# Tagged Questions

Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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### How to do fixed point iteration with matrices?

I am trying to follow solution to solve $$\min[\mathbf{z},\mathbf{q+Mz}]=0$$ by fixed point iteration. If $\mathbf{M=C+B}$ then a recursive algorithm with $k$ showing the iteration can be written as ...
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### Minimizing nonsmooth single variable functions?

What options is available if one wants to minimize a nonsmooth convex function of one variable? Subgradients would work, but there has to be some nice way of utilizing that we're only searching in 1d. ...
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### What is the weak formulation of this problem?

Find $u\in H_D^1(\Omega)$ such that $-\nabla\cdot(a\nabla u)=0$ in $\Omega$, $\dfrac{\partial u}{\partial n}=g$ on $\Gamma_N$, $u=0$ on $\Gamma_D$. The function $a(x,y)$ is ...
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### computing the area of a region using Monte Carlo integration

Suppose that I am interested in estimating the area of $\Gamma \in \mathbb{R}^2$. I do not know the exact shape of $\Gamma$ but I have a sufficiently large number of sample points $(X,Y) \in \Gamma$ ...
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### Adjust the data up curve φ(x) = α1e^(α2x) by the method of least squares

Adjust the data up curve φ(x) = α1e^(α2x) by the method of least squares: Here's what I've done so far but I think it is wrong(and sorry for the bad english) --x | 0    | 1    | 2   | 3   | 4   ...
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### Newton- fixed point iteration

The formula for Newton iteration (which is a zero-finding problem) is $x_{k+1}=x_{k}-f(x_k)/f'(x_k)$. I read in my textbook that this can be also be seen as a fixed-point iteration; where the zero ...
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### Why does my answer depend on the starting values?

I'm trying to find the zeros of f(z)=c(z)-2500=0 using the secant method. I get the correct values (8.9, -2.6, 7.7, 12.3) but only if I put the starting values close. For instance this version of the ...
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### What is the general idea of Nitsche's method in numerical analysis?

I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ...
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### Simple Implementation of QZ-Algorithm fails in MatLab [closed]

i am still very new to numerics, but i have a question concerning a very simple Implementation of the QZ-Algorithm in Matlab. My Code Looks like: ...
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### Mollifiers and Rates of Converegnce

I am interested in how quickly a.e. convergence happens to say: $|f(x) - f(x+h)|$. Originally, I thought I had proved something way too strong, but smoothed that out while typing this question up. ...
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### Improper integral over product of exponentials: $\int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx$

I'm looking for a way to evaluate following integral $$\int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx$$ f(x) resembles however a complex simulation and can therefore ...
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### Solving for n in the equation $\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$

Solving for $n$ in the equation $$\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$$ Can anyone show me a numerical method step-by-step to ...
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### Separating the Complex Error Function into Real and Imaginary parts

I'm trying to do a numerical integral of the following form: $$\int_a^b (\mathbb{R}\left[\operatorname{erfi}(z)\right])^2 \, dz$$ That is, I would like to integrate the square of the real portion of ...
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### Question about an boundary integral equation with a jump in the boundary

I have the following problem: $$\Delta u = 0\;in\;\Omega$$ with several boundary conditions. Applying Green's second identity the representation formula can be derived: ...
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### How to find $\log{x}$ close to exact value in two digits with these methods?

I'm trying to find the result of $\log{x}$ (base 10) close to exact value in two digits with these methods: The methods below are doing by hand. I appreciate you all who already give answers for ...
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### How is optimal coordinates change chosen for Chebyshev expansion?

I'm looking into SLATEC implementation of Bessel function $J_0$ computation (readable in C in GSL), namely at its part for arguments in interval $(0,4)$. There a Chebyshev expansion is used, but the ...
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### The effect of the CFL number in the numerical solution in this conservation law

I've been studying the very basics of numerical methods applied to conservation laws, and I'm having trouble understanding the role of the CFL number in the upwind scheme. I want to understand it (if ...
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### Accuracy of angular difference: direct difference or difference identity

Can anyone point out if there is a difference in the accuracy of the result of calculating an angle difference when using the difference between two arctan values or when using the difference formula ...
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### Please recommend a book/article for Newton-Raphson method

There are so many search results I'm a bit lost. I would like to read an article to fully understand it, including the math end, and the appliction side, please recommend one that contains also ...
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### Simulating an orbit - numerically solving $M(E) = E + \sin(E)$

