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

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3
votes
1answer
25 views

Explicit Finite Difference Scheme For Approxating a p.d.e

$\frac{du}{dt} = \frac{d}{dx}[\frac{1}{x^2+1}\frac{du}{dx}]$ I am trying to approximate this pde with a finite difference scheme but I am confused with the d/dx. Do I just take the derivative of ...
1
vote
0answers
23 views

FitzHugh–Nagumo system with diffusion

I was studying the FitzHugh-Nagumo model with diffusion and I quite do not understand the meaning of it. If we consider the system without diffusion, \begin{equation}\label{FHN}\begin{cases} ...
1
vote
0answers
20 views

Finite difference method and division by zero problem with no flux boundary condition

I am trying to implement an angionesis model described by Anderson and Chaplin in 1998. The model is based on a set of PDEs defined on an unit square with the following no-flux boundary condition ...
1
vote
1answer
49 views

Error for Trapezoidal Rule in multi-variable integrals

For one dimension integrals $\int_{a}^{b}f(x)dx $, we know the global truncation error goes like$\ \approx\mathcal{O}(h^2)$ where $h=\frac{b-a}{N}$ and N is the number of intervals. Also knowing how ...
0
votes
0answers
11 views

Truncation Error of Adams-Bashforth 3 step Method

I'm attempting to derive the truncation error for the 3 step Adams-Bashforth method. I know that to derive the truncation error for the 2 step Adams-Bashforth method we proceed as follows. Suppose ...
0
votes
0answers
24 views

polynomials/numerical analysis

Suppose that $n ≥ 1$. The function $f$ and its derivatives of order up to and including $2n + 1$ are continuous on $[a, b]$. The points $x_i, i = 0, 1, \ldots , n$ are distinct and lie in $[a, b]$. ...
0
votes
0answers
26 views

Matlab ODE solving

So I have an ODE that needs to be solved a few thousand times on MATLAB and am wondering what the most efficient method to use would be. I am changing a constant term each time. My ODE is of the form ...
0
votes
0answers
29 views

Find the steady-states of the system of differential equations using sympy (in python) and determine their local stabilities.

The system is given by: $\frac{dx}{dt} = r x(1 - x) - \beta x y$, $\frac{dy}{dt} = \beta x y - \gamma y$. Analytically, I have found the Jacobian is given by: $J(x,y) = \begin{bmatrix} r(1 - ...
1
vote
0answers
19 views

Find formula with Richardson Extrapolation based on centered difference formula

I'm preparing for my exams next week, and I'm making exercises as a preparation. Now, I'm asked to derive the following formula using Richardson Extrapolation based on the centered difference formula: ...
2
votes
2answers
69 views

Finite Difference Approximation of Derivative [closed]

I want to build a finite-difference approximation of this derivative: $\frac{\partial^2T }{\partial x^2}$ There are given an error of approximation: $O(\Delta x^{4})$ and nodal values of function:$ ...
0
votes
0answers
34 views

Chebyshev polynomial derivatives proof

I've been going through this pdf on Chebyshev polynomials and am stuck on section 1.3 (page 4) which sets out to prove the expression for the derivatives by induction (a fact they state should be ...
1
vote
1answer
26 views

Of significant figures and truthworthy computation

I have a question I picked on the internet, but I am not sure about the term truth-worthiness part of the question. Find the product of 346.1 and 865.2. State how many figures of the result are ...
0
votes
1answer
30 views

Richardson extrapolation for second derivative

So we know what the Richardson Extrapolation for a first derivative looks like using a recursive formula like this: $$ D_{m\Delta x}^{1}=\frac{f(x+m\Delta x)-f(x-m\Delta x)}{2m\Delta x}$$ $$ ...
1
vote
2answers
58 views

Taylor of second order for System of Differential Equations

I need to solve the next system \begin{eqnarray} x' &=& y+x(x^2 + y^2) \\ y' &=& -x + y(x^2 + y^2) \end{eqnarray} with $x(0) = 4$, $y(0) = 0$ I don't know how to start so I know use ...
1
vote
2answers
20 views

Laguerre's Method

Given that, polynomial $P(z) = \sum\limits_{i=0}^{n} a_i z^i$ where $a_i$ are the real coefficients and $P(z_0) = 0$. With the help of Laguerre's Method we find the rest of the complex solutions ...
1
vote
1answer
25 views

Numerical solution of heat equation on periodic domain

Consider the steady heat equation $\nabla\cdot(k(x) \nabla u)=f$ in two dimensions on a periodic domain, say $[0,1]\times[0,1]$. My goal is to solve it numerically with standard central 5-points ...
1
vote
1answer
26 views

Gaussin Elimination preserves S.P.D.

