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

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1answer
187 views

How to find the unknown values in this Numerical Integration type?

Given the following type of numerical integration: $$I(f)=\int_0^1 f(x) \, dx \approx \frac 12 f(x_{0}) +c_1 f(x_1) $$ a) Find the values ​​of: the coefficient $c_1$ and points $x_0$ and $x_1$ so ...
2
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1answer
142 views

Solution to second order nonlinear ODE

I need to find and exact solution for the following ODEs $$y''=-3y'+2y+2x+3,\qquad y(0)=2$$ $$y(1)=-4+5\exp\left(-3/2+\left(\sqrt{17}\right)/2\right)$$ and $$y''=2y^3-6y-2x^3;$$ $$1\leq x\leq2;$$ ...
1
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1answer
189 views

Some questions about variations of fixed point method

I'm doing some excercises in Fixed Point Iteration methods with Matlab. I have to find roots for $f(x)=e^x -x -1.9\cos x$ by using $x_{n+1}=g(x_n)$. I know how to choose $g(x)$ such that I can find ...
3
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1answer
527 views

What is the order of convergence of Newtons root finding method? And when does it converge?

Given a function $f(x)$, we can approximate $x_r$ where $f(x_r)=0$ , by using Newton's method: $$x_{n+1}=x_n -\frac{f(x_n)}{f'(x_n)} $$ The method only works when 'you choose an $x_0$ near enough to ...
3
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1answer
82 views

Evaluating order of convergence

I think this is quite a simple question, I just want to make sure I understood all correctly. Here's the problem: I have a numerical method, which is in some way dependent on its spacing $h$ (like ...
3
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2answers
93 views

Aproximating rational with fraction with “smallest numerator and denumerator possible”

For example $0.795=\frac{159}{200}$. But is there a way to find fraction with smaller numerator and denumerator that will represent number $0.795xyz...$ i.e. it will approximate our given number? I ...
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1answer
104 views

Signal approximation using linear combination of functions

How I can approximate the signal $x(t)=0.001\,t^3 \exp(-0.1t)$ in the interval $[0,100]$ using a linear combination of the following functions: $f_1(t)=A_1$ $f_2(t)=A_2\cos(0.05t)$ ...
2
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1answer
78 views

How to compute the second derivatives?

Motivation: In isogeometric analysis, state variables(e.g. displacement) are defined in the parametric domain, which can be mapped to the physical domain by $\boldsymbol{\xi}\mapsto \boldsymbol{x}$ ...
3
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0answers
48 views

Timestepping PDE with positive eigenvalues

I'm trying to numerically solve a PDE, namely: $$ \partial_t \binom{u(x, t)}{v(x, t)} = x \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right) \cdot \partial_x \binom{u(x, t)}{v(x, t)} ...
3
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2answers
85 views

Fixed point iteration problem of $f(u)=u^3-u-1$

I was thinking about the following problem: Let $f(u)=u^3-u-1$. Then I have to verify whether the following statements are true/false? 1.Starting with the initial guess $u^{(0)}=1.5,$ the ...
4
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1answer
84 views

Finite-Element Method: Question on stability for equation $u_{t}+au_{xxx}=f$

I am trying to determine the stability of the PDE $$u_{t}+au_{xxx}=f$$ Given the finite-element scheme $$\frac{u_{j}^{n+1}-u_{j}^{n}}{\Delta t}+a ...
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2answers
59 views

$g \colon [0,1] \to [0,1]$ be a continuous map and consider the iteration $x_{n+1}=g(x_n)$.

I came across the following problem: Let $g \colon [0,1] \to [0,1]$ be a continuous map and consider the iteration $x_{n+1}=g(x_n)$.Then Which of the following maps will yield a fixed point for ...
2
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1answer
185 views

Reference request: Finite difference methods on curvilinear (body fitted) grids

I was wondering if someone may be aware of some form of detailed summary (book, tutorial paper) about the use of finite difference methods on curvilinear (body fitted) grids. I was only able to ...
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0answers
222 views

Multiplication and Division of functions

Suppose that you have two continuous functions, $f(x)$ and $g(x)$. Suppose that you have numerical approximations for these functions, stored a vectors, $f^*$ and $g^*$. If I want to approximate ...
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1answer
3k views

Euler's method for second order differential equation

Not really homework but sample exam. The question is to use Euler's Method to approximate Y: $Y''(t) = Y'(t) - 2Y(t)$ $Y'(0) = Y(0) = 1$ with $t_0 = 0$ and $h=0.2$ So what I did: First ...
0
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1answer
96 views

Better than Runge-Kutta-Fehlberg 4(5) at high order?