Well for a given kepler orbit (which is a ellipse) $0 \leq e < 1$. There are several functions to describe the motion of an object. $$r(\nu) = \frac{a (1 - e^2)}{1 + e \cos(\nu)}$$ Where $a$ is ...
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### Fast computation of integral of Gaussian pdf

Which methods/algorithms for computation of the function $F$, where $$F(a,b) = \int_a^b e^{-t^2}dt,\quad a\leq b,$$ are the best, i. e. fast and accurate? I need to compute those integrals ...
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### $Az + B\overline{z}$ as a linear operator

Given two matrices $A,B \in \mathbb{C}^{n\times n}$ with fixed $n\in\mathbb{N}^+$, let us consider the operator $$L:\mathbb{C}^n \to \mathbb{C}^n,\\ L(z) = Az + B\overline{z}.$$ This operator is not ...
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### Marking iterations on Cobweb / Staircase diagrams

In the below cobweb diagram I am interested in why the iterations $( x_1,x_2,x_3)$ are marked when the 'cobweb' intersects the line $y=x$. Image: http://i.stack.imgur.com/iiYST.png For example, why ...
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### How to tell that 2 set of data are not so difference by using statistical method?

How to find stable point of these data? 2.0, -3.5, 0.0, 1.5 1.3, 6.3, 0.1, -3.4 3.3, -1.1, 3.0, 4.1 -2.5, 4.3 -1.0, 2.2 The example data is random ...
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### Computing wavenumbers for discrete Fourier transform

I'm trying to implement a Fortran program to compute the derivative of a function using the FFT. To begin with, just to test my installation of fftpack, I computed the Fourier transform of ...
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### Deferred Corrections vs Multigrid

I've been looking at the method of Deferred Corrections (see page 9 of this presentation) to numerically solve ODE IVPs. To me, the process looks identical to a V-cycle in a multigrid method if the ...
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### Numerical or analytical or exisistence: Inverse Laplace Transform

Edit 1: With the hint of Ron, we can simplify the question to : $$\bar{f}(s)=\frac{1}{(s^2+1)\arctan s }$$ So what about this function's inverse Laplace Transform? Or can anyone tell me that the ...
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### How do I get geographic coordinates of a point if I know the distance from two other points?

The keyword here is geographic. I am assuming the solution has something to do with a spherical triangle. I know that this problem has one, two, infinite or no solution at all. My specific problem ...
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### turn $L\{u(t-3)(t^2)\}$ to $L\{u(t-3)[(t-3)^2+6(t-3)+9]\}$?

I was given looking at one of the examples in my textbook and it took this laplace transform $L\{u(t-3)(t^2)\}$ and turned it into this $L\{u(t-3)[(t-3)^2+6(t-3)+9]\}$ in the next step. I'm ...
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### Upper bound, supremum norm [closed]

How to find upper limit for this: $\sup_{f \in C[a,b],f \neq 0} \frac {||p_f||_{\infty}}{||f||_{ \infty }}$ $p_f$ is interpolation polynomial of f.
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### Reading list to master Numerical Analysis' research literature

As of lately I have been going through many research papers in my current job, and even though I have a Mathematics background at Masters level in Mathematical Finance, I sometimes struggle to follow ...
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### Estimate order of convergence given error table

I looked through some other posts and didn't quite understand what was going on... Given a numerical method of calculating the solution to, say, a DE, we typically get an error table. The errors are ...
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### Increase of precision in numerical Hessian computation?

In a function for numerical calculation of the Hessian [http://grizzly.la.psu.edu/~suj14/programs/Jacob.m] I saw the following 3 lines (here dh, eps, x0 and xdh are all vectors of the same size): ...
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### How to determine whether a point is inside a closed region or not?

Take the following parametric equation of an implicit curve as an example: $$\left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right.$$ ...
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### Cholesky decomposition and rotation matrix inverse

I implemented three methods for inversion of a matrix, all are classic. I wanted to test for the most generalized method, while taking efficiency into account. For Cholesky decomposition, which is ...
### Chebyshev spectral derivation with 16 nodes for $\,f(x)=e^{\,\text{sin}^{2}\,(x)+\cos(x)}\,$ defined in $\,[0,2\,\pi].\,$
I'm making the following exercise in Matlab, and I'm having trouble expresing my result in $x\in[0,2\pi]$ not in $x\in[-1,1]$. I first done this (as shown below) in Gauss-Lobatto points, but I don't ...