# Questions tagged [quadrature]

The quadrature tag has no usage guidance.

18
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### Symmetry of weights in Newton-Cotes quadrature formula

Why do the weights of a generic Newton-Cotes quadrature formula show symmetry w.r.t. the midpoint of the integration interval? I understand that, given $f:[a,b]\mapsto\mathbb{R}$ and being $n>0$ ...

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113 views

### For noisy or fine-structured data, are there better quadratures than the midpoint rule?

Only the first two sections of this long question are essential. The others are just for illustration.
Background
Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, ...

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**1**answer

85 views

### Proof Reference - Polynomial interpolation at quadrature points

If $\left( p_n \right)_{n=0}^{\infty}$ is a family of orthogonal polynoamials with respect to a measure $\mu$ on $[-1,1]$, and $\left( x_j, w_j \right)$ are the quadrature points and weights for the ...

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111 views

### Clenshaw-Curtis integration without Fourier

The Clenshaw-Curtis quadrature rule approximates an integral $I=\int\limits_{-1}^{1} f(x) \, dx$ by $$I\approx I_n = \sum\limits_{j=1}^N f(x_j)w_j \, ,$$
where the $x_j$'s are the roots of the $N$-th ...

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154 views

### When (if ever) are the weights from Smolyak (sparse grid) cubature positive?

Are there any $1$-dimensional quadrature rules of arbitrary accuracy, on either $[0,1]$ or $\mathbb{R}$, with any non-trivial weight function, such that the associated $N$-dimensional cubature rule ...

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**2**answers

225 views

### Numerical Computation of Orthogonal Polynomials Recurrence Relations

Background and notations: Given an interval $I\subseteq \mathbb{R}$ and a continuous finite measure $d\mu = w(x)dx$, and denote $p_n(x)$ the orthogonal polynomials with respect to $d\mu$. We have the ...

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556 views

### Recurrence of Legendre polynomial roots/ quadrature points

Consider Legendre polynomials $p_n (x)$ on $[-1,1]$. For each $n \in \mathbb{N}$ we denote the zeros of $p_n (x)$ by $\left( x_j ^{(n)} \right) _{j=1} ^n$.
We know that these roots are distinct, and ...

**1**

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**1**answer

266 views

### PDF and CDF using Gauss-Legendre quadrature

Consider the unit interval $I$ with a continuous probability measure $\mu$, and consider a smooth random variable $f:I\to \mathbb{R}$. We can define its cumulative distribution function and ...

**6**

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422 views

### Symmetric matrix formula for Gauss-Legendre quadrature

While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...

**4**

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**1**answer

550 views

### Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate.
Longer: If $p_n$ is the $n$-th Legendre polynomial, ...

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**1**answer

129 views

### Does Gaussian Quadrature actually refer to Gauss-Legendre Quadrature？

When the term Gaussian Quadrature appears in most Literatures, does it actually refer to Gauss-Legendre Quadrature.
In other words, do they implicitly admit that they use the Legendre orthogonal ...

**14**

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841 views

### Computing Gauss Legendre quadrature for large $N$

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...

**4**

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**3**answers

501 views

### Quadrature formula max accuracy

I'm looking for a maximum accuracy quadrature formula:
$$
\int_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A_1f(x_1)+A_2f(x_2)+R(f)
$$
I don't know exactly if it's Trapezoidal rule which has the ...

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132 views

### 2d quadrature weights for an arbitrary set of nodes

I need to estimate the value of a 2d integral
$\int_{y_{min}}^{y_{max}}dy \int_{x_{min}}^{x_{max}} dx \, f(x,y) P(x,y)$
I have the an explicit analytical form for $P(x,y)$.
I have samples of the ...

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968 views

### Area Under Generalized Parabolas and Hyperbolas without Calculus

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...

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**5**answers

2k views

### Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...

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**1**answer

601 views

### Approximation of an integral of a concave function

I suspect this is a homework question somewhere, but I've not seen it elsewhere and it seems like it should be easy: let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) =...

**1**

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**1**answer

368 views

### gaussian quadrature

Gaussian quadrature allows us to integrate polynomials up to order $2 n-1$ using only $n$ function values.
$\int_{x_0}^{x_1} ( \sum_{i=0}^{2 n-1} a_i x^i ) dx = f(a_0, \dots , a_{2 n-1}) $
thus, the ...