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

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Name of numerical methods for second-order differential equation

Numerical methods that try to solve first-order differential equations of the form: $$ \frac{\partial}{\partial t} y = f(y,t) $$ are often Runge-Kutta methods, and there is a whole family of ...
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17 views

Ruling span derivation?

I have recently read a paper about the ruling span for electrical wires and they have an approximation that looks like it can be derived with mathematical intuition only. I'd like to find a derivation ...
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16 views

Implicit numerical integration: error bound

Suppose I'm solving this equation numerically with a time step $h$: $$x''(t) = f(x)$$ Discretizing it and using implicit integration: $$x^{n+1} - 2x^n + x^{n-1} = h^2f( x^{n+1})$$ $x^{n-1}$ and ...
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29 views

The equivalent of least squares, but for vectors

Given a set of poins, one can use a fitting method such as least squares to find the straight (or the parabola, or the 3rd grade equivalent) that's closest to all points at the same time (via ...
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Gauss-Legendre quadrature error

I'm trying to evaluate the error in Gauss-Legendre quadrature formulae on $[a,b]$. So far I have that the error is less or equal to $$ \frac{f^{(2n)}(\xi)}{(2n)!}\langle p_n,p_n \rangle, \enspace ...
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11 views

Composite trapezoid rule and trigonometric functions

I am trying to solve the problem talked about in: Trapezoid rule over trigonometric polynomials Show that the composite trapezoid rule over an equidistant partitioning with interval size ...
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39 views

Is there any practical use of this algorithm?

Example The exact solution of a DE $\frac{dp}{dv}=-1.4\frac{p}{v} $ with initial condition $(P_1,V_1)=(1,1)$ can be obtain by solving the integral ...
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22 views

Finding area by integration, increasing inaccuracies with complex functions?

I am looking for an explanation as to why the method of integration to find the area of function using limits provides a greater % difference between other methods (In this example Simpsons) with ...
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19 views

Error in Gauss-Legendre quadrature

I've tried Googleing this, but so far I haven't succeeded. Can someone point to me a webpage or book in which I can find the error estimate (detailed, not just the final formula) of the Gauss-Legendre ...
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8 views

Thresholding in spectra of partial traces of random symmetric matrices

I found an interesting behavior while looking at partial traces of random matrices. This is something I was studying numerically, and I haven't completely ruled out the possibility of numerical ...
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17 views

Gauss elimination: Difference between partial and complete pivoting

I have some trouble with understanding the difference between partial and complete pivoting in Gauss elimination. I've found a few sources which are saying different things about what is allowed in ...
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24 views

When is this iteration quaranteed to converge

I have a nonlinear $N$-component equation of the form $x_n = \sum_m f_n(x_m),$ where $f$ is some function and the goal is to find a set of $x_n$ that satisfies this equation. I have experimented ...
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21 views

Is it possible to construct a green function of the Dirichlet problem from the green function of the Cauchy problem?

For the heat equation. Is there a method to obtain the green function of the Dirichlet problem in a rectangular 2D domain from the green function of the Cauchy problem (infinite domain) PDE's?
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14 views

How should i apply Richardson Extrapolation?

I trying to understand how the Richardson Extrapolation works, and what it is good for. The internet has lots articles about the this, but they all seem to lack in what it is useful for. I wanted ...
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17 views

How to turn the integro-differential equation into an ODE

I want to get the numerical solution of the integro-differential equation by Mathematica but failed. Maybe the first step should be turning that into an ODE, is there some method? {0.01+10 (0.01 ...
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10 views

Nonuniform partition - euler method

Consider a nonuniform partition $a=t_0< t_1< \dots < t_{\nu}=b$ and assume that if $h_n=t^{n+1}-t^n, 0 \leq n \leq N-1 $ is the changeable step, then $\min_{n} h_n > \lambda \max_{n} h_n, ...
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12 views

