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

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7 views

Numerically solving a transport equation

I would like to solve this transport PDE numerically : $$ \partial_t f + v(f) \partial_x f = 0 $$ What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by ...
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0answers
15 views

Grand canonical derivative.

I've been trying to work out how to find the density in the thermodynamic limit of a nearest neighbour magnetic lattice gas in the grand canonical ensemble. I'll with hold the Hamiltonian for the ...
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1answer
9 views

Complexity of Newton iteration problem for a d-dimensional problem

If we assume that we have $f:\mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and we want to use the Newton iteration method to solve $f(x)=0_{\mathbb{R}^{d} }$. Is there any theorem regarding the ...
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3answers
70 views

Calculate an integral depending on n

Is there a way (simple or not) to calculate the following integral? $$\int_{-1}^{1} \sqrt[n]{1-x^n} dx$$ Thanks
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3answers
15 views

Efficient algorithm for maximum of a differentiable function

Is there an efficient algorithm which can be used to find the global maximum of a differentiable function (of one variable) on a given interval?
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1answer
48 views

Complex stationary point of $\frac{z}{1-e^{-z}}+z$?

I apply the method of steepest descents I need to know the stationary points $z_0$ of the function $$ p(z)=\frac{z}{1-e^{-z}}+z, $$ such that, $ 0 <\mathrm {Im} (z)<2 \pi$. That is, I want $z_0$ ...
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34 views

How to Find the pointwise limit of $(f_n)$

For $x \in [0, \pi/2]$, if $$f_n(x) = \frac {nx} {1+n\sin(x)}$$ how do you find the pointwise limit of $(f_n)$ ?
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13 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
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3answers
40 views

How to solve these simultaneous equations using numerical methods?

How to solve these simultaneous equations for $\alpha$ and $\lambda$ using numerical methods? $\lambda * [(\frac{3}{4})^\frac{-1}{\alpha} - 1] = 11$ $\lambda * [(\frac{1}{4})^\frac{-1}{\alpha} - 1] ...
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0answers
23 views

When to use iterative methods for solving systems of linear equation

Iterative methods such as Jacobi, Gauss-Seidel method and successive over relaxation have a very limited field of use - for diagonally dominant matrices. So how they could be used on practice? What is ...
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0answers
28 views

Is this inequality true for every polynomial

Suppose $f(n) , g(n),h(n)$ are polynomials and for all $n \in (a, b)$ we have $f(n) + g(n)=h(n)$ such that $\frac{g(n)}{f(n)} \approx 0$ and $ \frac{g(n)}{h(n)} \approx 0$. Also let $n_0 \in (a,b)$ ...
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10 views

Spectral Differentiation using FFT on an arbitrary domain( python) [on hold]

I am trying to write a python script for spectral differentiation on a domain of arbitrary length . The function I'm trying it on is the gaussian, $f(x)=e^{-x^2}$. The program works for the domain ...
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1answer
12 views

Numerical method with a time derivative boundary condition

I'm trying to reproduce a result from a paper I'm reading using a numerical scheme that I'm coding myself. The equation is a reaction diffusion PDE. $$\frac{\partial M}{\partial t}=\frac{\partial^2 ...
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0answers
9 views

Error estimation - spline interpolation

I got a question regarding error estimation and spline interpolation. I got a parabola shaped graph that I've used spline interpolation on to get more accurate data. I've used a much smaller step on ...
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0answers
13 views

Iterative methods monotonically decreasing of the residual

For a question on Iterative Methods I have to show that the 2-norm of the residual is monotonically decreasing. We are given the following formula: $r^{(k+1)} = r^{(k)} - \alpha^{(k)} A z^{(k)}$ where ...
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1answer
26 views

Does any numerical diff.eq. solver give correct results given small step-size?

