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

learn more… | top users | synonyms (2)

0
votes
0answers
26 views

Non Linear Systems : Broyden's Method

I am trying to implement Broyden's method for solving systems of non-linear equations following these documents http://heath.cs.illinois.edu/scicomp/notes/chap05.pdf http://web.mit.edu/jmartin3/...
0
votes
4answers
38 views

Pipe and Cistern

Pipes $A$ and $B$ can fill a tank in $9$ hours and $12$ hours respectively. Both pipes are opened together to fill the tank, but pipe $B$ is closed after some time. If the tank is full in $6$ hours, ...
1
vote
1answer
159 views

Have I found ALL the solutions to this diff eq & boundary conditions?

If we find a solution to a differential equation and its boundary conditions, how can we know if we have found ALL the solutions? For example, let g(x) be a smooth continuous function of x: (Eq 1) $$...
0
votes
0answers
15 views

Ask for reference convergence of implicit euler method for initial value problem with dissipative source term

I am considering the convergence of implicit euler method for solving the following initial value problem: \begin{cases} u'(t)=f(t,u(t)),t\in[0,T]\\ u(0)=u_0\in \mathbb{R}, \end{cases} where $u:[0,T]\...
0
votes
1answer
60 views

Can $\int_0^1 \frac{1}{x} e^{-x} dx$ be integrated?

I have an integral with a singularity at $x = 0$. $$\int_0^1 \frac{1}{x} e^{-x} dx$$ It's not a removable singularity so is it possible to perform the integration? For example could some complex ...
-1
votes
1answer
37 views

using Euler's method to solve this question ($\frac{dv}{dt}=-kA$) [closed]

Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area. dv/dt= -kA where V=volume (mm3), t =time (min), k =the evaporation rate (mm/min), and A =...
0
votes
1answer
20 views

Computing $PAQ = LU$ using Gaussian elimination with complete pivoting

Suppose $PAQ = LU$ is computed via Gaussian elimination with complete pivoting. Show that there is no element in $e_i^{T}U$ i.e., row $i$ of $U$, whose magnitude is larger than $|\mu_{ii}| = |e_i^{T}U ...
1
vote
0answers
54 views

About a geometric algorithm to compute $\sin$ based on the unit circle

In an old post I have found a user which claims to have a geometric algorithm to compute trigonometric  functions for an angle between $0^\circ$ and $90^\circ$ based on the unit circle. Here's the ...
0
votes
0answers
30 views

What does “order” exactly mean in numerical methods?

I am trying to understand the concept of order in solving numerical differential equations of the form $\frac{dx(t)}{dt}=f(t,x(t))$. Let's start from the local discretisation error at $t$: $$L(t,h)=\...
0
votes
1answer
34 views

The Runge - Kutta method and two-body problem

Is it possible to get an approximation of the two body problem: $$\left\{\begin{array}{lll} x''(t)=-\frac{x}{(x^2+y^2)^{3/2}}, & x(0)=1-\varepsilon, &x'(0)=0\\ y''(t)= -\frac{y}{(x^2+y^2)^{3/2}...
1
vote
0answers
37 views

Complex Roots (Numerical Methods)

I was given the following question in my Numerical Method exam and I think it is related to Newton's Basis Polynomial, but couldn't solve it. Could anyone guide me to the solution? Show that for ...
0
votes
1answer
46 views

Numerical integration in Matlab (Gaussian 3 point quadrature)

Write a Matlab function that applies the Gauss three point rule to N sub-intervals of $[a, b].$ The input parameters should be the name of the function being integrated, $a, b,$ and $N$. Attempt: ...
0
votes
0answers
24 views

Truncation Error of 2-stage Runge-Kutta Method

I'm trying to derive the truncation error for the 2-step Runge-Kutta Method given by $$k_1 = f(x_n,t_n)$$ $$k_2 = f \left(x_n+\frac{2\Delta t}{3}k_1,t+\frac{2\Delta t}{3} \right)$$ $$x_{n+1}=x_n + ∆t(\...
0
votes
0answers
41 views

