# Tagged Questions

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### 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 ...
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### 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: ...
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### 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| $\le$ ...
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### 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: $$S(2n) = \frac{4T(2n) - T(n)}{3}$$ ...
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### Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, $$y''(x) + p(x)y(x) = 1$$ in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
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### 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 ...
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### 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 ...
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### 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 $d^{2}y/dt^{2} = -y \sin(y)$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...