equal to $n$?
Background
It is understood that a function $q$ is $w$ orthogonal to a function $p$ over $[a,b]$ if there holds:
$$ \int_{a}^{b} q(x)w(x)p(x)dx = 0$$
For example, for $w(x):=1$, $[a,b]=[-1,1]$, the ...
- 79
1
vote
1
answer
140
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How to derive closed form expression for the zeros of Legendre's Polynomials (Gaussian-Quadrature formulas)?
In Gaussian quadrature formula of integration we need to have the zeros of Legendre's polynomials. Although we may find the zeros numerically, I got closed form formulas such as $$\pm\frac{1}{3}\sqrt{...
- 422
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Solving 2nd order ODE with variable coefficients
ODE:
$$X''(t)+A(t)X'(t)+B(t)X(t)=F(t),$$
IC's: $$X(0)=U_1, $$
$$X'(0)=U_2$$
where
$X(t)=[X_1 X_2...X_n]^T$ and $U_1, U_2,F(t)$ are $n\times1$ matrices,
$A(t), B(t)$ are $n\times n$ matrices.
...
- 968
1
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1
answer
384
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Find the Legendre polynomial
Let us consider the numerical integral $ \ \int_{-1}^{1}w(x) f(x)dx=\sum_{i=0}^{N} f(x_i)w_i$, where $w_i$ are the weights and $w(x)$ is the weight function.
Legendre polynomials, denoted by $ \{p_n \...
user612508