Numerical method for solving equation with $u \frac{\mathrm{d}u}{\mathrm{d}x} + u$

I'm looking for a finite difference method to solve $$a(x) u \frac{\mathrm{d}u}{\mathrm{d}x} + u = b(x)$$ where $u(0) = c$. I tried to do a lagging convergence on the $u$ ie $$a(x) u^{(n)} \frac{\...
0
votes
0answers
16 views

Applying Boundary Condition to Finite Element Matrix

Several times now I have seen the following done without justification and I cannot figure out why it can be done: Consider the 1 dimensional "pde" $-u'' = f, u(0) = a, u(1) = b$ over $[0,1]$. We ...
0
votes
0answers
28 views

Trapezoidal Rule Mathematical Error

I want to find the absolute error for Trapezoidal rule numerical error,so I have this function: $\displaystyle f(x)=\frac{1}{1+x^2}$, the type of error is: $\displaystyle \epsilon \leq \left| \frac{(...
0
votes
0answers
20 views

Example of “no analytical solution”

Is there a good test for no analytical solution? How can I learn the difference between equations that have an analytical solution and the ones that need numerical methods ("unsolvables" in analysis)?
0
votes
0answers
19 views

Gauss Seidel - Finite Element Method

I am solving an equation using finite element method, and for that I have to use Gauss Seidel to invert a matrix. In Gauss Seidel I am using a "while" which breaks if the absolute error reaches the ...
0
votes
2answers
71 views

Approximate solution of a trigonometric equation using only pen and paper

I found an exam question that I managed to solve via calculator but not by using only pen and paper. Is there a solution to this? Prove that there is an $x$ satisfying $10x-9 = 9\sin x-10\cos x$ and $...
0
votes
1answer
16 views

Gauss transforms to factor $A = LU$

Consider a symmetric matrix $A$, i.e., $A = A^{T}$. Consider the use of Gauss transforms to factor $A = LU$ where $L$ is unit lower triangular and $U$ is upper triangular. You may assume that the ...
2
votes
0answers
39 views

Numerical methods for ODE: Implicit, explicit, stability, stiffness

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
2
votes
0answers
30 views

Numerical methods for ODE: Taylor vs. Interpolation approaches

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
0
votes
1answer
52 views

Gauss-Legendre three point rule

Use the change of variables $$x=\frac{a+b}{2}+\frac{b-a}{2}t,$$ to show that $$\int^b_a f(x) \ dx = \frac{b-a}{2} \int^1_{-1} f\left( \frac{a+b}{2} + \frac{b-a}{2}t \right) \ dt.\tag{1}$$Hence obtain ...
0
votes
0answers
25 views

Solving Poisson Equation Finite-difference using Python

Hi I'm trying to compute numerically the solution to the next Poisson equation: $$ \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^{2}u}{\partial y^{2}} = 4 $$ with the boundary conditions $$ u(x,...
0
votes
2answers
49 views

Condition number for each variable

Condition number of a matrix tells us how viable it is to solve $Ax=b$ $$A= \begin{bmatrix} 1.001&1\\ 1&1 \end{bmatrix} $$ Is a matrix that would be difficult to solve numerically. However ...
3
votes
1answer
26 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/(x^...
1
vote
0answers
25 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} \dot{u}=...
1
vote
0answers
24 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
54 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
13 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
27 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
35 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
20 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
73 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
36 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
35 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: $$ D_{m\Delta x}^{1}=\frac{f(x+m\Delta x)-f(x-m\Delta x)}{2m\Delta x}$$ $$ D_{m\...
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
23 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
34 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
27 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
39 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
34 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
27 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)\\ \end{...
0
votes
0answers
17 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
44 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 $f(\tilde{\alpha})$...