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

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2
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1answer
100 views

Numerically solving 1D Heat

I'm looking at page 10 of http://www4.ncsu.edu/~zhilin/TEACHING/MA402/notes1.pdf What happened to the boxed term (it doesn't seem to appear in the matrix equation). (I'm just implementing this and ...
1
vote
1answer
85 views

Queries regarding Newton's method

I am currently trying to study the Newton's method of optimization through this wiki article http://en.wikipedia.org/wiki/Newton%27s_method_in_optimization. However, I didn't get this concept about ...
3
votes
1answer
349 views

Determining eigenvalues and eigenvectors from a symmetric matrix

Let $A \in \mathbb{R}^{N \times N}$ be symmetric. a) The respective Eigenvalue $\lambda$ to an approximately defined Eigenvector $0 \neq x \mathbb \in {R}^n$ from $A$ has to be calculated, ...
2
votes
1answer
2k views

Solve Falkner Skan Numerically?

how would we go about solving the Falkner Skan equation numerically? The equation is $$f'''+ff''+\beta\left(1-f'^2\right)=0$$ $$f(0) = f'(0) = 0$$ $$f'(\infty) = 1$$ I tried using ODE45 to solve ...
2
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2answers
209 views

Numerical Analysis over Finite Fields

Notwithstanding that it isn't numerical analysis if it's over finite fields, but what topics that are traditionally considered part of numerical analysis still have some substance to them if the reals ...
2
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0answers
132 views

Numerical Integration in Laplace domain

I need to calculate two different integrals containing a Bessel function in the Laplace domain. I have tried different kinds of methods but didn't have any luck. I don’t know how to treat the Laplace ...
2
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0answers
119 views

Solution for this matrix equations (closed form or approximate solution)

Given a system of equations, I'm curious whether I can find the closed form solution for $P$, Here, $G$,$H$ are known $N \times N$ matrix, $\lambda$ is a known scalar; $s$,$t$ are two $N \times 1$ ...
1
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1answer
147 views

Interpolation of vectors with quadratic polynomial

I have following points (-|b-a|,a), (0,b), (|c-b|,c) with a, b and c as two-dimensional vectors. These should be interpolated component-by-component with a second-degree polynomial p. My problem now ...
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0answers
415 views

solving Bessel function equation by hand

I have a Bessel function of the first kind given by the equation $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m+\alpha}$$ I am trying to ...
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0answers
157 views

What methods can be used to numerically solve a system of linear non-differentiable equations?

I'm writing an implementation of Backwards Differentiation Formula, it's a multistep method of solving a system of ODEs. The four-step method looks like this: $$ y_{n+1} - y_n = h f(t_{n+1}, ...
1
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2answers
209 views

Computational efficiency of Machin-like formulae

From what I have read, it appears that the most efficient methods of calculating $ \pi $ are Machin-like formulae. And it is known that certain formulas are more efficient than others. Are there any ...
6
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1answer
137 views

Given the error in the cg-method, calculate a lower bound for the condition number

Edit: If any information is missing, please tell me and I'll edit the question. Thanks again! The conjugate gradient (cg) method was applied to a positive definite Matrix $A$. It is only known ...
3
votes
1answer
111 views

Root bracketing in complex space

I have some function $F(\omega): \mathbb R\to\mathbb C$. The function $F(\omega)$ has both roots and singularities. Fortunately, I can calculate positions of singularities analytically. So my ...
1
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1answer
195 views

Jacobi method and HPD Matrices

Let $A$ be HPD. Denote by $D$ the diagonal matrix obtained by observing the diagonal elements of A, i.e. $D = \operatorname{diag}(a_{11},a_{22},\ldots,a_{nn})$. I would like to show that if the ...
0
votes
1answer
2k views

How to correctly apply Newton-Raphson method to Backward Euler method?

I'm solving a system of stiff ODEs, at first I wanted to implement BDF, but it seem to be a quite complicated method, so I decided to start with Backward Euler method. Basically it says that you can ...
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0answers
213 views

solving two systems of equation implicitly

I have been trying to solve the following two systems of equations simultanously and I'm very hesitant on how to go about it. Whether I need predictor-corrector methods, if I need to linearize the ...
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0answers
137 views

a minimum and maximum value problem

First and foremost, I greatly appreciated the prior attempts made by the excellent mathematicians Robert Israel, and mixedmath on the related problem. Now I have the following problems of the ...
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1answer
145 views

Four numerical methods example

Consider the quadratic polynomial $f(x) = x^2 − 6x + 2$. The two roots of this function are $R_1 = 3 + \sqrt 7$, $R_2 = 3 − \sqrt 7$. Consider the following 4 different iterative processes. ...
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1answer
165 views

use Numerical methods for solve help [closed]

The reciprocal formula $F(x) = x (2 − Ax)$ $F'(x) = (2 − Ax) + x(−A) = 2 − 2Ax$ $⇒ F' = (1/A ) = 2 − 2A -(1/A) = 0$ $⇒$ We have to calculate the second derivative at the root $X =1/A$ $F''(x) = ...
4
votes
2answers
705 views

How does one fit the curve $y = ae^{bx} + c$?

How does one fit the curve $y = ae^{bx} + c$? The method of taking logarithms of both sides does not simplify to allow linear regression. I can take the three equations derived from setting the ...
3
votes
1answer
209 views

Remainder term for Gauss-Laguerre quadrature

I need to calculate a quadrature rule with maximum degree of accuracy that looks like this: $$ \int_0^\infty e^{-x}f(x)dx = \sum_{i=0}^n A_if(x_i) + R_n(f) $$ where $n=2$. For $R_n(f)$ I have this ...
0
votes
1answer
171 views

Numerical method for system of linear and quadratic equations in several variables

I am looking for a numerical method that can solve a system of non-linear equations. The non-linear equations are polynomials that are in one of the following forms: $$ ...
4
votes
2answers
189 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
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1answer
222 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
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4answers
934 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 ...
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0answers
61 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
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1answer
180 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 ...
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2answers
213 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
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1answer
99 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
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2answers
242 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
855 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
219 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
133 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
158 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
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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
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0answers
33 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: $h_j(x)=0$, $x\in X$, $j=1..m$. ...
1
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2answers
5k 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 ...
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0answers
238 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
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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 ...
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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 ...
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2answers
661 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 ...
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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
161 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
773 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
124 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
716 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
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0answers
86 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 ...
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1answer
159 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.