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

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4
votes
2answers
186 views

Refraction equation, quartic equation

Given two points $P$ and $Q$, a line ($A$, $B$ - orthogonal projection of $P$, $Q$ onto the line) and a coefficient $n$, I want to find out such point $C$ that $\frac{\sin{a}}{\sin{b}}=n$ (in fact, ...
1
vote
1answer
216 views

ODE15s matlab function problem

I have been trying to use the ODE15s built-in function of Matlab to solve the following system of equations: $\frac{dy_{1}}{dt}=f_{1}\left(y_{1},y_{2}\right)$ ...
1
vote
4answers
905 views

How to solve $Ax = \lambda x + b$ efficiently?

Let $A$ be a real symmetric invertible matrix and $b$ a real non-zero vector. Consider the problem of finding a real number non-zero $\lambda$ and a real valued vector $x$ such that $$Ax=\lambda x ...
0
votes
0answers
60 views

Percentage variation dependance of a function of two variables

The language is a sort of barrier in this case (even in my native language) so I'll try to make an example here to clarify the question. Given a function $f(a,b)$ I want to answer the question: to ...
1
vote
1answer
174 views

Globally Convergent Methods for Nonlinear Systems of Equations

We recently switched from the basic Newton-Raphson to a more advanced globally convergent Newton’s method with Line Searches and Backtracking (see Numerical Recipes, Chapter 9.7). For some special ...
1
vote
2answers
207 views

Numerical integration given a derivative of a function of two dependent variables

I want to solve the following equation of an integral valued function: $Q = \int_{0}^{x_p}f(t_p,x)dx$ for some particular $x_p$ at a fixed time $t_p$, given some known $Q$ and an initial $f(0,x)$. ...
1
vote
1answer
96 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 ...
0
votes
1answer
73 views

Weighted quadrature formula

I need to calculate a quadrature rule with maximum degree of accuracy that looks like this: $$ \int_0^\infty e^{-x}f(x)dx = A_1f(x_1) + A_2f(x_2) + R(f) $$ where $f(x) = cox(x)$, presumably. ...
1
vote
2answers
233 views

How do I numerically calculate a function from its gradient?

I know the gradient of a function t on a cartesian grid: $\vec{g}(xi,yj,zk)=\nabla t(xi,yj,zk)$. I know t for the center pillar: $\ t(xc,yc,zk)$. For each node in the cartesian grid I want to ...
6
votes
2answers
804 views

How do you determine the closest whole number ratio for a given real number?

I've been messing around with formulas for musical notes (trying not to read anything that would help me unless I get stuck) and currently I'm at a point where I'm trying to get a function that ...
5
votes
1answer
214 views

how to solve $aX+bX^2=e^{cX}$

I build a model for our problem, but i cannot get a result from my model. Could anyone give me some idea to solve this formula: $aX+bX^2=e^{cX}$ Thx in advance!
2
votes
0answers
132 views

Numerical Integration

For $r=1$, how to calculate the following integral numerically. $$\frac{8}{\sqrt{3}r^2}\int_{x=0}^{\frac{r}{2}}\int_{y=0}^{\sqrt{3}(\frac{r}{2}-x)}\prod_{i=0}^2\left(1-\frac{2}{\pi} \cos^{-1} ...
3
votes
1answer
154 views

Cutting of the Delaunay triangulation

I am working on terrain rendering tool currently. I have to cut a piece from a given Delaunay triangulation. Suppose following triangulation is given: The red square depicts area to cut from the ...
23
votes
5answers
3k views

How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
0
votes
1answer
1k views

Need to understand question about not-a-knot spline

I am having some trouble understanding what the question below is asking. What does the given polynomial $P(x)$ have to do with deriving the not-a-knot spline interpolant for $S(x)$? Also, since ...
0
votes
0answers
31 views

How to test emptiness of constrained sub-set.

