Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.
3
votes
1answer
31 views
Solving 3 simultaneous cubic equations
I have three equations of the form:
$$ i_1^3L_1 + i_1K +V_1 + (i_2+i_3+C)Z_n = 0 $$
$$ i_2^3L_2 + i_2K +V_2 + (i_1+i_3+C)Z_n = 0 $$
$$ i_3^3L_3 + i_3K +V_3 + (i_1+i_2+C)Z_n = 0 $$
where $ ...
0
votes
1answer
26 views
Prove that if $A$ is symmetric and has a LU-decomposition then $A=LDU' \Rightarrow U'=L^T$, where $L^T$
Suppose the matriz $A$ has a LU-decomposition, in other words, suppose there exists matrices $L$ and $U$ such that $A=LU$ where $L$ is lower triangular and $U$ is upper triangular.
We can to prove ...
1
vote
0answers
34 views
Optimize fill factor by move objects between areas
I have a optimization problem which is about several small rectangles inside one outer rectangle. We have, let say, three outer rectangles which are in following order (similar to weeks).
Each ...
26
votes
0answers
302 views
+100
Manual proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ is a noninteger
Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than 1, and ...
2
votes
1answer
250 views
Runge-Kutta 4 - solving system of 6 differential equations (BVP)
I'm facing a tricky problem. I need to solve a system of 6 differential equations numerically, but I don't have 6 IVP (initial value problem) conditions, instead I have 6 BVP (boundary valye problem) ...
1
vote
1answer
35 views
Some questions about variations of fixed point method
I'm doing some excercises in Fixed Point Iteration methods with Matlab. I have to find roots for $f(x)=e^x -x -1.9\cos x$ by using $x_{n+1}=g(x_n)$. I know how to choose $g(x)$ such that I can find ...
1
vote
1answer
30 views
What is the order of convergence of Newtons root finding method? And when does it converge?
Given a function $f(x)$, we can approximate $x_r$ where $f(x_r)=0$ , by using Newton's method:
$$x_{n+1}=x_n -\frac{f(x_n)}{f'(x_n)} $$
The method only works when 'you choose an $x_0$ near enough to ...
1
vote
1answer
48 views
How to find the unknown values in this Numerical Integration type?
Given the following type of numerical integration:
$$I(f)=\int_0^1 f(x) \, dx \approx \frac 12 f(x_{0}) +c_1 f(x_1) $$
a) Find the values of: the coefficient $c_1$ and points $x_0$ and $x_1$ so ...
2
votes
0answers
40 views
Solution to second order nonlinear ODE
I need to find and exact solution for the following ODEs $$y''=-3y'+2y+2x+3,\qquad y(0)=2$$ $$y(1)=-4+5\exp\left(-3/2+\left(\sqrt{17}\right)/2\right)$$ and $$y''=2y^3-6y-2x^3;$$ $$1\leq x\leq2;$$ ...
3
votes
1answer
27 views
Evaluating order of convergence
I think this is quite a simple question, I just want to make sure I understood all correctly.
Here's the problem: I have a numerical method, which is in some way dependent on its spacing $h$ (like ...
3
votes
2answers
43 views
Aproximating rational with fraction with “smallest numerator and denumerator possible”
For example $0.795=\frac{159}{200}$. But is there a way to find fraction with smaller numerator and denumerator that will represent number $0.795xyz...$ i.e. it will approximate our given number?
I ...
0
votes
0answers
24 views
Signal approximation using linear combination of functions
How I can approximate the signal $x(t)=0.001\,t^3 \exp(-0.1t)$ in the interval $[0,100]$ using a linear combination of the following functions:
$f_1(t)=A_1$
$f_2(t)=A_2\cos(0.05t)$
...
3
votes
3answers
160 views
Why does Newton's method work?
I find many sites explaining how to use Newton's method, but none explaining why it works. Could someone give me the intuition behind it? Thanks.
0
votes
0answers
36 views
Projection Methods
I found out that in fact two classes of numerical algorithms are called "Projection Methods". Projection methods á la Yousef Saad to solve a linear system Ax=b (f.e. krylov subspace methods) and ...
2
votes
1answer
218 views
How to determine the N-smallest eigenvalues of a symmetric matrix using the Power Method?
I was assigned to make a program that finds the largest, the N-largest, the smallest and the N-smallest eigenvalues of a symmetric matrix, using the Power Method. So far, I've been able to succesfully ...
1
vote
1answer
53 views
How to compute the second derivatives?
