Questions tagged [quadrature]

Quadrature refers to techniques in numerical integration, such as Riemann sum approximations, Simpson's rule and Gaussian quadrature.

Filter by
Sorted by
Tagged with
0
votes
0answers
20 views

Is the trapezoidal rule equivalent to Gauss-Chebyslev quadrature?

I encountered a paper in which the author derives a quadrature rule for integration of a function $f(x)$ over a domain $[0,L]$, $$\int_0^L f(x) dx = \sum_{i=1}^{N-1} w_i f(x_i).$$ I'll give a quick ...
0
votes
1answer
28 views

Quadrature for numerically evaluating Cauchy integral formula using unit circle as closed contour

I am currently trying to evaluate the derivatives of a function $F(z)$ in z=0, which is only known numerically on the unit circle ("$UC$") in the complex plane. My question is this: given $z=...
0
votes
1answer
54 views

Proof of generalized open & closed Newton-Cotes formulas

I am looking for proof of generalized open & closed Newton-Cotes formulas. I couldn't find any reference which properly proves both theorems. Most books just state the theorem and do not provide ...
0
votes
0answers
91 views

Show that a polynomial of degree 2n+2 exists such that the integration is not exact

Let $\displaystyle{I_n(f)=\sum_{i=0}^na_if(x_i)}$ be a quadrature formula for the approximate calculation of the integral $I(f)=\int_a^bf(x)\, dx$. Show that a polynomial $p$ of degree $2n+2$ exists ...
0
votes
1answer
39 views

Unclear choice of polynomial in Gaussian Quadrature method?

Recently, I have been reading a paper regarding the evaluation of an integral using the Gaussian quadrature method but I cannot understand the rationality behind the weight and orthogonal polynomials ...
0
votes
0answers
42 views

Chini's equation or Bernoulli equation? How to use quadrature?

$y' - c y^{1/2} = f(x) $ $\quad \quad $ for c : constant and $f$ continuous I am trying to solve the above equation that is presented to me as a Bernoulli equation. However, during my searches, I ...
0
votes
0answers
27 views

Gauss-Lobatto quadrature and nodal points for FEM

By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.) ...
0
votes
0answers
25 views

Gaussian Quadrature Proof

Let $I_G\left[a,\:b\right]$ be the value of the Gaussian Quadrature from $a$ to $b$. There are $n$ $+$ $1$ parameters. Prove that for all $f(x)$ with degree less than or equal to $2n$ $+$ $1$, $I_G\...
0
votes
0answers
35 views

Gauss Quadrature and arclength of a parametric curve

I have $x=6t+t \cos(\pi t)$, $y=-\sin(\pi t)$ and $x'(t)=6+\cos(\pi t)-\pi t\sin (\pi t)$, $y'(t)=-\pi \cos(\pi t)$. Arclength $L(0, 4)$ of a parametric curve between $t = 0$ and $t = 4$ is ...
0
votes
0answers
41 views

Has anyone performed the error analysis for Trapezoidal Rule but with non-uniform grid?

I've taken courses in Numerical Analysis, but all the proofs we've examined for the error of the Trapezoidal method hinge on a uniform grid. However, I'm working on a quadrature method which depends ...
1
vote
0answers
21 views

Convergence of Gaussian-Quadrature measure

Suppose I have a measure $\alpha$ defined by $$ \alpha(x) = \sum_{i : \lambda_i < x} w_i $$ for nodes $\lambda_i$ and positive weights $w_i$, $i=1, \ldots, n$ (if we need $\alpha(x)$ to be ...
0
votes
0answers
11 views

Bound Lagrange interpolating function constructed with nodal values on Gauss points to a minimum and a maximum value

I need to keep a Lagrange interpolating function constructed on $P+1$ Gauss quadrature nodes to a minimum and a maximum values by tweaking nodal values of the exact function at Gauss nodes. The ...
0
votes
1answer
50 views

If $2 \sum_{j=0}^{n}a_jf(x_j)$ is a quadrature for $\int_{-1}^1f(x)dx$ exact for polynomials of degree $\le n$ and $x_j=-x_{n-j}$, show $a_j=a_{n-j}$.