Let $A \in \mathbb{R}^{n \times n} $ be symmetric positive definite with positive diagonal entries. I'm trying to show that at each step $m$ of gaussian elimination $$ a^{(m+1)}_{ij} = a^{(m)}_{i,j} ...
1
vote
1answer
37 views

Iterative trapezoidal method for differential equations

I am studying numerical methods for differential equations. I came accros the trapezoidal method in two forms, an explicit and an iterative one. I would like to know the advantages and disadvantages ...
0
votes
1answer
28 views

Laguerre polynomials and Gram Schmidt

Last two days I was trying to solve the following problem But I couldn't. It's a problem (#5.2.2) from Mathematical Methods for Physicists by George B. Arfken and Hans J. Weber, 7th Edition. I tried ...
0
votes
1answer
17 views

Discretisation of Euler's method

I don't quite understand the discretisation error of Euler's method $$L(t,h)=\frac{x(t+h)-x(t)}{h}-f(t,x(t))$$ What I don't understand is: $\frac{x(t+h)-x(t)}{h}$ is the "gradient" and $f(t,x(t)$ is ...
1
vote
1answer
24 views

Rounding error of trapezoidal method

I'm working with the Modified Euler method sometimes called Heun's method or explicit trapezoidal method. I have a book on ordinary differential equations numerical analysis that claims: The ...
4
votes
0answers
101 views

Large system of nonlinear equations

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ ...
0
votes
0answers
15 views

Dynamic Programming problem with extremely high dimension over 1000

I am dealing with a dynamic programming problem with extremely high dimension. Currently, I know some methods like Smolyak algorithm and Adaptive sparse grid method. They can solve problem with ...
1
vote
3answers
42 views

Find a matrix $A \in \mathbb{R}^{2 \times 2}$ such that $ \|Ax\|_{2}=\|x\|_{2}$ for every $ x\in \mathbb{R}^2 $

How to find a matrix $A \in \mathbb{R}^{2\times 2}$, $A\neq I_{2}$ such that for every $ x\in \mathbb{R}^2$ we have $\|Ax\|_{2}=\|x\|_{2}$. Is that even possible?
-1
votes
1answer
38 views

Solving a numeric statistics problem - R [closed]

I am quite stuck with solving some complicated numerical equation I would like to solve the following equation: $(1-k)\tilde{\alpha}+kf(\tilde{\alpha})=C$ where $0<k<1$ and ...
0
votes
0answers
23 views

Truncation error on finite differences method

Suppose that we have the following FTCS method to solve the heat equation, $$\frac{u_{n}^{k+1}-u_{n}^{k}}{\Delta t}=D\frac{u_{n+1}^{k}-2u_{n}^{k}+u_{n-1}^{k}}{(\Delta x)^2}$$ I am asked the following: ...
0
votes
0answers
18 views

Find the smallest maximized absolute error in polynomial interpolation

Given $$ f(x)=\begin{cases}1&,0\le x \le1 \\2x&,1<x\le2 \end{cases} $$ I found that the interpolating polynomial $p \in \mathbb{P}_{2} $ at $x_{0}=0,x_{1}=1,x_{2}=2$ ...
1
vote
2answers
21 views

Differential equations with Euler's method

A differential equation y' + 2y = 2 - e^(-4*t) With starting point y(0) = 1 and increment ...
2
votes
1answer
50 views

Prove uniqueness of linear spline

Let $f: \mathbb{R} \to \mathbb{R}$ be linear spline (continuous, piecewise polynomial of degree $\le 1 $) at knots $x_0 < x_1 < … < x_n$. Prove $f$ can be uniquely represented at form: ...
0
votes
1answer
32 views

Making these Term Equal / In Terms Of

Here is my question: 30x + 15 19x + 95 Ok, in this question, x is infinite but my main question is what would the second expression be if x's value was in terms ...
0
votes
0answers
14 views

Estimate an Taylor approximation II

i am doing some exercise for my numerical analysis course. And i found myself wondering if the following argument is legal. The context of this exercise is the smoothend newton algorithm, especially ...
-1
votes
0answers
16 views

Condition number plane fit

I fit a 3D plane to 3D points. I setup the corresponding linear system $Ax=0$ by removing the mean of all the points and stacking them as rows into $A$, and solve for a non-trivial ($x\neq0$) using ...
0
votes
1answer
12 views

find estimation of interpolation error for non differential function

Given $f(x)=|x|^{1/2}$ , $-1\le x\le 1$ , I have found the interpolating polynomial $ p(x)=x^2$ for $x_{0}=-1,x_{1}=0,x_{2}=1$. How to estimate $$\max_{-1\le x\le 1}|f(x)-p(x)|$$ now that $f$ is not ...
0
votes
1answer
22 views

interpolation error using higher derivatives

Given $x_{0},x_{1},x_{2}\in[a,b] $ each one different from the others,$f \in C^{4}[a,b]$ and $p\in\mathbb{P}_{3}$ so that $$p(x_{i})=f(x_{i}), i=0,1,2 $$ and $$p'(x_{1})=f'(x_{1})$$ prove that: ...
-1
votes
1answer
57 views

