3
votes
2answers
48 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 ...
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 ...
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
35 views
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
2answers
52 views

exact or numerical value of an improper integral

i am dealing with an improper integral which has been arised in my research. i will be greatful if you have any idea about the numeric value of this integral. $$ \int_{0}^{\frac{1}{4}} \frac{u^{4} ...
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 $x=x_j$. ...
2
votes
1answer
61 views

How to solve integrate(exp(x^2)) from 0 to 1 using by Numerical integration/Gauss-Legendre Quadrature?

I tried to implement Numerical integration/Gauss-Legendre Quadrature using by Python. And follwing this example I could get the correct answer but anyway if the function changes to exp(-x^2) ...
1
vote
0answers
36 views

Comparing Methods For Solving Double Integral

For my assignment I'm asked to compute a given double integral using these methods: ...
1
vote
1answer
132 views

two point gaussian quadrature to approximate $\int_0^1(1-x)f(x)\text{ dx}$

I want to use two point gaussian quadrature to approximate $$\int_0^1(1-x)f(x)\text{ dx}$$ Because $(1−x)$ is a linear polynomial, polynomials $f$ of degree at most $2n−2=2$ (because we use two point ...
0
votes
0answers
28 views

Convolution of Brownian function with characteristic function

Given Brownian function defined on the interval $[0,1]$. Our aim is to filter this function, one may use the low band pass filter. The idea is to cut the high frequency of its Fourier transform by ...
2
votes
1answer
363 views

error of the composite trapezoidal rule

Let $f\in C^2[a,b]$. The composite trapezoidal rule is given by $$T_n[f]:=h\left(\frac{f(a)+f(b)}{2}+\sum_{k=1}^{n-1}f(x_k)\right)\;\;\;\;\;\left(h:=\frac{b-a}{n},\;x_k:=a+kh\right)$$ First, I've ...
0
votes
2answers
108 views

How do we prove the error estimation of the rectangle method

Let $f\in C^2[a,b]$. An approximation of the integral over $[a,b]$ is given by $$I[f]:=\int_a^bf(x)\text{ dx}\approx \frac{b-a}{n}\sum_{i=1}^nf\left(a+\frac{2i-1}{n}(b-a)\right)=:M_n[f]$$ I've spent ...
2
votes
2answers
70 views

Scale-invariance of Simpson's rule approximations to log

If I was trapped on a desert island and needed to compute $\log(2)$, the natural logaritm of $2$, one thing I could do is use the equality $$\log(2) = \int_1^2 \frac{1}{x} \ dx$$ and approximate the ...
3
votes
1answer
89 views

Integral of 2D Gaussian over triangle

I am interested in computing (numerically) the integral of a 2D Gaussian (with arbitrary mean and covariance matrix) over a triangle in the plane. I would like the solution to work over a wide range ...
0
votes
1answer
73 views

Volume of an airfoil

I need to know the volume a NACA 0010 airfoil , because there is nothing on the internet and i am still in high school , i have no clue on how to do it , so i decided that if i had the shape on a ...
1
vote
1answer
104 views

Error estimates for the trapezoid rule

Say we require $$ \left|\int_{a}^{b}f(x)dx - T_{n}\right| < 10^{-4} $$ where $T_{n}$ is the composite trapezoid rule with $n$ subintervals. To guarantee that $T_{n}$ satisfies this error bound, we ...
2
votes
1answer
247 views

Integrating $\sin(n\theta(x))/\sin(\theta(x))$ for some function $\theta(x)$

I have an indefinite integral of the form: $$ \int \frac{\sin(n\theta(x)))}{\sin(\theta(x))} dx. $$ $\theta$ is a function of $x$ (and actually a complicated one). Is it possible to integrate it ...
0
votes
0answers
926 views

Trapezoidal integration rule for double integrals with non-equally spaced points

I've found this very useful link about 2D Trapezoidal Rule (or Composite Trapezoidal Rule) that works when the points are equally spaced. I want to implement an analogous rule for non-equally spaced ...
0
votes
1answer
45 views

Ship wake modeling and definite integrals

In the study of the ship's wake, it needs to calculate the following definite integral: ...
3
votes
2answers
146 views

approximate $\int_{u=0}^b e^{-\frac{u^2 + ac}{au}}du.$

I'm trying to find an approximation (or exact solution if possible) for an integral of the form: $$\int_{u=0}^b e^{-\frac{u^2 + ac}{au}}du.$$ I was thinking of somehow applying a Gauss Hermite ...
1
vote
1answer
240 views

Can a chi squared distribution with a huge number of degrees of freedom be computed with a good precision?

Let $X$ be a chi squared variable with $121$ degrees of freedom. So the density $f_X$ of $X$ is defined by $$ ...
1
vote
0answers
92 views

How to evaluate $I=\iiint dr_{12}dr_{13}dr_{14}$ analytically/numerically?

How to evaluate this integral analytically or numerically: $$ I=\iiint dr_{12}dr_{13}dr_{14} $$ constrained by $$ r_{12}<r_0,\\r_{23}<r_0,\\r_{34}<r_0, $$ where $r_0$ is a given real ...
1
vote
2answers
207 views

Numerical integration given a derivative of a function of two dependent variables

I want to solve the following equation of an integral valued function: $Q = \int_{0}^{x_p}f(t_p,x)dx$ for some particular $x_p$ at a fixed time $t_p$, given some known $Q$ and an initial $f(0,x)$. ...
2
votes
1answer
481 views

What is the best state-of-the-art numerical integral algorithm?

I'm trying to implement a numerical integrator that should have the minimum relative error and is not slow. So I was looking for the best accepted state-of-the-art algorithm to do so but there seems ...
7
votes
3answers
532 views

integration method

I want to calculate an integral by using the hit and miss method. I can not understand how this method works. I would be grateful if someone could explain me and help me to calculate the value, with a ...