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)

2
votes
3answers
13 views

Fixed Point Iterations - Root Finding

Given any function $f(x)$, how can you come up with the corresponding $g(x)$ such that $g(p)=p$ (where p is the root)? Say, $$f(x)= sinx -\frac{x}{1.4}$$ my professor told me to simply isolate for ...
0
votes
0answers
16 views

Distance from a point to the involute of a circle

I know that the involute of circle of radius $r$ centered at $(0,0)$ is given by the following parametric form: $$\begin{cases} x(\theta) = r \big(\cos(\theta) + \theta\ \sin(\theta) \big),\\ ...
0
votes
2answers
48 views

Showing that an equation has a root in an interval

Show that the equation $x^4 - 7x^3 + 1 = 0$ has a root in the interval $[0,1]$. How would I go about working this out in steps?
0
votes
1answer
21 views

Finding the solution to an equation using trial and improvement.

Using trial and improvement to find this solution to 2 decimal places. The equation $x^3=10-3x$ has a solution such that $1 \le x\le 2$.
2
votes
6answers
101 views

Showing that $\sin(x) + x = 1$ has one, and only one, solution

Problem: Prove that the equation $$\sin(x) + x = 1$$ has one, and only one solution. Additionally, show that this solution exists on the interval $[0, \frac\pi2$]. Then solve the equation for x with ...
1
vote
1answer
18 views

Origin of divergence in a divergent field (2D)

I have a field of measured vectors, see example of four vectors in image below. If there was no noise they would all point outward exactly from one "central point". i.e. there would be a circle whose ...
0
votes
0answers
19 views

Order of convergence of nonlinear iterative solver

I'm given a sequence $x_n \rightarrow \alpha$ of a nonlinear solver such that $$\lim_{n\rightarrow\infty}\frac{x_{n+1}-\alpha}{x_n-\alpha}=c$$ converges linearly (i.e. $c\in(0,1)$). Now, I need to ...
0
votes
0answers
7 views

How to find out transient response of z-transform (discrete)

Given z-transform transfer function $H(z) = \frac{Y(z)}{X(z)}$, with the corresponding linear ODE, how does one find out transient response of such a transfer function given a certain input?
0
votes
0answers
8 views

How to calculate u1-modified parametric iteration method

Please refer to paper "A modified parametric iteration method for solving nonlinear second order BVPs, link: http://www.scielo.br/pdf/cam/v30n3/a02v30n3. In example $5.1$, I am unable to understand ...
0
votes
1answer
19 views

Unsure of 2d finite difference method with second order term?

I have the following equation for the price of Black Scholes Euro option - (1) $$-\frac{\delta C}{\delta t} = -\alpha\frac{\delta^2 C}{\delta x^2} + [r - \delta + \frac{\sigma^2}{2}]\frac{\delta ...
0
votes
0answers
23 views

Region of absolute stability

We have the problem $$\left\{\begin{matrix} y'=\lambda y &, t \in [0,+\infty), \lambda \in \mathbb{C}, Re(\lambda)<0 \\ y(0)=1 & \end{matrix}\right.$$ Applying the Backward Euler method ...
0
votes
0answers
18 views

Is the system stiff or not?

Let the problem of initial values $\left\{\begin{matrix} y_1'(t)=-0.5y_1(t)+0.501 y_2(t), t \in [0,10^3]\\ y_2'(t)=0.501y_1(t)-0.5y_2(t) \\ y_1(0)=1.1 \\ y_2(0)-0.9 \end{matrix}\right. \\ $ The ...
1
vote
0answers
20 views

How to avoid integrating across a singularity in numerical integration?

I'm trying to evaluate expressions of this form: $$f(x)=a(x)\int_0^x\frac{dx'}{a(x')^2}.\tag1$$ Here $a$ is twice differentiable and has some first order zeros, and $f$ is supposed to also appear ...
-2
votes
0answers
18 views

Orthogonal polynomials and aproximation of the least - square sense [on hold]

I was asigned this problem in class which (unfortunatly) I don't know how to solve. I was hoping someone could shed some light on the problem. Thanks :)
0
votes
0answers
19 views

