0
votes
0answers
16 views

How do you obtain the version of Simpson's rule required as well as deduce the composite integration rule? [closed]

Consider the function $$g(x)=f(a+(x−1)h)$$ and obtain a version of Simpson’s rule applicable to an integral $$\int_{a+h}^{a−h}f(x)dx.$$ Then deduce the composite integration rule ...
0
votes
1answer
37 views

How do you solve the second part of the question where i am required to derive Simpson’s integration rule?

When $v(x) = A + Bx + Cx(x − 1)$ show that $$\int_0^2v(x)dx= 2A + 2B + \frac23.$$ By choosing A,B and C so that $y = v(x)$ fits a given curve $y = g(x)$ at $x = 0$, $x = 1$ and $x = 2$ derive ...
1
vote
1answer
51 views

What function to use to get geometric mean in trapezoidal rule?

When deriving a trapezoidal rule an integral of $f(x)$ is switched to integral of new function $g(x)$ approximating the first one $$\int_a^b {f(x)dx}\approx \int_a^b {g(x)dx}$$ where $g(x)$ is a ...
2
votes
1answer
65 views

How does one find the area of an implicit function?

For example we have the equation $y^2+\sin({4y\cos{x}})=4$ You can see the graph here at: https://www.desmos.com/calculator/1sxvfl2amd So far I know it is split into top and bottom. I'm trying to ...
1
vote
2answers
41 views

Euler's method for first three approximations?

I have tried variations of the problem for an hour at least and cannot get around to sloving this one. Thank you for input!
0
votes
1answer
27 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?
3
votes
2answers
49 views

Comparison of Newton-Cotes Quadrature and Gaussian Quadrature formulas

Newton-Cotes quadrature formulas are a generalization of trapezoidal and Simpson's rule. The trapezoidal rule involves $2$ points, Simpson's rule involves $3$, and in general Newton-Cotes formulas ...
0
votes
0answers
34 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
1
vote
0answers
18 views

Quadrature methods: even order?

I noticed that all quadrature methods I know (Newton-Cotes and Gaussian quadrature) have always even order in the sense that a quadrature method is of order $n$, if all polynomials of degree $n-1$ are ...
3
votes
1answer
109 views

Numerical integration of $\sin(p_{m})$ and $\cos(p_{m})$ for a polynomial $p_{m}$

I was wondering if anyone knew about any numerical methods specifically designed for integrating functions of the form $\sin(p_{m})$ and $\cos(p_{m})$ where $p_{m}$ is a polynomial of degree $m$. I ...
1
vote
0answers
12 views

Order of Romberg's method

We call a method(numerical integration) of $n-$th order, if it can integrate any polynomial of degree $n-1$ without any error. In this sense: The simpson rule is of $4$-th order and the trapezium ...
0
votes
1answer
9 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
1
vote
0answers
36 views

Error bound by the Simpson's rule

My lecture notes have a little exercise. Two functions are given: $$ f(x) = \cos(x) \ \text{and} \ g(x)=\sqrt{x+1} $$ And we're asked about the error bound of the Simpson's rule to estimate the ...
0
votes
1answer
41 views

An integration formula using Gauss-Laguerre method

Using Gauss-Laguerre method show that: $ \int_{0}^{\infty}\frac{e^{-x}}{x+a}dx=\frac{a+3}{a^2+4a+2}+\frac{4}{\theta^5} $ where: $ 0<\theta<1, a>0 $
0
votes
1answer
32 views

Degree of Precision Effect on Quadrature Accuracy

For an $n$ point Gaussian quadrature, one can show that it has degree of precision $2n - 1$ meaning it will exactly integrate polynomials of that degree or lower. Is it always true that a quadrature ...
0
votes
1answer
27 views

What numerical quadrature algoritm can be use to handle $\int_b^c K_0 (x-a)-K_1(x-a) dx$?

