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

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0answers
18 views

Anyone knows a Good Textbook in Numerical PDES

I am planning on taking a course on numerical PDEs next semester. The course covers the following topics listed below. I am looking for a good book that covers these topics (or at least most of them). ...
4
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1answer
171 views

Approximating an integral with another integral with finite limits

I came across the following integral in my work $$\int_{-\infty}^{\infty} \frac{\frac{1}{(1- \ \ 2 \pi j s \theta)^{m}}-1}{2\pi j s }\ e^{-2\pi j s\sigma^2}\ ds $$ Assuming $\theta,m,\sigma^2$ are ...
277
votes
7answers
7k views

“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
3
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1answer
449 views

Fitting a sine function to data

I have a sequence of $n$ points $(x_i,y_i)$, for $i=1,\dots,n$. I would like to find the function, of the form $y=V\sin(x+\phi)$, which best fits the points. Which numerical method could I use? I have ...
1
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1answer
46 views

The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho, $$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
5
votes
1answer
180 views

How can I tell which matrix decomposition to use for OLS?

I want to find the least squares solution to $\boldsymbol{Ax}=\boldsymbol{b}$ where $\boldsymbol{A}$ is a highly sparse square matrix. I found two methods that look like they might lead me to a ...
3
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1answer
401 views

Riemann sum error and the integral

It is a well known, that we have the following approximation error: $$ ...
0
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0answers
23 views

Lagrange Newton Method Singular Matrix

i implemented the lagrange-newton method in python to find the problem to nonlinear optimizing problem for learning purposes. But every guess i made a guess for the initial values the resulting ...
0
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1answer
18 views

How to express a system of differential equations in a form suitable for numerical methods?

I am modeling rocket thrust equations using some of the formulas and derivations on page 37 & 38 here. For my Rocket model, I have the following two equations: $$dv/dt = 383v^2$$ $$dA/dt = 635.14 ...
5
votes
1answer
58 views

Using numerical methods to calculate integral

$$ \mbox{How can I go about calculating}\quad \int_{0}^{\infty}\,{\rm e}^{-100\,x^{2}}\,{\rm d}x\quad \mbox{to}\ {\sf\mbox{five}}\ \mbox{decimal places of accuracy ?.} $$ Do I use Simpson's Rule ?. ...
4
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1answer
84 views

Need some facts about Newton-Schulz iterative method and its application to sparse matrices

I am studying Newton-Schulz iterative method for obtaining an approximate inverse , which is given by $V_{k+1}=V_{k}(2I-AV_{k})$, wherein $I$ is the identity matrix and it converges, when the ...
0
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1answer
13 views

Meaning of indices for cubic hermite splines

While digging through some code about Perlin noise, I noticed, that a Cubic Hermite Interpolation polynome is used at some point. At this point, I wanted to know, which of the Hermite basis ...
2
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5answers
120 views

Is there a proof that $\int \frac {dx}{x}=\ln |x|+c$?

Is there a proof that $$\int \frac {dx}{x}= \ln|x|+c$$ for $x\neq 0$ I would be interest for any replies or any comment.
1
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2answers
67 views

How to prove that trigonometric functions form a Chebyshev system?

How can be proven that $$\{ \operatorname{cos}(kx)\}_{k = 0}^n \text{ and } \{ \operatorname{sin}(kx)\}_{k = 1}^n$$ are Chebyshev systems in the interval $(0, \pi)$? Any ideas will be appreciated. ...
0
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0answers
15 views

order of convergence for approximations

Let $u \in L^{2}(0,1)$ and $0 < x_{1}< x_{2}<... < x_{n} = 1$, where x$_{k}$ = k$\cdot$h, n$\cdot$h = 1, a partition of the interval [0,1]. Define I$_{k}$(x) = 1 if x $\in$ [x$_{k}$, ...
1
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2answers
22 views

minimum number of iteration in Bisection method

One root of the equation $e^{x}-3x^{2}=0$ lies in the interval $(3,4)$, the least number of iterations of the bisection method, so that $|Error|<10^{-3}$ is (a) 10 (b) 6 (c) 8 (d) 4
7
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3answers
438 views
+100

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
1
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0answers
40 views

Newton method for maps between Banach spaces

I am trying to understand the following theorem, which can be found in Kolmogorov and Fomin's (p. 509 here): Let map $F$ [$:X\to Y$ where $X,Y$ are Banach spaces] be strongly differentiable in a ...
1
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1answer
27 views