Absolute stability Euler method

I am looking at the following exercise. We suppose that the explicit Euler method is applied at the differential equation of second order $\left\{\begin{matrix} x''(t)+(\lambda+1)x'(t)+ \lambda ...
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10 views

what are stability analysis of numerical methods

which of the stability in numerical methods (A, A(0), A(alpha), A nut, stiffly stable, L or L nut) is stringent
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how should a numerical solver handled hybrid points either as a single scheme or block method

how should I handle the hybrid point in a hybrid scheme for solving IVP problems when treating the numerical examples (both for direct method and block method)
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9 views

lattice variation, cylindrical discretisation of PDE

Given an energy functional $ E=\int_{0}^{\infty} \,dr.r\left[\frac{1}{2}\left(\frac{d \phi}{dr}\right)^2 - S.\phi\right] $, I am told that discretizing on a lattice $ r_{i}=ih $ (h=lattice size, i is ...
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11 views

Order of error Verlet integration

I have a simulation of moving particles. The integration method I'm using is Velocity Verlet. Wikipedia states that the order of the error of this method is $2$. However, if I calculate the order of ...
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58 views

How do I solve a Second order differential equation that is a variation of the Sine Gordon Equation?

$$0.1 \frac{d^2 \varphi}{d\tau^2}+\frac{d\varphi}{d\tau}+\sin\varphi-1.1=0$$ Im not quite sure how to reduce this equation. The inclusion of $\sin \varphi $ throws me off some what. If it helps $\tau ...
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25 views

Numerically solving the diffusion-reaction equation with boundary values

I want to solve a nonlinear PDE (steady-state diffusion reaction): $\Delta u = f(u)$ That has the following boundary conditions: $u_y(x,0) = 0$ $u(x,h) = m$ I am trying to solve it via newton's ...
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31 views

Approximation of the coefficients of the Fourier Series via the FFT

Is there literature on the approximation of the coefficients of the Fourier Series via the FFT? The approach I'm interested is merely numerical, consisting of computing the integrals with the ...
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22 views

Proof of the Lax-Wendroff theorem

The Lax-Wendroff theorem says that, if a conservative numerical scheme for a hyperbolic system of conservation laws converges, then it converges towards a weak solution. In the book "Numerical ...
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18 views

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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20 views

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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23 views

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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16 views

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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23 views

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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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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Solving modular inequalities efficiently

Is there an efficient algorithm (polynomial in $n$ and $N$? What about subexponential in $n$ and $N$?) to find the set of all solutions of the equations \begin{eqnarray} a_1 < &k_1x& < ...
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colvolution function

I am trying to understand the equaliti denoted in the attache picture. Any help? Thank you!
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35 views

Can any one help me to check this question is right or wrong?

Taylor method in Numerical analysis to solve this question(4th order): Question is $y'=2t^2 +y^2 -1 $ $$ 0 \lt t \lt 2 \qquad y(0)=0 \qquad h=1$$ Solution : I solve this question is it right ...
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29 views

Relation between parallel transport and Jacobi field II

Before I asked a question here: Relation between parallel vector field along a geodesic and Jacobi field along that same geodesic The current question is related, and actually arise from numerical ...
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33 views

Newton backward and forward interpolation (for ODEs) intuition.

For Newton's backward and forward formulas, I understand everything algebraically, but can someone please explain me this formula intuitively, especially intuition how "powers of the forward ...
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44 views

upper bound of an $L^\infty$ function's derivative

Consider a function $u:\mathbb{R} \longrightarrow \mathbb{R}^n$ that is essentially bounded, i.e., $u \in L^\infty$. There is an upper bound of its derivative? I think there is not allways ( i.g. ...
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12 views

Which method of estimating graph can be used here?

I am making an experiment and I need to estimate a graph about the results I get. The problem is, I don't know what results my experiment will give me. For example these are my experiment results (x = ...
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21 views

Derive by Double False Position

The equations for this problem are $3x = y$ and $2(x + 15) = y + 15$. I know the answers from doing algebra. $x = 15$ and $y = 45$, but I'm not sure how to calculate that using the method of double ...
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23 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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14 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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34 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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11 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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18 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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27 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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12 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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53 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 $g = ...
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Is convex or non convex function?$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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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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23 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 ...