Let $X$ be subset of $\mathbb{R}^n$ which is $n$-dimensional space. This subset is defined by k inequalities: $g_{i}(x)<0$, $x\in X$, $i=1..k$ and m equalities: $h_j(x)=0$, $x\in X$, $j=1..m$. ...
0
votes
2answers
4k views

Solving coupled 2nd order ODEs with Runge-Kutta 4

I'm having a hard time figuring out how coupled 2nd order ODEs should be solved with the RK4 method. This is the system I'm given: $x'' = f(t, x, y, x', y')$ $y'' = g(t, x, y, x', y')$ I'll use the ...
1
vote
0answers
235 views

Semi implicit integration - stability issues

I am trying to decide whether to use semi-implicit integration vs. explicit integration (particularly Position Verlet over Semi implicit Euler). Although the Verlet approach is widely used and is ...
0
votes
1answer
1k views

Runge-Kutta 4 for systems of equations

This question is part of an assignment in numerical methods class. I am supposed to find the position and velocity of a spaceship flying around the Earth and Moon. I am given initial values of the ...
0
votes
0answers
139 views

How can I show that a given system of differential equations is stiff?

I was writing a program to solve a system of equations with Runge-Kutta method, but it doesn't work well and I decided to check what Mathematica can say about it. It turns out that I have a stiff ...
1
vote
2answers
613 views

Help with Chebyshev Economization of $\exp(x)$?

This may be a stupid question, so I apologize in advance if it is. This is a very common example of Chebyshev Economization, but I still do not understand how the coefficients are found. I want to ...
1
vote
1answer
63 views

Proof that infinite functions can fit a table of numerical values

Suppose while conducting experiments, I measure a finite number of variables with some constants like temperature, etc. We get a table of finite number measurements (numerical values to some decimal ...
2
votes
3answers
154 views

Solve equations using the $\max$ function

How do you solve equations that involve the $\max$ function? For example: $$\max(8-x, 0) + \max(272-x, 0) + \max(-100-x, 0) = 180$$ In this case, I can work out in my head that $x = 92.$ But what is ...
2
votes
1answer
734 views

Is it possible to determine if this matrix is ill-conditioned?

I want to better understand ill-conditioning for matrices. Say we're given any matrix $A$, where some elements are $10^6$ in magnitude and some are $10^{-7}$ in magnitude. Does this guarantee that ...
2
votes
1answer
123 views

What representation should I choose for numerical computation of hypergeometric function ${}_2 F_1(1+i\eta, 2; 2+i\eta; x)$ where $|x|=1$

I have a task - to plot graphics of the function: $$ I(E) = \frac{16i \pi k \mu}{(\beta - ik)^{4}} \frac{1}{1 + i\eta} {}_2 F_1(1+i\eta, 2; 2 + i \eta; x) $$ where $$ x = \left( \frac{\beta + ...
4
votes
2answers
694 views

Integral of $x^2\ln(x)$ using Simpson's rule

This is my homework question: Calculate $\int_{0}^{1}x^2\ln(x) dx$ using Simpson's formula. Maximum error should be $1/2\times10^{-4}$ For solving the problem, I need to calculate fourth derivative ...
2
votes
0answers
85 views

Need little hint to prove a theorem .

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
votes
1answer
158 views

Quasi-modified equation for harmonic oscillator

Could you help me with this question please? Find quasi-modified equation of 2nd order for solution of harmonic oscillator equation with semi-explicit Euler(also called symplectic Euler) scheme.
4
votes
1answer
68 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
902 views

Error analysis - Bisection algorithm

I have a brief question related to an example in my textbook. In my book, the following theorem on Bisection Method is presented: If $[a_0,b_0], [a_1,b_1],. . .,[a_n,b_n]. . .$ denote the intervals ...
1
vote
1answer
172 views

Solving a system of differential equation including non-linearity

I am trying to solve three equations which are of the form shown below numerically: $\frac{\partial v}{\partial t}=f(v,y)$ $\frac{\partial}{\partial t}\left(vy\right)=f(v,y,z)$ ...
0
votes
1answer
259 views

Solving a system of equations that vary in time and space implicitly

I am writing a code for a system of equations for which the variables vary in time and space. I have written an implicit code to solve a system of equations before but in that case I could write the ...
2
votes
0answers
209 views

calculating the amplitude of a cosine function

I want to be able to be able to get the amplitude of the following function: $$||A||\cos(2 \omega t + a)+||B||\cos(3 \omega t +b)+||C||\cos(5 \omega t +c)$$ I am trying to find a way to get the ...
2
votes
1answer
335 views

Two-Point boundary value problem

To solve ${d^2y \over dx^2} =f(x)$, $0<x<1$ with $y(0)=\alpha, y(1) = \beta$. We can get a finite difference approximation by taking $$\frac{y_{j+1}-2y_j+y_{j-1}}{h^2} =f_j \\\Rightarrow ...
1
vote
0answers
39 views