Definite Integration of a multivariable function [on hold]

$$ \int _{-\infty }^{\infty }\!\int _{ -\infty}^{\infty }\!\int _{ -\infty }^{\infty }\!\int _{ -\infty }^{\infty }\! 4.0\, \left( 0.06003683241\,{\frac {{\it k2}\,\sin \left( 10^{-8}\,{ \it k1} ...
-3
votes
0answers
14 views

Faber polynomials in Matlab [on hold]

I want to compute some faber polynomials associated to an ellipse centered at a point \sigma (in the complex plane) in Matlab. Say the ellipse has minor axis a and major axis b. If someone know how to ...
1
vote
0answers
12 views

Reduction of matrix $A$ to $B$ to find eigenvalues by Power method [duplicate]

How to reduce matrix $A$ to $B$ such that it has all eigenvalues and eigenvectors of $A$ but the dominant eigenvalue (eigenvalue with largest magnitude) is replace by $0$ ? I am using Power method to ...
1
vote
1answer
25 views

Why does the midpoint method have error $O(h^2)$ [duplicate]

In solving an ode $$ y'(t) = f(t, y(t)), \quad y(t_0) = y_0 $$ the midpoint method estimates $$y_{n+1} = y_n + hf\left(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n, y_n)\right)$$ But why is the error ...
0
votes
0answers
17 views

How to define a unimodal function?

I am doing MATLAB assignment in which I need to find the minimum of a given 3D function. It is more or less guided, but I'm confused on what one step means: a. First, write a function that will ...
1
vote
1answer
26 views

Integrate Using Gauss Laguerre Quadrature

Can we integrate a function F(x) using Gauss Laguerre Quadrature when it is not of the form: $$\int_0^{+\infty} F(x) dx$$ An example in my numerical methods book seems to imply that it can; however, ...
0
votes
0answers
15 views

Implementing specific SVD algorithms

My goal is to learn to implement the two-sided Jacobi SVD, a method of SVD for bidiagonal matrices, and a method of SVD for tridiagonal matrices. Can anyone recommend a place to learn about these, or ...
1
vote
0answers
17 views

Linearization around an equilibrium point.

I am trying to understand linearization around an equilibrium point. This is new to me. So I would like to 'see' how it works basically and see how important it is to choose a right equilibrium point ...
0
votes
0answers
15 views

contour integration and analytically

I am trying to compute the following contour integration but am quite stuck I have to evaluate it analytically, by extending it to the complex plane and solving an appropriate integral involving a ...
1
vote
1answer
21 views

How do I turn this Crank-Nicolson type equation into three vectors representing the middle, upper, and lower diagonals in a tridiagonal matrix?

I have the following homework problem: I have calculated the Crank-Nicolson equation to be Equation 1 $$ -200.05u_{m-1}^{n+1}+400.9995u_{m}^{n+1}-199.95u_{m+1}^{n+1} = ...
0
votes
0answers
18 views

How to implement QR method for bidiagonal matrices?

My goal is to take the singular value decomposition of a (not necessarily square) matrix. I have a method to do bidiagonalization of a matrix, and I can chop the bottom rows of zeros. In order to find ...
0
votes
1answer
16 views

Backward-Euler implicit integration for multiple variables

I'm a bit confused how the Backward-Euler implicit integration method works for multiple variables (i.e., systems of differential equations). For single variables, we solve the implicit equation ...
1
vote
0answers
14 views

Do quasi-Newton methods check the second-order optimality condition?

I have a practical question about quasi-Newton methods. In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In ...
0
votes
1answer
41 views

N body ODE simulation

I have been staring at the below matlab code, which simulates planetary motion, for a while now and I could really use a fresh set of eyes. I cannot understand why it is giving straight line ...
0
votes
0answers
11 views

Observe approximation order of numerical solution of a Partial Differential Equation

When solving a Partial Differential Equation numerically, I estimated the approximation orders theoretically as follows, $$ u(x,t)= u_{h,k} + C_1 h^{p} +C_2 k^{q}, $$ where $ u_{h,k} $ is a numerical ...
0
votes
0answers
20 views

Estimates of $L^2$-orthogonal projection in $H^1$ and $H^{-1}$-norm

suppose we have a finite element space $M_k$ of $L^2(\Omega)$ and the orthogonal projector $Q_k$, defined by $(Q_k w,v)=(w,v)$ for all $w \in L^2(\Omega)$ and $v\in M_k$. My aim is prove the ...
1
vote
0answers
16 views

Ideas on how to improve stability in solving PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
2
votes
1answer
32 views

Inhomogeneous eigenvalue problem, the shooting method and constraints

In trying to solve a problem occurring in QM calculations I've encountered the following pickle, with which I hope you could help me. I am trying to solve an inhomogeneous eigenvalue differential ...
2
votes
2answers
45 views

How to compute ratios when the numbers are extremely small and numerical issue might arise?