I've seen that there are less stable numerical differential equation solving methods, like using plain Euler steps $y(x+h)=y(x)+hf(x)$. For a given $h$ there are better methods. But when solving ...
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0answers
17 views

Approximating roots

Given $n,r\in\Bbb N$, assume $a=n^\frac{1}r$. Assume that $a_d$ is $a$ truncated to $d$ digits ($d$ is total digits both before and after decimal Eg: truncating $412.243$ to $2$ digits is $410.000$ ...
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3answers
228 views

Rewrite trigonometric expression to be be numerically “stable”

Is it possible to write the following function: $$ f(x) = \begin{cases} \frac{x-\sin x}{1- \cos x}& x\neq 0\\ 0 & x=0 \end{cases} $$ as a composition of elementary functions (including ...
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0answers
15 views

Optimal initial guess in Newton-Raphson method for nonlinear systems [on hold]

I would like to know what is an optimal initial guess for use with Newton-Raphson method for nonlinear systems. Thank you for your help.
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0answers
11 views

[Levenberg-Marquardt]What is the link between positive-definiteness and well-conditioning?

Working on optimization problems through neural networks, I use the Levenberg-Marquardt algorithm. I have read this assertion that I do not understand : A positive definite diagonal matrix is ...
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0answers
25 views

How to make 4 (parametrized) points (in the complex plane) concentric?

I consider a functional equation (see the earlier discussion at MO) $$ f(f(x))= f(x)^2 + x \qquad \text{where also} f(0)=0$$ In the following I write it in a more concise form (with $z$ instead of ...
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2answers
30 views

Solving $f(x) = x^5 +x + 1 = 0$ with halving the interval / bisection method

Question: Use halving the interval / bisection method to approximately solve: $$f(x) = x^5+ x + 1 = 0$$ with a precision of $\pm 0.1$ Attempted solution: The general idea, as I understand it, is ...
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0answers
11 views

Differentialequation with Eulers method

I have a problem with a differential equation that can be used Euler method in a digital manner. I use a program that is designed to excel. The entire task looks like this: Differential equations y ...
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1answer
18 views

Modified version of Simpson Rule

I'm supposed to use some different version of Simpson's Rule in my Numerical Methods homework to compute some areas, considering the non-uniform spacing case . Namely, I've got two equal length ...
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1answer
22 views

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? [on hold]

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? Any help or just tips will be appreciated, thanks.
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1answer
14 views

Learning spectral methods in numerical analysis

I'm trying to learn the theory about spectral methods without any specific ties to a particular program like MATLAB. I tried to search for some lecture videos but it seems very limited and I'm not ...
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0answers
48 views

Confusion regarding dF/dx=0, F=constant

I thought I found a theorem Given a curve in the $(y,x)$ plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of $F$ along this curve satisfies relation $g = ...
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0answers
27 views

Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
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0answers
17 views

How do I show that these three numerical methods are at least second order? [on hold]

$$w_{i+1} = w_{i} + \frac{h}{2}(3f_{i} − f_{i−1}),$$ $$w_{i+1} = w_{i} + \frac{h}{2}(f_{i+1} + f_{i}), $$ $$w_{i+1} = w_{i} + h f(t_{i} + \frac{h}{2}, w_{i} + \frac{hf_{i}}{2}) $$
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0answers
27 views

Is convex or non convex function?$J(u,c)=\int K(x).u.(f(x)-c)^2dx$

I have a function such as $$J(u,c)=\int K(x).u.(f(x)-c)^2dx$$ where $f(x):\Omega \to R$; c is constant; $0 \le u \le 1$; and K(.) is gaussian kernel. My question is that : Is J convex or non-convex ...
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1answer
20 views

Relationship between centralization and floating-point arithmetic

Suppose I have a problem of least squares where I have $n$ independents regressor variables $x_i$, and suppose I applied the process of centralization and scaling to this variable through ...
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1answer
43 views

Cholesky algorithm

Good afternoon everyone, I'm in need of a factoring algorithm cholesky and algorithms to solve upper and lower triangular systems, but I'm not finding any work in that octave. Recalling that need the ...
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1answer
47 views

fourth derivative

If I want to numerically compute a fourth derivative, I can adopt the central finite differences for the internal nodes, the backward finite differences for the last node, and the forward finite ...
2
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1answer
22 views

Find numerical minimum of a function with many parameters

I have a function $$f(\vec{r}_1\dots,\vec{r}_N)=\mathrm{The \ sum\ of\ square roots\ of\ the \ eigenvalues\ of\ }\Omega(\vec{r}_1\dots,\vec{r}_N)$$ And I want to find one of its local minima with ...
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1answer
30 views