Finite difference method for BVP (2nd order) [duplicate]

Possible Duplicate: Two-Point boundary value problem Given $-{d^2u \over dx^2} =0$ with $u(0)=\alpha, u(1) = \beta$. We can get a finite difference approximation by taking $U^0 =\alpha$ and ...
2
votes
1answer
253 views

Fourier integral/ Fourier transformation of an oscillatory function with FFT

$f(x) = \cos(x^2)$ and $g(k) = \sqrt\pi \cos((\pi k)^2 - \pi/4)$ are a Fourier pair. I want to reproduce $g(k)$ by Fourier integrating $f(x)$ using FFT, i.e. approximating ...
1
vote
1answer
77 views

1st order ODE problem (forward euler)

Let ${du\over dt} = u, \ u(0)=1$. Let the step size be $\tau = 1/3$. Let $0=t_0<t_1=\frac{1}{3}<t_2 =\frac{2}{3}<t_3=1$. Given $U^0=u(0) =1$, we find $U^{n+1}$ $U^{n+1}=U^n+\tau f(t,U^N) \\ ...
2
votes
1answer
1k views

2nd order ODE to 1st order ODE/Forward euler method

I have a 2nd ODE: ${d^2u \over dt^2} =5tu+\sin ({du\over dt})$, $u(0)=1$, ${du\over dt}(0)=0$. Iwas reading my notes and it asked to write the 2nd order ODE as a system of 1st order ODEs. And then to ...
6
votes
1answer
95 views

Formula for straight part of a slightly bumpy line

Given a straight line that deviates from the horizontal by at most 15 degrees. On this straight line there are bumps on top at random places on the line. The combined width of the bumps is at most ...
1
vote
1answer
117 views

Sampling & Value Prediction and Error Correction?

I am a programmer and I don't have much background in mathematics. I know this question might look much more clear to you if I could articulate it in mathematical terminologies. The problem is this is ...
0
votes
0answers
178 views

How a direct method can be compared with an iterative method?

How a direct method can be compared with an iterative method? I have an iterative method to compute Moore- penrose generalized inverse. There are some direct methods available to compute Moore-Penrose ...
1
vote
1answer
317 views

Linear interpolation for finding root of $f(x)$

When using linear interpolation, with similar triangles, to find the root of a function you narrow down the interval the root is in. If $f(1) < 0$ and $f(2) > 0$ then the root is in $[1, 2]$ ...
1
vote
3answers
723 views

Interval bisection to find a root of f(x)

I'm attempting to understand Interval bisection. I'm given a simple question in my textbook, and I can do the process easily, I just don't know when to stop. The question is "Use Interval bisection to ...
1
vote
0answers
56 views

Gibbs Sampling versus General Cases

Suppose we are given a prior distribution about an unknown parameter $\pi(\theta)$. Also we are given $f(x_{1}, \dots, x_n|\theta)$. We want to find $\pi(\theta|x_1, \dots, x_n)$. Now ...
4
votes
4answers
726 views

Calculation of Bessel Functions

I want to calculate the Bessel function, given by $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m}$$ I know there are some tables that ...
2
votes
1answer
68 views

transport equation FD methods

Please help me to understand the following: if I have a transport equation $u_t+au_x=0$ and I want to solve it using finite differences I can see a lot of info on the explicit with central differences ...
1
vote
1answer
105 views

Algorithms for finding closed form approximations for integrals (with no closed form solutions)

It is well known that many integrals have no closed form solutions, normally what you would do is solve them numerically. My question is if there are algorithms that give you good closed form ...
2
votes
3answers
822 views

what is “kink”?

Pleas tell me that what a "Kink" is and what this sentence means: Distance functions have a kink at the interface where $d = 0$ is a local minimum.
7
votes
3answers
683 views

What's the intuition behind “stiff systems”?

One of the mysteries that my numerics class left me with are "stiff systems". Our prof didn't really manage to explain what they are. Now that the class (and the test) are over, I'm still curious. ...
1
vote
2answers
145 views

Linear equation Numerical analysis consistency

\begin{align*} x - \alpha y &= 1\\ \alpha x - y &= 1 \end{align*} For which values of alpha does the system have an infinite number of solutions, no solutions and one solution. ...