# 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 field. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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### Solve heat transport equation numerically with forward finite differences and explicit timestep

I am working on numerical solutions to the diffusion equation and came across a counter-intuitive phenomenon. Let's stay in 1D for this. The diffusion / heat transport equation is ($f$ my state ...
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### How to efficiently compute the minimal polynomial of a number expressed in radicals?

Preamble: I want to calculate the minimal polynomial of a number of the form $$x=\sum_{i=1}^k \pm a_i^\frac 1{k_i}$$ Where the $a_i$ are algebraic numbers also of this form with a finite ...
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### Can we use the inverse quadratic interpolation for this equation's non-real roots with these points?

$$f(x)=x^4+2x^3+5x^2+5x-3=0$$ I've chosen these values as guessed values: $$(x,f(x)) --> (-1+4i, 172+116i), (1+2i, -42+2i), (-3+i, 14-69i)$$ Am I doing something wrong or can't we obtain a non-...
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### Laguerre's method explanation

Can anyone please explain the steps of Laguerre's method? I searched for it but I couldn't really understand them. I am a high school student and things in Wikipedia didn't really help me understand. ...
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### maximize sum of rational polynomial functions

I have sum of rational polynomial functions and I want to find the maximum of the expression over the unit circle. Assume that the polynomials $a_k(z),b_k(z)$ is real on the unit circle, and all ...
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### Recurrence relation in the bisection method

When beginning to talk about error bounds on the bisection method for root finding, my book states the following: Let $a_n$ $b_n$ and $c_n$ denote the $n$th computed values of $a,b,$ and $c$, ...
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### How do I avoid significant rounding error in evaluating $(\ln(x) - \sin(\pi x))(1-x)^{-1}$?

How do I avoid significant rounding error in evaluating $$\frac{\ln(x) - \sin(\pi x) }{1-x}$$ This function causes error as $x\to 1$. How can this be avoided? I tried using taylor's expansion but I ...
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### Numerical operations when numbers are very large?

Explain the best way to evaluate $f(x,y) = \sqrt{(x^2 + y^2)}$ numerically when $x$ or $y$ are very large. Does anyone have any insight to this? I'm lost. I usually know how to deal with these types ...
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Consider the diffusion-advection equation given by $$- \mu \Delta u + \mathbf{v} \cdot \nabla u = f \ \text{in} \ \Omega$$ with some appropiate boundary conditions. Here the velocity field $\mathbf{... 0answers 23 views ### Meaning of$\frac{\| {PA - LU} \|_{F}}{\| {A} \|_F}$and$\frac{\| {PA - LU} \|_F}{\| {L}\|_F \| {U} \|_F}.$I'm studying PLU decomposition and in one of the problem I was asked to implement the algorithm in MATLAB, then report$\frac{\| {PA - LU} \|_{F}}{\| {A} \|_F}$and$\frac{\| {PA - LU} \|_F}{\| {L}\|...
Burden and Faires define: If $\|\cdot\|$ is a vector norm in $\mathbb{R}^n$, then $$\|A\|=\max_{\|x\|=1}\|Ax\|.$$ In other words, the measure given to a matrix under such a norm describes how ...