This was an problem in today's exam in numerical methods and I would like to know if my solution is correct. For approximating the integral $I(f) = \int_{-1}^{1} f(x) dx$ we take the quadrature ...
1
vote
1answer
43 views

Gaussian-hermite quadrature points and weights, numerical accuracy/stability

I'm trying to implement a code using numeric integration over with Gaussian-Hermite quadrature, parametrized by number of points used. Recurrence relation makes it easy to find polynomial coefficients ...
1
vote
0answers
46 views

A parametric jump discontinuity of a definte integral

Related to the famous problem of the fastest path between two points (Brachistochrone) I created the following integral $$I(k)=\int_{0}^{1}\sqrt{\frac{x^{2(k-1)}+(1-x^k)^{2(1-k)/k}}{x^{2(k-1)}[1-(1-x^{...
0
votes
1answer
35 views

Numerically solving integrals with singular endpoints

I'd like to write a small library which uses the boundary element method to compute the charge distributions on bodies. This requires me to solve the following integrals $$\int_{0}^{1}{\log{\sqrt{[x_i-...
0
votes
1answer
46 views

Given the following composite quadrature rule find the error

Currently I'm seeing a numerical methods course, I have been trying to understand the topics with the following book: G. Phillips M. and P. Taylor J., Theory and Applications of Numerical Analysis. ...
0
votes
2answers
117 views

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 ...
1
vote
0answers
26 views

Piecewise Gauss-Legendre quadrature order of convergence

Given a definite integral $\int_a^bf$, If we increase the number of nodes and weights of the Gaussian quadrature, we would get closer to the exact integral. But another way to get more exact is to ...
1
vote
1answer
42 views

Can I use the Netwon-Raphson method to solve a function that is a quadrature approximation of an arc-length integral?

I am using an arc length integral $\int_{t_0}^x \sqrt{(|ds|^2)} dt$ to solve for the $x$ that would give me a specific arc length. Assume that I know $t_0$, etc. and the only unknown is $x$. That is, ...
1
vote
0answers
16 views

Find weights and coefficients such that quadarature formula is exact

I'm struggling with the following problem: What is the maximum degree of exactness that we can obtain with the following quadrature >formula $$\int_0^1 f(x)\frac{1}{\sqrt{x}}dx \approx w_0 f(x_0) +...
0
votes
0answers
13 views

Arbitrarily large degree of exactness for a precise quadrature formula

I don't know how to solve the second part of the following exercise: Consider the quadrature formula $Q(f)=w_1 f(x_1)$ for the computation of the weighted integral $$W(f)=\int_0^1 x^{\alpha} f(x)dx$$ ...
0
votes
0answers
22 views

Quadrature Rules Proof

Consider the interpolatory quadratures with an odd amount of equally spaced nodes, then the quadrature is exact for all polynomials of degree n or less. I am trying to find a proof online in order to ...
2
votes
1answer
56 views

Numerical quadrature for an improper multiple integral

In my numerical analysis course, my professor asked us to evaluate the integral $$2 \int_{0}^{1} \cdots \int_{0}^{1} \prod_{i<j}\left(\frac{u_{i}-u_{j}}{u_{i}+u_{j}}\right)^{2} \frac{d u_{1}}{u_{1}}...
1
vote
1answer
36 views

Why are the weights of gaussian quadrature so smooth?

