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

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10 views

Solving implicit theta-method function numerically faster than using fixed point iteration

When we are using the theta-method to solve an IVP. We have the equation: $$y(x_{n+1}) \approx y(x_n) + h[(1-\theta) f(x_n, y(x_n)) + \theta f(x_{n+1}, y(x_{n+1})]$$ where $f(x,y)=y'(x)$ The only ...
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1answer
18 views

Initial value problems with known solutions?

I'm trying to find a list of IVPs with known solutions to test my implementation of some numerical techniques. The only one I know of is: $$f(x,y)= y' =-\lambda y\;,\;\;\; y(0)=1$$ with the ...
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0answers
5 views

Zero-stability for numerical methods, only applies to LMM's?

I'm trying to get a better grasp of the notion of zero-stability. Mainly I'm using a book by Leveque (Finite Difference Methods for Ordinary and Partial Differential Equations). Anywho, Leveque ...
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1answer
34 views

Where is the symmetry of Fourier transform in its implementation in Maxima and Wolfram Alpha?

From Wikipedia I saw that there is a symmetry of the Fourier transformation $F(F(f))(x) = f(-x)$ This matches the graphical explanation of the (German) Youtube video (9:15 to 9:45). I tried to see ...
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9 views

Does p-order consistency for a linear multistep method imply lower-order consistency?

In some worksheet for college it says: A multistep method $$\sum_{m=0}^s a_s y_{n+m} = h \sum_{m=0}^s b_mf(t_{n+m}, y_{n+m})$$ with $a_s = 1$ is consistent of order 1 iff the following algebraic ...
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1answer
9 views

Condition number of a matrix bounded from below and above?

Is condition number of an invertible matrix bounded from below? And is condition number always bounded from above for an invertible matrix?
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19 views

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
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1answer
95 views

Can I turn $Ax=b$ into $Ax=0$?

For a system of equations $$ \begin{bmatrix}d_1 & d_2 & \dots & d_n \end{bmatrix} \begin{bmatrix}u_1\\u_2\\ \vdots \\ u_n \end{bmatrix} = d_{n+1} $$ where each $d$ is a column of ...
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1answer
33 views

How to generate a point cloud with known symmetry?

So I would like to know if there are any published algorithms to generate point clouds with known symmetry groups, such as $D_{3h}$ or $O_h$ and stuff like that. I know lots and lots of point clouds ...
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0answers
11 views

Why does QUADPACK only enforce the least strict error boundary?

According to this reference (which is in agreement with my own numerical experiments), QUADPACK tries to fulfill the following accuracy requirement on the approximation error: |RESULT - I| $\le$ ...
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1answer
29 views

Algorithm for retrieving all the permutations (randomized) for a vector sequence 1…N with only unique values

Here is the problem: I have a vector of $N$ elements long (containing only unique values from $1...N$). I am searching for an algorithm to obtain all the (randomized) combinations possible, where ...
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0answers
26 views

interpolating polynomial

Let the function f(x) = ln (1/|x-1|) be approximated by an interpolating polynomial of degree 15 with the nodes equally spaced in the interval [-1, 0]. What bound can be placed on the absolute ...
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1answer
68 views

loss of significance errors in mathematical expressions

Find a way of computing $$f(x) = \dfrac{e^{3x} - \cos (2x)}{3x}$$ without serious loss of significance near $x=0$.
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0answers
27 views

Finite Difference - Forward Difference with 2nd order Accuracy: What to do at the boundary?

I implemented a BVP using a first-order finite difference scheme after the shooting method did not work reliably. Its the first time I have worked with this. The code works but I would like to move to ...
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0answers
15 views

traslate information-technology [on hold]

i want to translate this text in frensh, any one can help me please On the one hand, DuMu x modules can be freely combined, on the other hand, dynamic polymorphism is avoided for reasons of ...
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1answer
26 views

finite element-question

can any any one help me to understand this paragraph First, the model domain G is discretized with a FE mesh consisting of nodes i and corresponding elements $E_ k$ . Then, a secondary FV mesh is ...
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0answers
10 views

fehlberg mthod and control step size

want to use Fehlberg method to solve a system of ode. But I don't how can I control step size? for example I use h=.5 and then approximate $y_{n+1}$ by RK4 and RK5 then what am I doing?
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0answers
13 views

Mapping from $\left(-\infty,\infty\right)$ to $[a,b]$ to reduce numerical error

Suppose $A = \left[\begin{array}{cc} \exp\left(x_1\right)&\exp\left(x_2\right)\\ \exp\left(x_3\right)&\exp\left(x_4\right) \end{array} \right]$, where each of $x_i\in\left(-1000,1000\right),$ ...
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1answer
34 views

Simpson's rule and Trapezoid Rule?