Am I doing this approximation correctly? (least squares method)

Here is the problem. Find the function $f$ of the type $f(x) = a\cos x + b\sin x$ which best approximates the function $g$ in the points : $$ \begin{array}{ c | c | c | c | c | c | c } x & ...
2
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0answers
42 views

Convergence of Newton method under some assumptions

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 508 here) that if $x^\ast$ is the unique root of equation $$f(x)=0$$on interval $[a,b]$ and if the function has ...
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2answers
465 views

How to find upper bound on absolute error with composite trapezoid rule

Obtain an upper bound on the absolute error when we compute $\displaystyle\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points. The formula I'm ...
2
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0answers
38 views

Why does this nonlinear ODE solution not work?

I am relatively new to Python and trying to use it to solve a second order nonlinear differential equation, specifically the Poisson-Boltzmann equation in an electrolyte. $$\phi''(r) + \frac2 ...
1
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0answers
23 views

Rayleigh quotient ($|r(q)-\lambda|=O(||q-x||^2)$ ?)

how to show $|r(q)-\lambda|=O(||q-x||^2)$ $r(q)=q^*Aq/(q^*q)$ and $\lambda$ is an eigenvalue, A is a symmetric matrix. x is the unit eigenvector corresponding to $\lambda$. and q is a unit vector. ...
1
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1answer
22 views

Gauss quadrature on tetrahedron

I have the Weights ${\bf w}=[w_1\,w_2\, w_3\, w_4]=[0.25, 0.25, 0.25, 0.25]/6$ and the points ${\bf x_1}=[0.1381966011250105 , 0.1381966011250105 , 0.1381966011250105]$ ${\bf ...
1
vote
2answers
29 views

Domain for which this matrix is positive definite

What is the domain for which this matrix is positive definite? $$\left(\begin{array}{cc} 12x^2 & 1 \\ 1 & 2 \\ \end{array}\right)$$ I'm trying to figure this out. I know the ...
5
votes
3answers
971 views

Calculate Runge-Kutta order 4's order of error experimentally

The Problem Use the order 4 Runge-Kutta method to solve the differential equation $ \frac{\partial^2 y}{\partial t^2} = -g + \beta e^{-y/\alpha }*\left | \frac{\partial y}{\partial t} \right |^{2} $ ...
-9
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0answers
34 views

fixed point iteration code [closed]

So as I was working out my math with the following equations using the fixed point iteration, I stumbled to wonder if one can ever write a C program, MATLAB (or the likes) for such math. So my ...
0
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0answers
15 views

Determine the order of dissipation for a finite difference scheme

Consider the PDE $u_t + au_x = f$ and the finite difference scheme: $$ \frac{3v_m^{n+1} - 4v_m^n + v_m^{n-1}}{2k} + a\frac{v_{m+1}^{n+1} - v_{m-1}^{n+1}}{2h} = f_m^{n+1} $$ I need to determine the ...
0
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0answers
25 views

Guassian quadrature

Consider the polynomial $L \in P_5$ defined by $$L(x)=\frac{d}{dx^3}(1-x^2)^4$$ My question is to use a technique similar to the one used in the analysis of the Gaussian quadrature to show that Q has ...
10
votes
1answer
605 views

Bounding the basins of attraction of Newton's method

In general, Newton's method for root finding has a "bubbly" boundary between basins of convergence for different roots. This is where fractals are usually created from. But outside these "bubbly" ...
30
votes
1answer
536 views

What's the most efficient way to mow a lawn?

For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define $E_x(S)=\{y\in\Bbb R^2:d(y,S)<x\}$. ($E_x(S)$ represents the expansion of $S$ by $x$.) Given a path $\gamma:[0,1]\to\Bbb R^2$, denote its length as ...
2
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1answer
513 views

Derive error term by using Taylor series expansions.

Using Taylor series expansions, derive the error term for the formula \begin{equation} f''(x)\approx \frac{1}{h^{2}}\left [ f(x)-2f(x+h)+f(x+2h) \right ]. \end{equation} I've tried it on my own ...
0
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0answers
8 views

What is numerical flux function?