Motivation:
In isogeometric analysis, state variables(e.g. displacement) are defined in the parametric domain, which can be mapped to the physical domain by $\boldsymbol{\xi}\mapsto \boldsymbol{x}$ ...
0
votes
0answers
122 views
Universal Approximation Theorem — Neural Networks
Universal approximation theorem states that "the standard multilayer feed-forward network with a single hidden layer, which contains finite number of hidden neurons, is a universal approximator among ...
0
votes
1answer
47 views
finite difference equations
i havent had a response to this question in a while, could someone please help me. Im struggling to understand the concepts of forward/backward/central differences on finite difference equations.
i ...
2
votes
2answers
99 views
Numerical Methods for Linear Matrix Equation
How can I solve (numerically) the linear equation $AB=0$. where $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$?
How much is the computational cost?
0
votes
1answer
75 views
Solve: This System of equations for $X$ (does a real solution, exist?)
How can I solve $AX + diag(X)[I-c]=0$ for $X$?
All matrices have real entries, $diag(X)$ is a diagonal matrix with the diagonal entries being the diagonal entries of $X$, and $c$ is a constant, real ...
1
vote
2answers
614 views
Proving the inverse (if any) of a lower triangular matrix is lower triangular
The inverse of a non-singular lower triangular matrix is lower triangular.
Construct a proof of this fact as follows. Suppose that $L$ is a non-singular lower triangular matrix. If $b \in ...
0
votes
0answers
17 views
Can I detect repeated eigenvalue by inverse iteration?
Suppose all eigenvalues of $A$ are nonnegative. By using inverse iteration $A-\mu I$ for many values of $\mu\ge 0$, I can find eigenvalues of $A$. If $A$ is a $n\times n$ matrix and have different $n$ ...
-2
votes
0answers
64 views
Bisection, Newton’s and Secant Methods for solving $\sin(x) = \cos(2x^2)$ [closed]
Use the Bisection, Newton’s and Secant Methods to find the solution
(to at least 8 significant figures ) of the equation
$$\sin(x) = \cos(2x^2)$$ For Newton’s ...
3
votes
6answers
484 views
Matrix-Square Root
I was wondering about matrix square roots. What is the procedure to evaluate $(X^{T}X)^{-1/2}$? Is it by a spectral decomposition of $(X^{T}X)^{-1}$ as $U\lambda U^{T}$ followed by the square root $S$ ...
3
votes
0answers
26 views
Timestepping PDE with positive eigenvalues
I'm trying to numerically solve a PDE, namely:
$$
\partial_t \binom{u(x, t)}{v(x, t)}
= x \left(\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right) \cdot
\partial_x \binom{u(x, t)}{v(x, t)}
...
3
votes
2answers
46 views
Fixed point iteration problem of $f(u)=u^3-u-1$
I was thinking about the following problem:
Let $f(u)=u^3-u-1$. Then I have to verify whether the following statements are true/false?
1.Starting with the initial guess $u^{(0)}=1.5,$ the ...
0
votes
0answers
15 views
mathematical equations that represent the equilibrium model for CO2 absorption.
I want to mathematically model a carbon dioxide absorption column using the equilibrium model (not rate based or dynamic) but I am having trouble finding the same number of numerical equations as I ...
2
votes
1answer
428 views
Rewrite matrix equation for Euler method and Improved Euler method
Consider a system of the form:
(1) $x' = Ax + g$
For appropriate matrices $x'$, $A$, $x$, and $g$.
If we let $y_n$ be the approximation to the solution of (1) at time step $t_n$, what matrix ...
0
votes
0answers
16 views
Finite-Element Method: Question on stability
I am trying to determine the stability of the PDE
http://mathurl.com/cazterh
given the finite-element scheme
http://mathurl.com/cetadmr
and constant s
http://mathurl.com/bcfq5us
My problem is ...
0
votes
0answers
17 views
Shifted inverse power method in Octave.
EDIT: Ok, I've managed it. It was very stupid bug... I must write $p=L\(P*z0)$ etc....
I'm trying to write a function which returning vector $a$ (vector of eigenvalues of matrix $A=A^T \in ...
0
votes
1answer
25 views
Euler's method for second order differential equation
Not really homework but sample exam.
The question is to use Euler's Method to approximate Y:
$Y''(t) = Y'(t) - 2Y(t)$
$Y'(0) = Y(0) = 1$
with $t_0 = 0$ and $h=0.2$
So what I did:
First ...
4
votes
0answers
212 views
Simulating from a Multivariate Gaussian without Cholesky
I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
2
votes
2answers
34 views
$g \colon [0,1] \to [0,1]$ be a continuous map and consider the iteration $x_{n+1}=g(x_n)$.
