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

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Prove $\int_\Omega f(x) dx=f(x_B) \int_\Omega1dx+ \mathcal O(\int_\Omega1dx \cdot sup_{x,y\in\Omega}||x-y||_2^2)$?

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_\...
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1answer
18 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 ...
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1answer
8 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
7 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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10 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: $...
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2answers
582 views

Fixed-Point Iteration method unable to converge to any of a function's infinte roots

An equation is given to me which has to be solved by direct iteration method: $$\sin(x) = {x+1 \over x-1}$$ or $$f(x)=\sin(x)-{x+1 \over x-1} = 0$$ I follow the following procedure with reasons ...
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2answers
134 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 ...
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1answer
51 views

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

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 $...
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1answer
15 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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0answers
13 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'=λu with a negative λ; An equation ...
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0answers
9 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 ...
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2answers
33 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 ...
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1answer
394 views

Runge Kutta Method Matlab code

So I have a programming assignment with the following instructions: Consider the nth-order differential equation $$Ax^n (t) = x ^{(n-1)}(t) + x^{(n-2)}(t) + ... + x(t)$$ where $A$ is a real-...
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2answers
20 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 ...
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0answers
28 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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1answer
31 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 $...
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1answer
37 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 ...
2
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1answer
21 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 ...
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0answers
22 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 ...
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1answer
41 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} \...
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0answers
23 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 ...
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0answers
17 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 ...
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2answers
44 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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1answer
430 views

How to evaluate a condition number for a function of several variables.

I'm trying to get the condition number of a function $f(a,b,c)$ to see if it is stable. It is multivariate. I am reading the information here http://www.google.com/url?q=http%3A%2F%2Fen.wikipedia.org%...
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1answer
55 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 ...
3
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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 ...
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0answers
51 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 ...
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2answers
42 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 ...
3
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1answer
58 views

(Possible) Misunderstanding of perturbation method for finding solution of polynomial equation?

This is the strange moment that I get when I solve this equation: $$ \frac{w^4}{4} - \frac{w^3}{3} = \varepsilon, $$ where $\varepsilon$ is a small parameter. If I plot the graph $ w \mapsto \frac{w^...
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0answers
36 views

N-dimensional piecewise cubic spline fitting

I have $m$ points in $R^n$ - say, $\{P_1 \dots P_m\}$ - that were collected at different time points but I don't know in which order they were collected. I want to fit an $n$-dimensional piecewise ...
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1answer
36 views

Uniqueness of interpolation polynomial.

I am new to numerical analysis and this is the first thing I came across. It says on my textbook that interpolation polynomials are unique and to prove that it was assumed that let there be two such ...
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1answer
24 views

Floating point numbers

In a certain computer represents numbers in base2, if the distance between 7 and the next largest floating-point number is $2^{-12}$. What is the distance between 70 and the next largest floating ...
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0answers
37 views

Approximating Geometric Brownian Motion numerically

I am trying to generate a numerical solution to the SDE for Geometric Brownian Motion. The stochastic process is given by $S_t = \exp(\sigma W_t + \mu t)$, and by Ito's lemma, we have that the SDE is ...
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22 views

More precise trail function in Rayleigh–Ritz method

In order to obtain displacement field of an elasticity problem, say a plate structure, we approximate the solution using trigonometric series with unknown coefficients which satisfy the essential ...
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0answers
8 views

Find parameters and node so that quadrature approximating integration will have maximum order

Find parameters $\alpha, \beta$ and node $c$ so that quadrature $Q(f) = \alpha f(a) + \beta f(b)$ approximating integration $\int_{a}^{b}f(x)dx$ will have maximum order. I don't know how to solve ...
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0answers
16 views

Stopping criterion for adaptive Simpson's rule in 2D

On the Wikipedia page for Adaptive Simpson's method, the criterion for stopping the bisection of the integration interval $[a,b]$ when approximating an integral $\int_a^b f(x)\ dx$ is given: $|S(a,e) ...
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0answers
13 views

How to approximate the probleme non-linear by finies elements $\mathbb{P}_1$

I have to approximate $u^1$ by finies elements $\mathbb{P}_1$ (such that $h=\frac{1}{4}$ and $V_h=\{v\in C(I),\quad v(0)=v(1)=0 \}$) \begin{cases} \dfrac{\partial u}{\partial t}-\dfrac{\partial}{\...
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0answers
33 views

Finding the maximum of $|\widehat{f''}|$ for $f$ in terms of the Gaussian

Let $$f = \begin{cases} e^{-x^2/2} - e^{-2 x^2} &\text{if $x\geq 0$,}\\ 0 &\text{if $x<0$.}\end{cases}$$ I would like to find out $|\widehat{f''}|_\infty$. A good numerical bound -- of ...
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0answers
34 views

Extrapolating the sequence of derivatives of an analytic function

Suppose that $f(x)$ is an analytic complex or real function on the real domain and we only know $f(x_0)$ and the first $n$ derivatives of $f(x)$ at $x=x_0$. Is there an algorithm whose accuracy ...
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1answer
1k 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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3answers
72 views

Finding an approximation of a function's root

I have the polynomial function $f (x) = x^5+2x^2+1$. I am trying to find an approximation to its root in $[-2,-1]$, with the precision of $0.1$, and with a minimal number of steps. The answer I was ...
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2answers
29 views

Please explain this differentiation step

I don't get how they went from line 1 to line 2. Which one is treated as the variable and which the constant? I rearrange line 2 to get $0=\frac{3\varepsilon}{M}-h^3$, but I still cannot see how we ...
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2answers
56 views

Understanding power method for finding dominant eigenvalues

The power method aims to find the eigenvalue with the largest magnitude. Does magnitude still have the same meaning in this context? If so, can't we tell from the outset which eigenvalue is the ...
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1answer
50 views

Precision in Cubic spline interpolation

I am working on cubic spline interpolation with set of data points from CAD with following steps: Form piecewise spline equations between points. cubic equation : $ ax^3 + bx^2 +cx + d = P(x) $ ...
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4answers
978 views

Finding nonnegative solutions to an underdetermined linear system

Here's the environment of my problem: I have a linear system of 4 equations in 8 unknowns (i.e. $Ax = b$, where $A$ is $4 \times 8$, $x$ is $8 \times 1$, and $b$ is $4 \times 1$, with $A$ given and $b$...
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0answers
350 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
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1answer
27 views

Rate of convergence to 0 of integral

I have the following integral $\int_{1/h}^{\infty} f(x) dx$ such that $f(x) \rightarrow 0$ as $x \rightarrow \infty$ and $\int_{1/h}^{\infty} f(x) dx \rightarrow 0$ as $h \rightarrow 0$. Is there a ...
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2answers
70 views

How can I get the expression of x?

If there is $$x^2e^{A\sqrt x}=B$$ then what is the expression of $x$? If this cannot be solved, is there any approximation?
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14 views

Skew symmetric and Telescopic operator

I have been reading some papers that refer to Skew-symmetric and telescopic operators. For eg. this is a link to a presentation showing the concept. Presentation Now, I understand what one means by ...
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0answers
14 views

Convergence of jacobi eigenvalue method

Prove that jacobi eigenvalue method of $3\times 3$ symmetric matrix always converges . I know the condition for convergence of jacobi eigenvalue method and error term but I'm not so sure about "...