I am learning "Numerical Approximations of Hyperbolic Systems of Conservation Law". I can not find answers for the following questions: 1.What is the numerical flux function? 2.How can one find ...
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1answer
21 views

Tissue Deformation Simulation using FEM

I need to simulate tissue deformation using FEM. Is it advisable to represent the object as a triangle mesh or a ...
0
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1answer
65 views

Interpolation of polynomials

let $f(x)=2^x$ and $x_0=1$, $x_1=2$, $x_2=3$. Use divided differences to compute the interpolation polynomial $P(x)$ satisfying $P(x_i)=f(x_i)$, i=0,1,2 and $P'(x_1)=f'(x_1)$ and estimate error ...
0
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0answers
14 views

Polynomial Interpolant [duplicate]

I need to find the polynomial using divided differences with respect to these conditions: $ p(1) = 2 $ $ p(2) = 4 $ $ p(3) = 8 $ $ p'(2) = 4ln(2) $ How do I deal with the condition on the first ...
0
votes
0answers
17 views

Constructing integration rules

Consider the polynomial $L \in P_5$ defined by $$L(x)=\frac{d}{dx^3}(1-x^2)^4$$ i. Find the roots $(ξ_i)i=0,...,4$ of $L$ and show that they are distinct and lie in $[−1, 1]$. (Hint: −1, 0, 1 are ...
0
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1answer
24 views

Show that the following iteration rule fulfills the condition of the Banach-fixed-point theorem

Given the following system of differential equations: $$\dot{\textbf{v}} = \left( \begin{array}{cc} -800.2 & -399.6 \\ -399.6 & -200.8 \\ \end{array} \right) \textbf{y} - \textbf{c}$$ Show ...
3
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1answer
32 views

Finite differences and conservation law

I am using a Finite Difference scheme to solve a simple PDE in conserved form: $$\partial_t u = \partial_x (\partial_x u +au\partial_x u) = (1+a)\partial_x^2u +a(\partial_x u)^2 $$ $$\frac{u_{n+1,j} ...
63
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22answers
12k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical. Numerical computations, to my understanding, never deal ...
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3answers
28 views

How to turn a decimal into a number to divide something by into it.

So here is what things will convert to: 0.5 = 2; 0.25 = 4; + MILLIONS MORE 1 = The whole of a number ( / 1 ) 0.5 = Half of the number ( / 2) But what is the math to convert decimals into only a ...
0
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1answer
33 views

Distributed Newton methods for large scale problems

I am keen to know about the literature landscape for distributed convex optimization methods which use second order information like the Newton step. This is as such a less evolved area compared to ...
1
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0answers
18 views

Matlab Newton's Method Non-linear system

There is something wrong with this program and I cannot seem to find it. I am trying to calculate the solution of a non-linear system using Newton's method. Matlab keeps saying there is a problem with ...
0
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0answers
31 views

How to solve one differential equation with two independent variables in heat transfer.

$$A\frac{ \partial T_a}{\partial t}=B(T_p-T_a)+C(D-T_a)-E\frac{\partial T_a}{\partial x}$$ Where $A, B, C, D, E$ are constants, $t$ is time and $x$ is $x$-axis of the box in which heat transfer is ...
0
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1answer
802 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
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0answers
55 views

Algorithm to numerically solve this system of three polynomial equations of degree $6$ [closed]

Mathematica Nsolve gave all $6$ solutions without an initial guess, whereas sympy Nsolve gave one solution closest to the ...
0
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1answer
24 views

Solving numerically a non-linear equation.

How is the more appropriate numerical method to solve the equation $$\cos(2\pi x)+\cos \left(\frac{2\pi N}{x}\right)=2,$$ for a given $N$? Notice that if $N \in \mathbb{Z}$, then $x\mid N$.
2
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1answer
36 views

matlab program help

Wanting to write a matlab program to solve the following iteration: $x^{(k+1)}=b+\alpha\begin{bmatrix}2&1\\1&2\end{bmatrix}x^k,k=0,1,2,\cdots$ where alpha is a real constant. Find the values ...
2
votes
0answers
35 views

solution of multidimensional PDE

I'm looking for a way to find a solution 'f' to the following PDE. $$ y \frac{\partial f}{\partial r} + g_1(r)\left(z\frac{\partial f}{\partial y} - y\frac{\partial f}{\partial z}\right) + ...
0
votes
2answers
61 views

Is there a proper way to prove that $f:[a,b] \to[a,b]$

Is there any proper way to know whether a function has the same domain and range $[a,b]$ where $a,b<\infty$ i.e. $f:[a,b] \to [a,b]$ ? For example: $$ f(x) = e^{−x} ,\qquad [\ln(1.1), \ln(3)] $$