I was watching this video. (https://www.youtube.com/watch?v=cKKrGr93f6c&t=616s) In minute 10:16 he shows a plot of the weights of gaussian quadrature. Why are they so close by each other - and not ...
2
votes
0answers
63 views

Transforming a square to regular triangle [duplicate]

I am trying to find the mapping for a square [(0,0),(1,0),(0,1),(1,1)] to a right angled triangle [(0,0),(1,0),(0,1)] then to a arbitary triangle in 2d. The issue I am having is a) finding the mapping ...
0
votes
1answer
34 views

Saving nodes in iterative Adaptive Simpson quadrature method (Matlab)

Implementing on Matlab the Simpson adaptive rule to approximate an integral (the following code), I am struggling with saving nodes correctly. I have tried different solutions, but none of them seems ...
0
votes
1answer
21 views

Compute integral of periodic function numerical spectrally

I have a $2\pi$-periodic function $f(x)$, and I want to calculate numerically the integral $\int_{0}^{\alpha}f(x)dx$ where $\alpha$ is a point in the interval $[0,2\pi]$. I have the function evaluated ...
0
votes
0answers
19 views

Spectral integration scheme

I'm working in a problem where I need to reconstruct numerically the (x,y) coordinates of a deformed circle from the tangent angle to the surface($\theta$) of the shape and its lenght($L$). Then I ...
0
votes
0answers
44 views

Find values of $p$ such that trapezoidal rule gives second order approximation

I have the following question, which should be easy, but I'm puzzled now: Consider $f(x)=|x|^\frac{p}{2}$, where $p$ is a non-negative integer and the integral $\int_{-1}^{1}f(x)dx$. For which ...
1
vote
0answers
57 views

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 $n>0$ ...
0
votes
1answer
59 views

Gauss-Lobatto and Gauss-Laguerre quadrature

I'm using MATLAB and this function: ...
0
votes
2answers
39 views

Closely minimize error bound $\frac{4 \pi \exp(\cosh(a))}{\exp(a N) -1}$

I'm trying to minimize an error bound $$\frac{4 \pi \exp(\cosh(a))}{\exp(a N) -1},$$ where $N$ is the step size for the trapezoidal rule and $-a < Im < a, a > 0$ is a strip bound which may ...
0
votes
1answer
20 views

How to find x's for representing a specific type of integral using a quadrature formula?

I have this integral which I want to approx with quadrature formula for some fixed $n$: $$ \int\limits_0^{+\infty} x e^{-x} f(x) dx \approx \sum\limits_{k = 0}^n A_k f(x_k) $$ I found info about how ...
0
votes
0answers
8 views

Exact error formula for numerical quadrature by midpoint rule

How can I show that for any function $g(x) : [a,b] \rightarrow \mathbb R$ with second derivatives there exists $z \in [a,b]$ so that $\int_a^b g(x) dx = g\left( \frac{a+b}{2} \right) + \frac{(b-a)^3}{...
0
votes
1answer
21 views

Quadrature for a cumulative distribution function

I have a univariate cumulative distribution function (CDF) that has the form $$\Pr (X \leq x) = F_X(x) = \int_{-\infty}^xh(s) e^{-s^2}ds$$ Namely it very much looks like having the appropriate form ...
0
votes
1answer
44 views

Evaluate $\int_{0}^{2}(x^2 + x^{3/2} + x + x^{1/2})dx$ using Simpson's Rule

I'm running into some troubles while trying to evaluate $$\int_{0}^{2}(x^2 + x^{3/2} + x + x^{1/2})dx$$ using Simpson's Rule Simpson's Rule states $$Q(f) = \frac{b-a}{6}(f(a) + 4f(\frac{a+b}{2}) +f(...
1
vote
0answers
33 views

(Feedback to) Two-point Gaussian Quadrature for $\int_{0}^{2}(x^2 + 3x -1)dx$

May I ask you for feedback ? Thanks ! We're asked to evaluate the integral $$\int_{0}^{2}(x^2 + 3x -1)dx$$ using the Gaussian-Quadrature Formula with $n=2$ points. We're not allowed to use that the ...
0
votes
1answer
33 views

