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

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1answer
21 views

Approximate a solution of a system of non linear equations

I have a system of non-linear equations of the form $$A x_1^B \exp \bigg(\frac{- C}{x_1} \bigg) = k_1$$ $$A x_2^B \exp \bigg(\frac{- C}{x_2} \bigg) = k_2$$ $$A x_3^B \exp \bigg(\frac{- C}{x_3} \bigg) ...
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0answers
20 views

Romberg integration problem [duplicate]

Romberg integration is used to approximate $\int_0^2 f(x) dx$ if $f(2)=0.51342$ and $f(3)=0.36788$ $R_{3,1}=0.43687$ $R_{3,3}=0.43662$ find $f(2.5)$ I am quite confused how to proceed this
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0answers
9 views

Standard methods for self-consistent field theory

I'm attempting to replicate a neural-network model that comprises the following self-consistent equation: $$ R(x_i)=\bigg[I_s(x_i)+I_A(x_i)+\frac{1}{N}\sum^N_{j=1}J(x_i-x_j)R(x_j)-T\bigg]_+ $$ The ...
0
votes
1answer
36 views

Are S.I. Euler and Verlet the same thing?

Verlet as given by Wikipedia: set $\vec x_1=\vec x_0+\vec v_0\,\Delta t+\frac12 A(\vec x_0)\,\Delta t^2$ for ''n=1,2,...'' iterate $\vec x_{n+1}=2 \vec x_n- \vec x_{n-1}+ A(\vec x_n)\,\Delta t^2.$ ...
0
votes
1answer
28 views

An asymptotic numeric problem.

Given a large enough integer $N$ is there always a $c\in(0,1)$ such that $$(N+ N^{1-c}){c\ln(e N)}>\ln( N+( N)^{1-c})(N+2 N^c)$$ holds? What is this $c$ explicitly (at least a close approximation ...
2
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0answers
37 views

Stable resolution of a $2\times2$ linear system

Cramer's method for the resolution of linear systems is known to be unstable, even in the $2\times2$ case. For general systems, stability can be improved by partial or full pivoting. When you ...
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5answers
82 views

Numerical Value for $\lim \limits_{n \to \infty}\frac{x^n}{1+x^n}$

Let $$f (x) := \lim \limits_{n \to \infty}\frac{x^n}{1+x^n}$$ Determine the numerical value of $f(x)$ for all real numbers $x \ne -1$. For what values of $x$ is $f$ continuous? I honestly do ...
2
votes
2answers
48 views

describe odd number series

How to solve odd's 1 - > 1 3 - > 2 5 - > 4 7 - > 5 9 - > 7 11 - > 8 13 - > 10 15 - > 11 17 - > 13 19 - > 14 21 - > 16 23 - > 17 25 - > 19 27 - > 20 ......... ......... ......... 127 ...
2
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1answer
44 views

Gauss Transform with $LU$ factorization

Consider a symmetric matrix $A$, i.e. $A = A^T$. a.) Consider the use of Gauss Transforms to factor $A = LU$ where $L$ is unit lower triangular and $U$ is upper triangular. You may assume that ...
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0answers
30 views

Euler's method (ODE) code implementation

I am trying to write a program in Python to solve a simple initial value problem with Euler's Method for ODEs. Programming is not my forte at all, so I am having trouble with implementation. I would ...
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0answers
8 views

DFT of a square signal

$f_n = 1, n = N/4, ... , 3N/4 -1$ and $0$ otherwise (the signal being N-periodic). I'm trying to arrive at a simplification for $F_k$. $$F_{2k} = \frac{1}{N}\sum_{n=N/4}^{3N/4-1}W_N^{-2kn} = \frac{1}...
10
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0answers
125 views

Finding $1/x^2 + 1/x^3 + 1/x^5 + \dots $

The following function came up in my work: $$ f(x)=\sum_{p\text{ prime}}\frac{1}{x^p}=\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+\cdots. $$ Naturally, this converges for ...
0
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0answers
13 views

Cubic B-Spline Step function transformation

Let $B = (x - x_i)^2(x_{i+2} - x) + (x - x_i)(x_{i+3} - x)(x - x_{i+1}) + (x_{i+4} - x)(x - x_{i+1})^2$ over the interval $x_{i+1} \leq x < x_{i+2}$ Suppose $i \equiv i - 2$, and set $h = x_{i+1}...
2
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1answer
25 views

Properties of matrix stable (numerical) rank

I happened to notice that there is concept "stable rank" that people used a lot in matrix computation theories, such as the work of Rudelson & Vershynin (2005). It is defined to be the ratio ...
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2answers
24 views

Growth rate of a function

I am having some trouble determining the growth rate of the function $m(n)=\inf\{m: \frac{1}{2^m}\le \frac{1}{n} m^{3/2}\}$. This comes up in problem 2.2.8 in Durrett's probability book. Essentially, ...
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0answers
38 views

How to integrate discrete data by Gaussian quadrature method

I'm trying to numerically integrate discrete data by Gaussian quadrature method. The file attached test.mat is a discrete data set taken from a finite-element mode ...
0
votes
1answer
18 views

A $C^1$-function, s.t. approximation by the Trapezoidal rule is more accurate than by Simpson's rule?