I am curious what numerical algorithm can be used to handle $$\int_b^c [K_0 (x-a)-K_1(x-a)] dx$$, where $a\lt b\lt c$ and K is the modified bessel function of the second kind. From plotting the ...
0
votes
0answers
25 views

Numerical Integration Over Two Regions of an Ellipsoid

I would like to perform a numerical integration over the surface of an ellipsoid $D$. The domain must be split in two by a plane intersecting the ellipsoid (the intersection is arbitrary), so that we ...
0
votes
1answer
31 views

Two point Gaussian Quadrature rule

I want to use the two point Gaussian Quadrature rule to approximate (evaluate) $\int_0^1 \! 6x^2-2x+1 \, \mathrm{d}x $ Since, with the two point Gaussian Quadrature rule, n=2 and the integral of ...
0
votes
1answer
36 views
0
votes
4answers
67 views

Numerical integration - $\int_{-1}^{1} f(x)dx$

I'm currently studying numerical integration, and ive come across a question i'd like help answering. We are given an integration rule as follows: $I(f)=\int_{-1}^{1}f(x)dx = \frac{2}{3} ...
0
votes
1answer
28 views

Is the assumption $f \in C^4$ necessary for the composite Simpson's rule to be of order $p=4$?

In my introductory numerics class, we wanted to integrate a function $f \in C[a,b]$ numerically. After developing the Simpson's rule, we proved that if $f \in C^4$ then the composite Simpson's rule ...
1
vote
1answer
29 views

Integration Rule Exact Degree

Given the integration rule $Q(x) = \alpha_1f(0)+\alpha_2f(1)+\alpha_3f'(0)$ for interpolating the integral $\int_0^1f(x) dx$ , I need to find $\alpha_1,\alpha_2,\alpha_3$ values s.t Q has exact degree ...
0
votes
1answer
50 views

How to numerically handle a double integral with a singular endpoint on the outer integral

I am trying to numerically integrate $$\int_0^a f(x) \int_{\sqrt{x}}^\infty \frac{\exp(-u^2)}{\sqrt{u^2-x}}du dx$$ where a is some positive real number and f(x) is some well behaved function. The ...
0
votes
0answers
15 views

Calculate the weights and the node in the integration formula

The problem is the following. Calculate the weights $w_1$ and $w_2$ and the node $x_1$ in the weighted integration formula $\int_0^1x^{\frac{3}{4}}f(x)dx\approx w_1f(x_1)+w_2f(\frac{3}{4})$ The ...
0
votes
1answer
25 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
0
votes
1answer
56 views

Double Integrals & Expected Value Monte Carlo Method

Tell me if I'm wrong Let $\Omega = [a,b]\times[c,d]\subseteq\mathbb{R}^2$, then $$ \iint_\Omega ...
2
votes
0answers
40 views

Are there high performance computing applications for symbolic integration?

Currently there are a number of applications for numerical integration in applied mathematics and physics. Many of these are integral transforms (often Fourier or Laplace), or solving definite ...
0
votes
1answer
18 views

Meaning of the unique up to a normalization factor

In the following text about Gaussian quadrature by Brian Bradie I cannot understand the meaning of the author: Associated with each weight function is a special family of polynomials, unique up ...
4
votes
3answers
78 views

Robust Numerical ODE Solver?

I made a little explicit Runge-Kutta 4th order solver a few days ago, but when testing it against various 1st and 2nd order ODEs chosen at random (for example $d^{2}y/dt^{2} = -y \sin(y)$, ...
2
votes
1answer
90 views

Can I compute this integral analytically?

I will give a small background and explain the variables and the system first. I have two images which are observed and are constant and we can treat them as continuous functions and I will call them ...
3
votes
1answer
82 views

Numerical integration fails

I am doing something wrong. This is my algorithm to evaluate the integral $$\int_0^1 \frac{1}{1+x}dx= \log(2).$$ with the Newton Cotes algorithm (Simpson and 3/8). Both give me that for large n ...
0
votes
1answer
17 views

Computational complexity of numerical integration of gaussian function

$ \int^{b}_{a} \exp(-x^2)\,dx$. I have the following two questions regarding the above integral expression of the Gaussian function: Is there a numerical method we can use to solve the above ...
2
votes
1answer
59 views

Change of variable

I have to approximate the following integral, using Simpson's Composite $1/3$ Rule: $\displaystyle \int\limits_{0}^1 \mathrm{\frac{e^{2x}}{\sqrt[5]{x^2}}}\,\mathrm{d}x$. The only problem is that ...
0
votes
0answers
31 views

Comparison of trapezoidal , Simpson's 1/3 ,Simpson's 3/8 and Boole's rules.