Find $\alpha, \beta, \gamma$ such that accuracy degree of $Q(f) = \alpha f(0) + \beta f(\frac{1}{2}) + \gamma f(1)$ is as high as possible

Given the Quadrature Formula $$Q(f) = \alpha f(0) + \beta f(\frac{1}{2}) + \gamma f(1)$$ for the approximation of $$I(f) = \int_{0}^{1} f(x)dx$$ We're asked to Determine the $\alpha, \beta, \...
0
votes
1answer
46 views

Evaluate $\int_{-1}^{0}(x^4-x^2+2)dx$ using Simpson's Rule

We're asked to evaluate the integral $$\int_{-1}^{0}(x^4-x^2+2)dx$$ using Simpson's Rule.
0
votes
1answer
25 views

Determine Weights of the Quadrature Rule $GL_2[f] := w_0f(-1) + w_1f(x_1) + w_2f(+1)$

We're given the following Quadrature Rule on the Reference Interval $[-1,1]$: $$GL_2[f] := w_0f(-1) + w_1f(x_1) + w_2f(+1)$$ Due to symmetrical reasons, we know that $x_1=0$ We're asked to find the ...
1
vote
1answer
49 views

How would one go about computing a Chebyshev-type quadrature problem with general integration limits of [a,b] instead of [-1,1]?

More specifically, a problem of the form $\int_a^b\frac{f(x)}{\sqrt{1-x^2}}dx = \sum_{i=1}^{N}w_if(x_i)$, where $a,b \in [-1,1]$, $w_i$ are the weights, and $x_i$ are the abscissa. A quick search ...
4
votes
1answer
81 views

What's the name for this numerical integration algorithm?

Suppose I estimate $\int_0^1 f(x)dx$ as $\frac12(f(a)+f(b))$, with $a,\,b$ chosen to achieve the lowest-order possible error. We assume $f$ equals its Maclaurin series and $\int$ commutes with $\sum$, ...
0
votes
1answer
43 views

Can anyone helps me understanding this integral solution?

I was reading a research article in which authors used Chebyshev-Gauss Quadrature on an integral which is given as in the attached image. Can anyone help me out with any citation to understand how $...
0
votes
1answer
33 views

Order/Degree of accuracy for multidimensional integration using Gauss-Legendre quadrature

Let we have continuous function $f(x,y)$ in all $(x,y)\in(-\infty,\infty)$, and we would take the finite integration of $f$: $$\int_{a}^{b}{\int_{p}^{q}{f(x,y) \,dx \,dy}}$$ and we take the ...
1
vote
2answers
112 views

Two-point Gaussian Quadrature Rule with weight function $w(x)=x$ [duplicate]

I am trying to construct a formula of the form $\int_0^1 xf(x)dx=A_0f(x_0)+A_1f(x_1)$ with degree of precision 3. A Gaussian Quadrature is the only interpolatory quadrature with degree of precision ...
2
votes
2answers
82 views

Surface area of an oblate spheroid using gaussian quadrature

I want to compute the surface area of an oblate spheroid using gaussian quadrature, the parametrization of the oblate spheroid is given by: $$x = a \cdot \sin\theta \cdot \cos \phi \\ y = a \cdot \...
1
vote
3answers
90 views

How does gaussian quadrature work for this example?

I'm trying to determine the weights for the following integral using gaussian quadrature: $\int_{0}^{1} x^2 f(x) \approx w_0 f(0) + w_1 f(a)$ I understand that you write polynomials from $\ x^n$ ...
1
vote
1answer
91 views

Magnus expansion and two point Gauss quadrature rule

In many papers about Magnus integrators, like this one at slide 7, one truncates the Magnus expansion and apply a quadrature rule to the integrals. My question is about the expression one gets if uses ...
1
vote
1answer
39 views

Gaussian Quadrature Points

I am trying to determine the transformation necessary to map a given set of Gauss points on the interval $[a,b]$ to the corresponding Gaussian quadrature points on the interval $[α,β]$. Here is what I ...