Find values $a, b \in \mathbb{R}$ and a function $f \in C^{1}[a,b]$, such that the approximation of $\int_{a}^{b} f(x)dx$ by the Trapezoidal rule $T(f)$ is better than the approximation by the Simpson ...
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0answers
19 views

Factoring out a variable from an unknown multivariable function

I have a data set that follows the behavior of a function f that depends on a lot of different variables. Let's call two of those variables $a$ and $b$. The specific behavior I'm interested in is $f(a)...
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0answers
8 views

Find the critical points (or roots) of many short Fourier series efficiently

I need to find the critical points of around 1000 Fourier/cosine series, each with length 4. More specifically, I need to do $$ m_j = \max_{0 \leq z < \pi} \sum_{m=1}^4 u_{j,m}\cos(mz), \quad \...
-2
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1answer
24 views

computational cost power matrix $A^k$

Can you help me? If $A\in\mathbb{R}^{n\times n}$, which it is the computational cost $A^{k}=A\cdot A\cdot\ldots\cdot A$?
-1
votes
1answer
10 views

Least squares method what is an good error [closed]

Hi I have started using the least squares method and I want to calculate the mean error of my approximation. I use the following formular for calculating the error: $\ \mathcal E = ||A\hat x - y||/\...
0
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0answers
21 views

How can we use the Lindley's method to approximate the following expression?

The Lindley's(1980) approximation is one of the most popular methods that is used to obtain Bayes estimates. In this method we need to maximum likelihood estimators(MLEs) of the unknown parameters. ...
0
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0answers
15 views

finite difference time domain grid question

The finite difference time domain method is a finite difference method for solving maxwell equations numerically. There are several pieces to it, but this is the root of my question $H_{i +1/2 , j+1/...
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2answers
37 views

Identify what value of $x$ may have issues with cancellation error

I am leaning numerical analysis and getting a hard time to understand cancellation error. For example, suppose we have $\ln(x+1)-\ln(x)$, this is the same as $\ln\left(\frac{x+1}{x}\right)$. Suppose ...
4
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1answer
60 views

Prove $\int_\Omega f(x) \,dx=f(x_B) \int_\Omega1 dx+ \mathcal O(\int_\Omega1 dx \cdot \sup_{x,y\in\Omega}\|x-y\|_2^2)$?

Let $\Omega \subset \Bbb R^n$ be a convex domain and $f: \Omega \to \Bbb R $ and $f \in \mathcal C^2(\Omega)$. Let $x_B $ be the barycentre of $\Omega$ with $$x_B:= \frac{\int_\Omega x \,dx}{\int_\...
1
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0answers
14 views

Piecewise-linear (or otherwise monotonic) interpolation as a matrix problem

Background: I'm hoping to find (or write) an algorithm to piecewise linear-interpolate large sets of unevenly sampled functions (10s of thousands of arrays of a thousand or so $x$ and $y$ pairs, where ...
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0answers
30 views

Solving a system of N-1 Ist order ODEs by Euler's Method

In order to solve a system of N-1 first order ODEs by Euler's Method For N = 4; t=0, h= 0.1, x= 0.1 should the Euler formula be? $U_n(t+h) = U_n(t) + h F_n(x_n, t_n)$ for n = 1, 2,..,N-1 but we ...
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0answers
22 views

numerical integration asymptotic relation

Let $Q\subset R^n$ be a convex subset and $f\in C^2(Q)\;$ We set $x_s:=\int_Q xdx$,$\;\;\;Vol(Q):=\int_Q 1dx$ and $diam(Q)=sup||x-y||_2$ Prove the following asymptotic relationship: $...
4
votes
2answers
145 views

Proving a contraction mapping is a Cauchy sequence

Let $\phi(x):[a,b]\rightarrow [a,b]$ be a continuous function. Show that if $\phi(x)$ is a contraction mapping on $[a,b]$ then the sequence $\{x^{(k)}\}$ defined by $x^{(k+1)} = \phi(x^{(k)})$ is a ...
2
votes
1answer
24 views

What is the necessary condition for ODE to have unique solution?

For the ODE: \begin{align} \dot{x}(t)&=f(x,t) \\ x(t_{0})&=x_{0} \end{align} If $\;\;f:\mathbb{R}^{n}\rightarrow{}\mathbb{R}^{n}$ is Lipschitz continuous on $\mathbb{R}^{n}$, then there exists ...
0
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1answer
19 views

Minimum error in floating point approximation of an elementary function.

I need a confirmation of a thing that probably is silly. Let $x$ a floating point number representable using $e$ bits for exponent and $m$ bits for mantissa, let $f$ a be an elementary function, you ...
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1answer
62 views

How to implement twice MATLAB integral build-in function for numeric integration? [closed]

Suppose we have a function $F(\lambda) = \int\limits_{\lambda}^1 f(x) dx$, where $f(x)$ has no formula for antiderivative. We can easily calculate it by means of build-in MATLAB functions. Let's use $...
0
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0answers
19 views

Explicit and implicit RK methods with stiff problems

Even if there isn't a precise definition of stiff equation, i think we can sum up (for the sake of the question) the concept in two cases: A linear equation $u'=\lambda u$ with a negative $\lambda$; ...
0
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0answers
10 views

How to understand the vector form of the Jacobi iteration?