These rules are often used in numerical integration. How do we analyze the given support points or function and select the most suitable one for best approximation?
0
votes
0answers
30 views

Integral with Inverse error function

I have a challenging integral to solve involving the inverse error function, $\rm Erf^{-1}$, $\mathcal{I}(x,\beta)=\int\,_{x_c(\beta)}^x\,{\rm d}x\,\exp\left[\sqrt{2}\sigma\,{\rm ...
1
vote
1answer
120 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
0
votes
0answers
14 views

Numerical integration and probability density functions

How to calculate the integrals of this type? Which method I can use? $$ I_1(t)=\int_{0}^{\infty} dy f(x,y,t)p(x_j,y,t)$$ where $p(x_j,y,t)$ is $p(x,y,t)$ for some $x=x_j$. ...
0
votes
0answers
28 views

like Gauss-Chebyshev integration formula using Lagrange polynomials

Suppose that $L_k(x)$ is Lagrange Interpolation Polynomial for points $x=1,0,-1$. How to show that: $$\int_{-1}^{1}\frac{f(x)}{\sqrt{1-x^2}}dx=\sum_{k=-1}^1C_kf(k)+E$$ where ...
4
votes
0answers
92 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
2
votes
2answers
98 views

Integration problem: $\int _ {-\infty} ^ {\infty} \frac {e^{-x^2}}{\sqrt{\pi}} e^x\ dx$

I have to integrate $$\int _ {-\infty} ^ {\infty} \frac {e^{\large-x^2}}{\sqrt{\pi}} e^{\large x}\ dx.$$ I've already done by numerical approximations, like Simpson's rule and Gauss-Hermite, but I ...
1
vote
1answer
119 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
0
votes
1answer
55 views

Modified Composite Quadrature formula

I'm trying to produce a modified composite trapezoid quadrature formula for $\int_a^b f(x)\ dx$ based on the Euler-MacLaurin formula. I know that it should involve $f'(a), f'(b), f'''(a)$ and ...
0
votes
1answer
41 views

Quadrature Rule “is exact for polynomials of degree n”

Could someone kindly explain what "a quadrature rule is exact for polynomials of degree n" means? Here is what I understand about numerical (Newton-Cotes) quadrature rules: Suppose we want to ...
1
vote
0answers
28 views

Numerical Integration of Highly Oscillatory Integral with Misbehaving Derivatives

I'm attempting to numerically handle an equation of the following form: \begin{equation*}f: x \rightarrow \int_{0.00001}^{2} d\omega e^{i \omega x} f(\omega)\end{equation*} where $f(\omega) = ...
2
votes
1answer
27 views

Least Squares Method Confusion

I'm learning about the Least Squares method. An exercise I am doing is "Find the constant c that makes the expression $$\int_{0}^{1} (e^x - cx)^2 dx$$ a minimum " Though, i'm not sure how to ...
0
votes
1answer
29 views

Numerical integration: Quadrature method which one to use?

Since “it depends” is the proper answer to a question about what quadrature method to use in evaluating an integral, what are the things that one should consider when making a choice.
0
votes
1answer
66 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
0
votes
1answer
30 views

Interpolation of Polynomial using Lagrange

$f(x) = x^3 + 2x^2 + x + 1$. Find a polynomial of degree $4$ that interpolates the values of $f$ at $x = -2, -1, 0, 1, 2$. I was trying to use the Langrange algorithm, but I think i'm doing it ...
0
votes
2answers
23 views

Interpolation of Polynomial

Let $f(x) = x^3 + 2x^2 + x + 1$. Find the polynomial of degree $2$ that interpolates the values of $f$ at $x = -1,0,1$. I was able to do the an initial part of this problem (not written), but I ...
1
vote
1answer
41 views

Trapezoid Rule - Number of Points

How many points should we use in the trapezoid rule in computing an approximate value of $\int_{0}^{1} e^{x^2} dx$ if the answer is to be within $10^{-6}$ of the correct value? I'm looking at the ...