I came across the following problem:
Let $g \colon [0,1] \to [0,1]$ be a continuous map and consider the iteration $x_{n+1}=g(x_n)$.Then Which of the following maps will yield a fixed point for ...
0
votes
0answers
28 views
X : Y :: P : Q to find Q. whats the name of this method?
This is a very simple relationship we often used but I don't remember the name of the method.
Thanks.
0
votes
0answers
9 views
Reference request: Finite difference methods on curvilinear (body fitted) grids
I was wondering if someone may be aware of some form of detailed summary (book, tutorial paper) about the use of finite difference methods on curvilinear (body fitted) grids.
I was only able to ...
0
votes
0answers
21 views
Multiplication and Division of functions
Suppose that you have two continuous functions, $f(x)$ and $g(x)$.
Suppose that you have numerical approximations for these functions, stored a vectors, $f^*$ and $g^*$.
If I want to approximate ...
0
votes
1answer
18 views
Better than Runge-Kutta-Fehlberg 4(5) at high order?
I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using ...
2
votes
1answer
38 views
How to prove Chebyshev–Gauss quadrature integrate polynomial of degree less than $2n-1$ exactly
What I want to ask is mentioned in the title.
For example: how can we show that ...
2
votes
2answers
31 views
Quadrature formula
How can we find a quadrature formula $\int_{-1}^1 f(x) dx=c \displaystyle \sum_{i=0}^{2}f(x_i)$ that is exact for all quadratic polynomials?
Thanks for help.
1
vote
0answers
21 views
Gaussian quadrature with arbitrary weight function
In class, our professor told us how to evaluate the integral $\int_a^bw(x)f(x) dx$ by finding the Gaussian nodes $x_i$ and weight $w_i$ with weight function $w(x)=1$ (also known as Legendre ...
0
votes
0answers
21 views
Von Newman stability analysis for 2D acoustic wave equation explicit
Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} ...
3
votes
2answers
44 views
How to verify the order of DOPRI Runge-Kutta method
I've written code in Fortran based on the RK8(7)-13 method by Dormand and Prince to solve the system $\mathbf{y}'=\mathbf{f}(t,\mathbf{y})$. The method is ...
1
vote
0answers
48 views
Divergence of Gradient Method
Is there any example of a continuous differentiable function out there, in which the gradient method with Armijo's stepsize-rule doesn't converge?
I found it pretty hard to create one myself because ...
0
votes
0answers
19 views
Are high dimensional cubic interpolation and cubic spline the same?
Hi I want to implement a 2d bicubic interpolation method. I checked the official matlab implementation code of imresize and interp2, and surprisingly found that bicubic interpolation method is ...
0
votes
0answers
24 views
stability of numerical method
For the one dimensional transport equation :$\frac{\partial{u}}{\partial{t}}+c\frac{\partial{u}}{\partial{x}}=0$, where $c$ is a positive number, consider the following two numerical schemes:
(A) ...
1
vote
0answers
26 views
Howuse this $R_{l}=\frac{1}{n}\left(\frac{(-1)^l}{2n}+\sum\limits_{m=1}^{n-1}\frac{1}{m}\cos{\frac{ml\pi}{n}}\right)$ and MATLAB get this four fig?
we consider Tikhonov's regularization method for $\delta =0.1, 0.01,0.001,$ and $\delta =0$
The Tikhonov's regularization method you can see:http://en.wikipedia.org/wiki/Tikhonov_regularization
and ...
4
votes
1answer
40 views
About parallel time computation
I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
4
votes
1answer
60 views
Need little hint to prove a theorem from a paper
I have an iterative method \begin{eqnarray}
X_{k+1}=(1+\beta)X_k-\beta X_k A X_k~~~~~~~~~~~~~~~~~ k = 0,1,\ldots
\end{eqnarray}
with initial approximation $X_0 = \beta A^*$ ($\beta$ is scalar ...
1
vote
1answer
63 views
Minimizing the norm related with iteration method
I am working on a iteration method to compute the generalized inverse of a matrix $A$ of rank $r$
Iteration method is
$X_{k+1} = X_{k} + \beta X_{k} (I - A X_{k}) $
where notations are as follows
...
2
votes
1answer
55 views
Convergence rate of Newton's method
Let $f(x)$ be a polynomial in one variable $x$ and let $\alpha$ be its $\delta$-multiple root
($\delta\ge2$).
Show that in the Newton's $x_{k+1}=x_k-f(x_k)/f'(x_k)$, the rate of convergence to ...