System of linear equations: and a small perturbation

If $Ax=b$ and $Ax'=b'$ where $x'$ and $b'$ are $x$ and $b$ with a small perturbation, the following inequality will always hold: $ (\left\lVert x-x' \right\rVert/\left / \lVert x \right\rVert) \le ...
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1answer
29 views

Calculate the (variational) derivative of the following equation;

Consider $ E[u]= \int^1_0 \big(u'(x)\big)^2+\big(u(x)\big)^2-2f(x)u(x) dx.$ Calculate the variational derivation for a function $v$; in other words, calculate $\frac{d}{d\epsilon}E[u+\epsilon v]$ at ...
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1answer
21 views

Romberg Integration: accuracy

I'm applying the Romberg method to numerically integrate a data set of equally space, numerically determined values. I would like some estimate of the uncertainty (or accuracy or error) in my answer. ...
0
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1answer
25 views

Runge Kutta method order

I have a Runge-Kutta method given by the Butcher tableau: $$ \begin{array}{c|ccc} 0 & & & \\ 1/2 & 1/2 & & \\ 1/3 & 0 & 1/3 & \\\hline & -1/3& 1/3 &1 ...
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0answers
49 views

That is My class work problem. but I don't understand how to calculate this problem. Can u help me?

A Fourier analysis of the instantaneous value of a waveform can be represented by $$ y = (t + \pi/4) + \sin t + 18 \sin 3t $$ Apply the appropriate method to determine the value of $t$ near to $0.04$ ...
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0answers
18 views

Numerical Methods: Mid Point Higher Order ODEs

I am taking a Numerical Methods class and the professor told us to find out how to solve Higher Order Ordinary Differential Equations using the midpoint method. As of right now, I only know how to use ...
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0answers
30 views

Nonlinear Differential Equation with Pure Neumann Boundary

Four governing equations concerning the reaction occurred in the porous electrode are \begin{equation} \nabla \cdot i_1 + \nabla \cdot i_2=0 \end{equation} \begin{equation} i_2 = -\kappa \nabla ...
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1answer
16 views

How do I use these multi-step methods to solve the IVP?

Use the two second-order multi-step methods $$ω_{i+1} = ω_i + \frac{h}{2}(3f_i − f_{i−1})$$ and $$ω_{i+1} = ω_i +\frac{h}{2}(f_{i+1}+ f_i)$$ as a predictor-corrector method to compute an ...
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1answer
19 views

2D Richardson Extrapolation

I know Richardson extrapolation can be used to estimate a parameter at a single point, but is there a 2D analogous of it where it estimates a parameter over a surface? For example, I have a 1 m by 1 ...
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21 views

Understanding golden section search

I don't understand it at all. The only thing I understand is that we have some interval in which we know a minimum lie and we know the function is unimodal. I also know that we are diving the interval ...
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0answers
16 views

Simulating a SDE

I have a question about simulating a SDE. I want to simulate $dS=\alpha(K-S)dt+\sigma S dZ$ with use of a Euler-marayama scheme. The numerical scheme becomes: $S_{i+1}=S_{i}+\alpha(K-S_{i})dt+\sigma ...
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1answer
36 views

How to calculate the shielding time and determine the time step

The problem is illustrated as follows. A shielding plate scans over a target plate at a constant speed $v_{scan}$ and dynamically shadows the target plate to adjust the exposure time of the light ...
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1answer
53 views

Algorithms For Large-Scale $\ell_{\infty}$ Minimization

The general problem I want to solve is well studied: $$ \min_x \Vert Ax\Vert_\infty \;\;\; \mathrm{s.t.} \;\;\; Bx=c, $$ which is equivalent to the following linear program: $$ \min_{t,x} \, t ...
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0answers
41 views

Closed form for integral of an error function

My question is similar to that posted here. I have the following integral that I want to determine in a closed form. My uncertainty arises due to the addition term within the Error function: ...
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0answers
45 views

what is transcendental equation? [closed]

What is transcendental equation? What is application,importance,and role of transcendent equation in math? how many transcendental equations? Bisection method, Newton–Raphson method, are use for ...
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21 views

Smallest square problem, $A^*A$ singular?

In our numerics class, we have to solve the smallest square problem $Ax = b$ with $$A = \left( \begin{matrix} 1 & 3 &-4\\ 3 & 9 & -2\\ 4 & 12 & -6\\ 2 & 6 & 2 ...