0
votes
0answers
37 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
5
votes
4answers
107 views
+50

Software, techniques and tricks of experimental mathematics to conjecture possible closed forms

It often happens that people conjecture possible closed forms of integrals, series, and so on starting from a numerical value calculated to very high precision. What are the techniques, tricks, ...
0
votes
0answers
16 views

How to estimate the error of a numerical multiple integration

I'm integrating over the wholes space the function $$f(\vec{r_1},\vec{r_2})=\exp{\bigg[-(r_{1\alpha}+r_{1\beta}+r_{2\alpha}+r_{2\beta})\bigg]} \cdot 1/r_{12}$$ where $\vec{r_\alpha}=(-R/2,0,0), ...
0
votes
0answers
22 views

Derive Runge-Kutta matrix with known weights and nodes

Can I derive a Runge-Kutta method by choosing freely the weights and the nodes? what are my constraints? So, if this is the general form of the explicit RK method: $$ y_{n+1} = y_n + \sum_{i=1}^s b_i ...
3
votes
2answers
32 views

Existence and uniqueness of weights for the rule $\int_a^b f(x) \ = \ \sum_{0 \leq k \leq n} w_k f(x_k)$

I want to establish this statement: If $a<b$ and $\{x_0,x_1, \cdots x_n\} \subset \mathbb{R}$ distinct, then there is one and only one set of weights $\{w_0, \cdots w_n \} $ such that $\int_a^b ...
2
votes
1answer
46 views

How to simplify this complex integral? [closed]

How to approximate this integral as a function of a and b? $$\int_0^\pi\int_0^{2\pi}\sqrt{(a-b\sin\varphi\cos\theta)^2+(b\cos\varphi)^2+(b\sin\varphi\sin\theta)^2}d\theta d\varphi$$ where a and b ...
3
votes
0answers
40 views

Approximating this definite integral

I ran into the following integral in my research that I believe has no closed-form solution: $$ I = \int_{s_0}^{s_1} \frac{(\alpha_x s + \beta_x)^{\lambda_x}}{(\alpha_y s + \beta_y)^{\lambda_y}} ds ...
0
votes
0answers
11 views

Numerically doing trapezoid rule and richardson extrapolation using haskell.

so im trying to numerically to solve some integrals using trapezoid and richardson methods. Ive implemented the code as followed in haskell. ...
0
votes
1answer
18 views

min/max in front of a sigma sign (Numerical Analysis)

I have an assignment in my Numerical Analysis class that involves an approximation of an integral with a sum. The sum looks to me like a Riemann sum, but I don't know what the "min" in front of the ...
1
vote
1answer
57 views

Find the coefficients in quadrature formula on $[0,1]$ with the nodes at $1/4$, $1/2$, $3/4$

In my worksheet I was given a question about numerical integration that says: Find the formula for $\int_{0}^{1}f(x)dx=A_{0}f(\frac{1}{4})+A_1f(\frac{1}{2})+A_2f(\frac{3}{4})$ I suppose the goal ...
1
vote
0answers
28 views

Will Gauss quadrature numerical integration work with a variable dx

The question kind of says it all, but I'm reading about Gauss quadrature from here: http://www.damtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/integration.htm which gives an equation of this form: ...
0
votes
0answers
13 views

Why does QUADPACK only enforce the least strict error boundary?

According to this reference (which is in agreement with my own numerical experiments), QUADPACK tries to fulfill the following accuracy requirement on the approximation error: |RESULT - I| $\le$ ...
1
vote
1answer
40 views

Simpson's rule and Trapezoid Rule?

Let $S(n)$ and $T(n)$ be the approximations of a function using $n$ intervals by using Simpson's rule and the Trapezoid rule respectfully. My book then states: $$S(2n) = \frac{4T(2n) - T(n)}{3}$$ ...
1
vote
1answer
34 views

Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, \begin{equation} y''(x) + p(x)y(x) = 1 \end{equation} in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
1
vote
1answer
27 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
0
votes
0answers
16 views

How to use Legendre Quadrature for Multiple Unbounded Integrals?

Legendre Quadrature as most other methods is designed for $[-1,1]$ interval and some variable change methods are used for extending them to $[a,b]$ interval. I found some method in a textbook for ...
2
votes
2answers
50 views

Why is Romberg integration usually based on trapezoidal rule?

The wikipedia article on Romberg Integration says that it's simply Richardson Extrapolation applied to either the Trapezoidal Rule or the Midpoint Rule. I'm reading out of a couple of textbooks on ...
1
vote
1answer
33 views

How to modify Gauss-Hermite quadrature rule when the weight function is slightly generalized

hope this is the right forum. Consider a slightly modified version of the Gauss-Hermite quadrature rule, where the weight function is not $\exp(-\frac{x^2}{2})$ as in the standard Gauss-Hermite rule, ...
0
votes
0answers
41 views

numerical calculation of an integral

I am having trouble finding the solution of this numerically and wondered if I could get some tips so that I can: $$ \int\limits^1_{0}\left[\min(ax, b) - \min(a x, c)\right] dF(x; p, \rho)$$ (1) ...
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
66 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
103 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
47 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
48 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
104 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
39 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
112 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
13 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
10 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
41 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
46 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
61 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
28 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
33 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
60 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
38 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
29 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
33 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
65 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
17 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
26 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 ...
2
votes
1answer
76 views

Using Monte Carlo method to estimate the value of $\int_0^1\int_0^1e^{xy} \ dx \ dy$

Monte Carlo, estimate the value of $$\int_0^1\int_0^1e^{xy} \ dx \ dy.$$ I am using matlab to solve this problem. My code is the following: ...
0
votes
1answer
91 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
57 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
91 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
95 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
84 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 ...