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

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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. I decided on the Crank-Nicolson method for the ...
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0answers
7 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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11 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
19 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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16 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
221 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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13 views

Optimal initial guess in Newton-Raphson method for nonlinear systems

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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7 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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23 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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9 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
40 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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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
42 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 ...
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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
28 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
20 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. ...
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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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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
51 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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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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43 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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0answers
20 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 ...
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1answer
78 views

Evaluate integral over complex path numerically to show that $C_\infty$ is equivalent to $-I$

I would like to evaluate $$C_\infty = \int_{R = -a}^{R = a} H_0^{(1)}(z) e^{-izt} dz $$ where $H_0^{(1)}(z)$ is the Hankel function of the first kind, $a \rightarrow \infty$, and $$ z = R - i ...
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8 views

Finding common roots to a variable number of functions

I am trying to solve the following problem. Given $a\in\mathbb R^n$, $u\in\mathbb{R}^n$, $m\in\mathbb{N}^\star$, Find the/some common roots $(t_1,...,t_m)$ of the $\frac{m(m-1)}{2}$ ...
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2answers
28 views

Secant method with two ODE's of degree 2 - matlab

$$\frac{d^2r}{dt^2}-r\left(\frac{d\phi}{dt}\right)^2=G\cos\alpha-g\frac{R^2}{r^2}$$ $$r\frac{d^2\phi}{dt^2}+2\frac{dr}{dt}\frac{d\phi}{dt}=G\sin\alpha$$ The two ODE's above are given. I have written ...
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49 views

Proof verification+proposition

Given 2 function $F(p,v)$ and $\frac{dF}{dv}=g(p,v)$ Differentiate F(p,v) with respect to v give $F_pf+F_v$ Formula 1 $$\frac{dF}{dv}=F_p\left(\frac{dp}{dv}\right)+F_v=g\\ ...
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16 views

How do I solve this system of PDEs numerically?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} (\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}) f(x,y) = F(f,g,h) \\ (\frac{\partial}{\partial x} - ...
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4answers
29 views

Soft absolute value

I'm looking for a "soft absolute value" function that is numerically stable. What I mean by that is that the function should have $\mp x$ asymptotes at $\mp\infty$ and behave smoothly in $[-1,1]$. ...
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131 views

how mental math is helpful to learn math? is it any scope for research or to improve new vedic math tricks? [closed]

Many peoples said vedic math is not math. its only collection of tricks but i have question that can we improve this tricks? is it any one try to improve that kind of tricks? if yes! what result they ...
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1answer
31 views

Help Obtaining Numerical Approximation of Lambert W Solution

I am studying a particular generating function $$\frac{2e^x}{e^{2x}+1+2x}$$ and I thought I would try to solve the equation $$e^{2x}+1+2x=0$$ to determine for what value of $x$ if any the function ...
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33 views

Why do I get a big error when I compute this integral with Gauss-Legendre Quadrature?

I'm using Gauss-Legendre Quadrature to solve the following integral: $\int_0^{1}x^xdx$ After I've compared the result with the MatLab vpa(int(...)) of the same integral I've noticed that the ...
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1answer
21 views

fitting by linear combination of exponential functions

Suppose that we have a set of points $(x_1,y_1), \ldots (x_n,y_n)$, and we want to fit a function of the form $f(x) = ae^{2x} + be^x + c$ to those points. If we make $z=e^x$, then our function becomes ...
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1answer
46 views

Are my results realistic or is there an error somewhere?

The background is that I'm solving a problem in Numerical Analysis which I asked about here: Is my derivate correctly programmed? Now if I use the new code, then I get a result that is along the ...
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1answer
56 views

Is my derivate correctly programmed?

I'm solving a problem in numerical analysis, page 9 of this text (soundwaves under water) and I think that I'm getting the correct result but I'm not sure if I programmed my derivate correct. My ...