When I read the book "Iterative Methods for Sparse Linear Systems" about Jacobi iteration, I can easily understand the component form for this iteration. However, since my background is computer ...
1
vote
2answers
38 views

Numerical analysis of wave equation in polar coordinates:

Is there a simple solution to deal with the problem of radial symmetry when solving a pde numerically. If so can someone provide some references/resources that explain this. Any help would be greatly ...
1
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0answers
12 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

$\newcommand{\P}{\mathbb{P}}$ I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 \P(A,B)}{\partial A^2}+\frac{\partial^2 \P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) \P(A,B)}{\partial ...
-1
votes
0answers
30 views

Romberg, trapezoidal rule exact for polynomials

My question is, how can I proof that the rombergs method of the summed trapezoidal rule is exact for polynomials with degree $(2n+1)$ or less. Thanks for helping, one or two tips can help me here. ...
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2answers
21 views

Does the rounding unit of a floating point system depend only on the mantissa?

The rounding unit (or machine epsilon) of a binary floating point system is usually represented as $\frac{2^{-(p - 1)}}{2}$ or simply $2^{-(p - 1)}$, according to this Wikepedia's article (if I'm not ...
0
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1answer
40 views

Does encountering a zero pivot during Gaussian elimination imply that the matrix is singular?

I was reading a problem about Gaussian elimination and pivots of a matrix, say $A$. The question is: During the Gaussian elimination process without pivoting a zero pivot has been encountered. Is ...
0
votes
1answer
33 views

Effects of Scaling on Matrix Norms

I feel as though is a very stupid question, but I'm struggling to muddle through it so here I am. For Gauss-Seidel methods one way to formulate the convergence requirement is that given the system $...
0
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0answers
27 views

Obtaining error between exact and finite element solution of a PDE when exact solution is not available

How does one obtain the error between the finite-element (FE) solution and the exact/analytical solution when the latter in not available? After all, isn't the purpose of a numerical method to find ...
2
votes
1answer
31 views

Least Squares Sensitivity to data

Let ($x_1$,$y_1$),...,($x_n$,$y_n$) be my data set. I have a function $f(x,{\bf c})$ where ${\bf c}=(c_1,...,c_m)$ is a vector of $m$ parameters. I want to fit to the data using non-linear least ...
3
votes
1answer
48 views

Solving a system of non linear equations

I have got a system of non-linear equations of the form $$A x_1^B \exp \bigg(\frac{- C}{x_1} \bigg) = k_1$$ $$A x_2^B \exp \bigg(\frac{- C}{x_2} \bigg) = k_2$$ $$A x_3^B \exp \bigg(\frac{- C}{x_3} \...
0
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0answers
24 views

Numerical method for fourier transform other than FFT/DFT

FFT relies on uniform samples, which cause aliasing, so FFT can be inaccurate in a certain case. Suppose you can obtain samples of $f(t): \mathbb{C} \to \mathbb{C}$ at any point ($t$ can also be a ...
2
votes
2answers
46 views

Computing a double integral with applications to prime numbers

I was reading the preprint [1] which contains on p. 7 the following formula (for $4<s\le6$): $$ f_1(s)=\frac{2e^\gamma}{s}\left\{\log(s-1)+\int_4^s\int_3^t\frac{\log(u-2)}{u-1}du\,dt \right\} $$ ...
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0answers
21 views

How to define a variable which is an integral involving cauchy principal value inside?

How to define a variable which is an integral involving cauchy principal value inside in any computer programming language? I want to know how to break down the procedure step by step from a ...
3
votes
1answer
32 views

Gauss quadrature infinite segment

Given a weighted integral $$I(f)=\int_{-\infty}^\infty f(x)e^{-x^2}dx.$$ How can I calculate the Gauss quadrature for two points. I know how to calculate the quadrature wih Legendre polynomials, but ...
0
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0answers
55 views

General issue when adding shocks on curves made of splines

Let us assume I have a "nice" curve and that I would like to introduce a small shock up/down of about 1% at a certain point along the curve. I am trying to find out what the best and most efficient ...
1
vote
1answer
57 views

An Approximation involving the Exponential Integral

Define for real $x > 0$ the function: \begin{equation} F(x)= 1 + x e^{x} Ei(-x), \end{equation} where $Ei(x)$ is the exponential integral. I found in a physics papers (Amaldi, Fluctuations in ...
1
vote
2answers
46 views

Logistic regression for football results - Estimating coefficient through maximum likelihood

Consider two football teams $V$ and $L$ with strengths $W_V$ and $W_L$, respectively. Let's assume that the draw probability $\mathbb{P}(Draw)$ is known. Then this model is supposed to give estimates ...