How to find the numerical error when we don't know the exact solution? [closed]

When some quantity $x$ (e.g., the values of a solution of a PDE, using a finite difference method) is calculated numerically, we get its approximate value $x^*$. The error is $|x-x^*|$. But since we ...
0
votes
1answer
22 views

Is $\frac{\rm d}{\rm d\omega(t)}\int_{t_0}^t\omega(t')\rm dt'=\int_{t_0}^t\frac{\rm d \omega(t')}{\rm d\omega(t')}dt'=t_0-t?$

I'm trying to calculate an error propagation, but the expression in the most LHS of the equation in the title crops up. Are you allowed to simply exchange the order so that the operations cancel? ...
-1
votes
1answer
39 views

How do you derive the backward differentiation formula of 3rd order using interpolating polynomials?

It was my exam question, and I could not answer it. How do you drive the backward differentiation formula of 3rd order (BDF3) using interpolating polynomials? I only knew how to derive it using the ...
2
votes
0answers
29 views

Newton-Raphson method on manifolds

Has anyone explored the notion of the Newton-Raphson method on manifolds? Or to put it another way, on $\mathbb R^n$, is there a natural coordinate free way of defining an iterate of the ...
0
votes
1answer
25 views

What is the appropriate way to use the Runge-Kutta method to calculate neural network node activities?

I am a cognitive scientist, and am modelling a particular kind of neural network called a "masking field", where the change in activation of a particular node in the masking field is: $${dy_i(t)\over ...
0
votes
2answers
144 views

Quadrature integration: calculating the weights

We've started today the Integration part on out Numerical Method course. Our professor wrote this exercise on the blackboard and we've been told to start thinking about it. It's the next one: ...
0
votes
1answer
12 views

Normalizing vector which causes overflow

Let's say I have a vector of the form $$[a_1e^\frac{r_1}{a_1},\hspace{2mm} a_2e^\frac{r_2}{a_2}],$$ which I can't compute because $\frac{r_i}{a_i}$ is large enough that when I raise $e$ to it, it ...
-1
votes
0answers
24 views

Numerical Analysis I: problem with interpolation

We want to build a table of values of the function $f(x)=\dfrac{x^4-x}{12}$, such that the linear interpolation for $0\leq x\leq a$ has an error inferior to $\delta$. Determine which will be the ...
-1
votes
0answers
5 views

How to program robin matrix in a laplace with mixed boundary condition !?

the problem is : -delta(u)=f in Omega. grad(u).n+u=g in Gamma (Robin boundary condition). where Gamma is the boundary of Omega Omega is a squar domain. my question is how i can programm the term ...
0
votes
0answers
33 views

Calculate number of trials reaching $p_k$ probability for $k$ successes given the $p_t$ probability of each trial success

Basically, I'd like to be able to answer questions in the form of "What is the number of trials needed to have at least $p_k$ probability of at least $k$ successes, given that on each trial the ...
0
votes
1answer
32 views

verification: Newton method convergence

P(x)=x$^2$-x-1 has a root of $\frac{1+\sqrt5}{2}$=1.61803398 My iterations are: x$_0$=2 x$_1$ =$\frac{5}{3}$=1.6666 x$_2$=$\frac{34}{21}$=1.619 x$_3$=$\frac{1597}{987}$=1.618 So it took me 3 ...
0
votes
1answer
22 views

Verification: fixed point formula

I have $2+\sin(x)-x=0$ and I need to formulate so that it converges in $[2,3]$. I have verified that the formula $x=2+\sin(x)$ gets a convergence in this region. Now to verify the assumptions my ...
1
vote
1answer
64 views

Root of function

Can you find root of the equation $$f(k)=1+(1-k^2)\ln(1+\frac{1}{k})?$$ I tried to use Matlab command but it does not give me any result. Can you suggest a method to find root of equation.
0
votes
1answer
42 views

Newton method iteration

I am trying to solve non-linear systems and since I can't download matlab on this device, I was wondering if there is a way I can set it up in excel. I know the formula for ...
0
votes
1answer
33 views

Matrix-free conjugate gradient

In the conjugate gradient method for solving $Ax = b$, to update the search direction $p$ you would need to evaluate the matrix-vector product $Ap$, i.e. making sure that each search direction are ...
0
votes
2answers
47 views

Solving $u'' - 5u = 6$ with finite difference methods.

I have a task: For an equation: $$u'' - 5u = 6, x \in (0, 1)$$ $$u(0) = 0, u'(1) - 3u(1) = 1$$ construct a recurrence relation("scheme" in the original) with second order approximation on a two-point ...