Let $S(n)$ and $T(n)$ be the approximations of a function using $n$ intervals by using Simpson's rule and the Trapezoid rule respectfully. My book then states: $$S(2n) = \frac{4T(2n) - T(n)}{3}$$ ...
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0answers
29 views

Numerical integration tolerance pitfalls

Consider that we want to estimate $$\int_{\pi/2}^{\pi/2+8\pi}sin(x)dx$$ (the value of this integrate is obviously zero) with the Midpoint rule. We start with the endpoints $a=\pi/2$ and $b=\pi/2+8\pi$ ...
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0answers
14 views

Is there a way to characterize the range of a chebyshev series through its coefficients?

Let $f$ be a Chebyshev series of order $n$ $$ f(x) = \sum_{i=0}^n a_i \cos\left( i \arccos\left(x\right)\right), x \in \left[-1, 1\right]. $$ Is it possible to characterize all the $\lbrace a_i ...
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1answer
57 views

Is Frobenius norm induced up to a scalar factor?

I know that the Frobenius norm is not induced since $||I||_F=\sqrt n\neq 1$. But what if we consider the norm $\frac 1 {\sqrt n} ||\cdot ||_F$? Thank you!
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1answer
94 views

How to solve $\tan x=x^2$ in radians?

How to solve $\tan x=x^2$ with $x \in [0, 2\pi]$? I try with trigonometry and many ways but the numerical solutions seems to be difficult.
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1answer
19 views

interpolate a function in a little interval

I want to interpolate a function that I have amount of it in this points: $0:.001:1$ and I have interpolate it in this points: $0:.0005:.001$. When I use lagrange method, I have a very bad result. I ...
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2answers
74 views

Solution of an equation involving even integers

If $x$ is any positive even integer $> 1$. I compute: $$ c = x + x! $$ Now assume instead $c$ (even integer) is given, and I want to get back the value $x$. Is it possible to find a simple ...
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1answer
24 views

Finding polynomial optimal in terms of least squares approximation

Find polynomial $w$ of degree at most $2$ optimal in terms least squares approximation for a function $f(x)=x^3$ in the norm $\|g\|=\sqrt{(g,g)}$, given that: $$ (f,g) = \int\limits^1_0 f(x)g(x)dx. ...
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1answer
44 views

Compute $\lim\limits_{x \to 1}[f(x)]$ and $\lim\limits_{x \to -1}[f(x)]$ for $f(x)=\frac{1}{1-x}-\frac{1}{1+x}$

Is it possible to rewrite expression $\frac{1}{1-x}-\frac{1}{1+x}$ in order to be able to find its values near $x=1$ and $x=-1$ more precisely? This is a question in a numerical methods course. Is the ...
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3answers
39 views

Definition of Global Convergence

I am confused by the notion of "global convergence" as used in numerical optimization literature, and did not find an exact definition for that yet. Now I try to double check my understanding here. ...
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1answer
39 views

Constrained Newton-Raphson method

Peace be upon you, I want to solve a system of two equations in which the existence of $ln\left(\frac{\alpha}{\alpha+\beta}\right)$ function makes some limitations in iterations of the Newton-Raphson ...
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0answers
30 views

interpolation a function in too small numbers [closed]

I have a problem that must interpolate function in very small points. I wrote this program by Lagrange method in matlab, but my result is NaN. please hep me? I take a photo from my a part of my ...
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0answers
34 views

interpolation the points that are so near [closed]

I must interpolate a function in [0,1] interval. I just have function in x=0:.01:1. so I use Lagrange interpolation method. But for h=.001, my results are false. I think this happen because of the ...
2
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0answers
32 views

What is the (currently) optimal root finding algorithm for multivariate functions? [closed]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
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1answer
30 views

Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, \begin{equation} y''(x) + p(x)y(x) = 1 \end{equation} in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
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0answers
21 views

numerical (script) fit of function with 2 arguments

I would like to find the least-square fit for a 1D-function that takes two arguments. m(x,y) = d * (x-x0)^2 / (y-y0)^2 I would like to write a c++ routine to ...
2
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1answer
32 views

Comparison of Adams-Bashforth and Runge-Kutta methods of order 4

I have a system of ODE, that must to solve with numerical methods. I solve it by Adams_Bashforth with order4 and Runge-Kutta with order4 methods. Do you know with same length step which methods ...
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1answer
16 views

discretization of mixed boundary conditions in the advection diffusion reaction equation, the Crank Nicolson method

Sorry, I have some doubts regarding the discretization of mixed boundary conditions in a PDE. I have discretized my equation, but I doubt about the boundary conditions. I dont know if you have to ...
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2answers
48 views

Solving a differential equation numerically to plot particle path

I'm trying to plot the evolution of a particle in an accretion disk by solving the equation $$2X\frac{\partial X}{\partial\tau}=V_R(X,\tau)$$ where I have found $V_R$ numerically to be ...
2
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1answer
28 views

Online resources to learn numerical methods for PDEs?

I would like to get into a career that uses alot of applied math. I took a numerical analysis course in undergrad and liked it, so I plan to self-learn numerical methods for PDEs. Other than the MIT ...
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1answer
34 views

Approximation of values in [0,1] by sums of unit fractions

Let $U = \{(-1)^k\cdot\frac{1}{n}: n\in\mathbb{N}, k\in\{-1,1\}\}$ be the set of positive and negative unit fractions. For a positive integer $m\in\mathbb{N}$ and a real $x\in [0,1]$ we set $d(m,x) = ...
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1answer
29 views

Finding spectrum with Schur decomposition

let $\alpha$ be a real number and $A_n(\alpha)$ the set of $n\times n$ matrices such that $$\alpha A+A^HA+A^H=I$$ I'd like to find the set of all eigenvalues for the matrices in $A_n(\alpha)$ using ...
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0answers
40 views

Careers in applied math with an MS other than in finance and data/machine learning?

Since I like math, I would like a career that uses alot of applied math. I'm about to complete my Master's and could do my thesis in numerical solutions of PDEs I'm already aware of careers such as ...
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1answer
71 views

Non linear ordinary differential equation

How to solve the ordinary differential equation $\frac{d^2y}{dx^2}+\sin(x+y)=\sin x,y(0)=0,y'(0)=1$ Then its possible to solve it by numerical methods?
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2answers
25 views

Iterative integration algorthm

By iterative algorithm, I mean a numerical algorithm that works by improving on a previous approximation to obtain a more accurate approximation. An example is Newton's Method. The numerical ...
2
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1answer
25 views

best approximation polynomial $p_1(x)\in P_1$ for $x^3$

I want to find a best approximation polynomial $p_1(x)\in P_1$ for $f(x)=x^3$ in $[-1,1]$ w.r.t. $||\cdot||_{\infty}$. I want to use Chebyshev polynomial to do that, but I don't know how to hang on.
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2answers
30 views

Constructing the Normal CDF $Z$-tables?

One topic that is always bypassed (in my experience, at least) in an undergraduate statistics course is the construction of $Z$-tables, such as the one provided at this link ...
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1answer
33 views

How to discretize mixed partial derivatives?

How to discretize $\frac{\partial^3 f}{\partial x\partial y^2}$ at mesh point $(i,j)$? We should use mesh points which are nearest to $(i,j)$.
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0answers
20 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
1
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1answer
27 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
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1answer
170 views

Bounding error when iterating a function

If I am iterating some function $f$ that goes to infinity as x goes to infinity with error $o(g(x))$, for example, is there anyway to bound the error? To be more specific, if I have some sequence ...
2
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0answers
34 views

Non-linear Hammerstein integral equation

I came across a problem that looks like a non-linear Hammerstein equation: $$ \displaystyle y(t)= v(t)+\int_{0}^{\infty} \frac{e^{\iota ts}}{y(s)}\mathrm{d}s $$ I tried solving it by collocation ...