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

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12 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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1answer
19 views

What does $\|u\|_{\mathcal{C}^2(\bar{\Omega})}$ mean?

What might $$\|u\|_{\mathcal{C}^k(\bar{\Omega})}$$ mean? $u$ is a sufficiently often differentiable function $\Omega \rightarrow \mathbb{R}$ and $\Omega \subset \mathbb{R}^n$ a bounded domain. It ...
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1answer
20 views

Uniqueness of function approximation over three points?

Given a function $f(x)$, we want to approximate $f$ using $P(x)$, such that: $P(x_0) = f(x_0)$, $P(x_2) = f(x_2)$, $P'(x_1) = f'(x_1)$. Prove that such a $P$ is unique $\iff$ $x_1 \neq ...
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0answers
19 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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0answers
30 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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23 views

Trying to use the method “Stiff” (Rosenbrock method implementation) from the book “Numerical Recipes in C”.

The program is compilable but I don't think it works correctly. According to the book, we need also method "odeint" for adaptive stepsize adjustment and fully implement Rosenbrock method. I used the ...
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0answers
11 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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1answer
36 views

What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$? I think it might mean that the ...
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1answer
37 views

Efficient algorithm to find the maximum of a sum of $m$ sines

Is there an efficient algorithm to find the maximum of a sum of $m$ sines? That is, find an $x \in \mathbb{R}$ such that $$f(x) = \sum_{k=1}^m \sin(\alpha_kx)$$ is maximized? By efficient, we mean an ...
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3answers
22 views

relationship between Big $O$ notation and limit

If I have a function $f(n)$ such that $f(n) \geq 0$ for all positive integers $n$ and that $\lim\limits_{n\to \infty} f(n) = 0$, then can I conclude that $f(n) = O\left( \dfrac{1}{n^k}\right)$ for ...
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19 views

Let $f:[-1,1]\to \mathbb{R}$ by $f(x)=x^4$. Determine the polynomial $p_2$ of degree less than or equal to 2 such that $||f-p_2||_2$ is minimal

also compute $||f-p_2||_2$. Write $p_2$ with respect to $\{P_0,P_1,P_2\}$ and $\{1,x,x^2\}$ I know its helpful to show what I have so far but I really don't know where to start. I'm looking at ...
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0answers
31 views

Some results on Robin boundary conditions

I have the following boundary problem $$ (P): \left\{\begin{array}{l} y''(t) = p(t)\, y'(t) + q(t)\, y(t) + r(t),\\ y(t_1) = \alpha, \\ y'(t_2)+\gamma \cdot y(t_2) = \beta, \end{array}\right. $$ ...
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1answer
26 views

Approximation formula for third derivative, is my approach right?

Derive by using Taylor approximation up to 4th degree (in $h$) of $f$ in $x_0 \pm h$, $x_0\pm 2h$ at $x_0$, an formula for approximation of $f'''(x_0)$ with an error term of order $h^2$. Could ...
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0answers
22 views

Scale vector in scaled pivoting (numerical methods)

In the scaled pivoting version of Gaussian elimination, you exchange rows/columns not only based on the largest element to be found, but rather the largest relative to the entries in its row. You ...
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19 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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1answer
22 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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21 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
25 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
87 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
17 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
53 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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0answers
46 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
45 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
30 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
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
13 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
10 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
232 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
16 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
13 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
19 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? [closed]

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
51 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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18 views

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

$$w_{i+1} = w_{i} + \frac{h}{2}(3f_{i} − f_{i−1}),$$ $$w_{i+1} = w_{i} + \frac{h}{2}(f_{i+1} + f_{i}), $$ $$w_{i+1} = w_{i} + h f(t_{i} + \frac{h}{2}, w_{i} + \frac{hf_{i}}{2}) $$
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0answers
29 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
21 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
44 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 ...
1
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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 ...