Consider that a set of numbers that can be written in the form $b^{x}$ (for some base): $$A = \{ a_1, ..., a_n\}$$ for example maybe $\{ 2^{-2000}, 2^{-2002}, 2^{-2500}\}$. Also assume we can ...
0
votes
1answer
29 views

Numerical Methods for ODE

I have been working through Iserles books A first course in Numerical Analysis of Ordinary Differential Equations. I am trying to figure out how to prove that the error constant for an s-step BDF is ...
0
votes
0answers
27 views

Local Truncation error

We have error formulas for Gaussian Quadrature and polynomial interpolation with 10 Gaussian points, use them to derive formulas for the local truncation error, where $$P = ...
2
votes
2answers
14 views

convergence of iterative methods for linear system

Here is a theorem about convergence of iterative methods for linear system in Burden and Faires' book "Numerical Analysis" For any $x_0 \in \mathbb{R}^n$, the sequence defined by $x^k = Tx_{k-1} + c$ ...
1
vote
1answer
61 views

weak convergence lim inf sequence example

I have been stumped by the following problem , I was able to answer the first part of the problem which involved the inequality by using weak convergence. But when it came to looking for such an ...
-1
votes
1answer
41 views

Set $T$ is Countably Infinite [closed]

How can it be shown that $$T = \{\,(i, j, k) \mid i, j, k \in\mathbb N\,\} $$ is countably infinite?
0
votes
0answers
12 views

Lobatto quadrature for $3$ and $4$ knots [closed]

Determine Lobatto quadrature formulas with $3$ knots, and with $4$ knots. Where can I find it?
0
votes
2answers
24 views

Finding the roots and the rescaling of an equation

This question is taken from Hinch's book on perturbation. I need to find the rescalings $x=\delta X$ and the roots of the equation $\epsilon^2x^3+x^2+2x+\epsilon=0$ I have found to possible ...
0
votes
2answers
19 views

What is the intuition behind matrix splitting methods (Jacobi, Gauss-Seidel)?

Descent Methods, like Gradient and Conjugate Gradient ones, have a nice geometric interpretation and I really love them. What about Jacobi, Gauss-Seidel or other matrix splitting methods? I can't see ...
0
votes
0answers
52 views

Possible to proof/disproof this statement?

Given 2 different function $E=f(p,v)$ $\frac{dE}{dv}=g(p,v)$. $E=f(p,v)$ $\frac{dE}{dv}=f_p(\frac{dp}{dv})+f_v=g$ $\frac{dp}{dv}=\frac{g-f_p}{f_v}=x(p,v)$ From this formula got the ...
-1
votes
0answers
20 views

Adomian method (how was the solution in this problem obtained?) [closed]

can someone please help explain to me how the y terms in problem 1 of this paper were obtained in detail paper: http://www.ccsenet.org/journal/index.php/jmr/article/view/45923/24853 thanks here is ...
0
votes
1answer
14 views

Spectral radius and convergence of fixed point iteration

Let $F: \mathbb{R}^n \to \mathbb{R}^n$ be a differentiable map. -editted- Let $x^\star$ be a fixed point of $F$. Then, is it true that the fixed point iteration $x_{n+1} = F(x_n)$ converges locally ...
-2
votes
0answers
26 views

Gauss Chebyshev formula [closed]

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral. $$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1. Also compare the result with true value, where the zeros and the ...
1
vote
1answer
73 views

Numerical series

Consider the series below that consist of 2 different formula $P_aV_a^{1.4}=P_bV_b^{1.4} $ and $P_aV_a=P_bV_b$ that keep repeating itself in the whole sequence. By assuming $P_1$ and $V_1$ both=1, ...
3
votes
0answers
52 views
+50

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
1
vote
1answer
33 views

Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented ...
0
votes
0answers
11 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
0
votes
1answer
23 views

Show that S is a cubic spline (natural or clamped)

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...