# Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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### Connection between collocation methods and quadrature rules

I know what a collocation method is and how it can be useful for the direct transcription of optional control problems, using specific collocation points. I also know what a quadrature rule is and how ...
1 vote
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### Numerically stable form for vector rejection between two nearly parallel vectors?

Given two vectors $\vec{M}$ and $\vec{N}$, I want to compute the rejection between them : $oproj(\vec{N}, \vec{M}) = \vec{N}-(\vec{M}\cdot\vec{N})\vec{M}$. For my purposes both input vectors are unit ...
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### Transform 2d finite element into standard 2d element

A certain three node triangular element has nodes with coordinate as follows: $$1:(0.13,0.01)\quad2:(0.25,0.06)\quad3:(0.13,0.13)$$ Deduce a transformation that will map this element into the standard ...
1 vote
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### Weird hexagons distribution of singular vectors of random matrices.

I ran the following numerical experiment: Sample 10⁶ random Gaussian 2×2 matrices ($A_{ij}∼𝓝(0, 1)$) Compute SVD $A = UΣV^⊤$. Let $u_\max, v_\max$ be the singular vectors of the maximum singular ...
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### BVP problem with conjugate gradient method

Life and death situation. I have the differential equaction $$1.1 \dfrac{d^2T}{dz} - 0.01\dfrac{dT}{dz} = 0, T(-1000) = 0.12, T(0) = -1.55$$ that im trying to solve with conjugate gradient method. So ...
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### Problem of numerical analysis of conditioning and PVI 2

"Let $a\in \mathbb{R}$, consider the PVI: $u'(t)=u_0e^{at}(a \cos(t) − \sin(t))$, $t > 0$ $u (0) = u_0$ with $u (t)=u_0e^{at}\cos(t)$ being the exact solution. Study the conditioning with ...
32 views

So during our numerics course we learned a few interpolation methods Aitken/Neville, divided differences,Lagrange and the Vandermonde matrix. How these work is clear to me for the most part, I'm just ...
1 vote
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### Approximating $\frac{1}{(1+x)^2}$ in $L^2$ norm with polynomial of degree $1$

I'm currently looking at an exercise where I'm supposed to get the best approximation of $\frac{1}{1+x^2}$ regarding $||f||= \sqrt{\int_{-1}^{1}f(x)^2dx}$ with a polynomial $p(x)=ax+b$. So my idea ...
### Runge Kutta Method for in $1+1$ dimension
Given a partial differential equation $\partial_t u(t,x) = F(t,x,u,u')$ Suppose I know the functions $u(t_0,x)$ and $u'(t_0,x)$ at some point $t_0$ for all $x$. In order to obtain the function $u$ at ...