I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using ...
2
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1answer
206 views

How to prove Chebyshev–Gauss quadrature integrate polynomial of degree less than $2n-1$ exactly

What I want to ask is mentioned in the title. For example: how can we show that ...
2
votes
2answers
88 views

Quadrature formula

How can we find a quadrature formula $\int_{-1}^1 f(x) dx=c \displaystyle \sum_{i=0}^{2}f(x_i)$ that is exact for all quadratic polynomials? Thanks for help.
2
votes
1answer
381 views

Gaussian quadrature with arbitrary weight function

In class, our professor told us how to evaluate the integral $\int_a^bw(x)f(x) dx$ by finding the Gaussian nodes $x_i$ and weight $w_i$ with weight function $w(x)=1$ (also known as Legendre ...
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0answers
97 views

Divergence of Gradient Method

Is there any example of a continuous differentiable function out there, in which the gradient method with Armijo's stepsize-rule doesn't converge? I found it pretty hard to create one myself because ...
3
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2answers
438 views

How to verify the order of DOPRI Runge-Kutta method

I've written code in Fortran based on the RK8(7)-13 method by Dormand and Prince to solve the system $\mathbf{y}'=\mathbf{f}(t,\mathbf{y})$. The method is ...
97
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1answer
11k views

Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger

Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than 1, and ...
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0answers
49 views

Howuse this $R_{l}=\frac{1}{n}\left(\frac{(-1)^l}{2n}+\sum\limits_{m=1}^{n-1}\frac{1}{m}\cos{\frac{ml\pi}{n}}\right)$ and MATLAB get this four fig?

we consider Tikhonov's regularization method for $\delta =0.1, 0.01,0.001,$ and $\delta =0$ The Tikhonov's regularization method you can see:http://en.wikipedia.org/wiki/Tikhonov_regularization and ...
4
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1answer
56 views

About parallel time computation

I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
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2answers
240 views

Linear shooting method for Second Order BVP

How can we use the linear shooting method to solve the boundary value problem $$y'' = 2y' - y, ~y(0) = 1, \text{and} ~~y(1) = 2?$$ I tried to convert it to a first order system, but didn't get what I ...
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1answer
189 views

reduce differential equations system of first order and using Euler

Given differential equations $$\ddot x=Gm_1\frac{y-x}{|y-x|^3}\hspace{2cm}\ddot y=Gm_2\frac{x-y}{|y-x|^3}$$ with constant $G,m_1,m_2$ I want to solve them with the Euler method. I know I have to ...
4
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1answer
4k views

Convergence rate of Newton's method

Let $f(x)$ be a polynomial in one variable $x$ and let $\alpha$ be its $\delta$-multiple root ($\delta\ge2$). Show that in the Newton's $x_{k+1}=x_k-f(x_k)/f'(x_k)$, the rate of convergence to ...
-1
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2answers
123 views

finite elements-exercice

We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$ Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), ...
0
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1answer
72 views

How to re-parametrize for quadratic minimization?

Given a real-rectangular matrix $S$ and inorder to solve this simple quadratic programming problem: Minimize $w'S'Sw = \|S w\|^2$ over $w$ subject to $e^Tw = 1$ and $w \geq 0$ using a solver I ...
5
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2answers
1k views

Numerical Approximation of the Continuous Fourier Transform

Given a function $F(k)$ in frequency space (sufficiently nice enough, eg. a Gaussian), I would like to compute its Fourier inverse \begin{equation}f(x) = ...
2
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2answers
319 views

resources to study PDE from

I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
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0answers
40 views

What are the available libraries or programs for finding extremes of a function with no symbolic definition?

In my current mathematical inquiry, I would like to gain insight on behaviour of a $d$-dimensional continuous function by locating its maximum over the hyperplane $\sum_{i=1}^d x_i = 1$ for $x_i$ ...
3
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1answer
128 views

Is it possible to solve initial value problem with central difference scheme?

Lets say I have an ordinary differential equation: $$\frac{dy(x)}{dx}=a \cdot e^x$$ I would like to solve this equation numerically. I could use for example Euler method or other explicit or ...
3
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1answer
126 views

About iterative refinement to the solution of the linear equations

I want to know what is iterative refinement for improving the solution to the linear equations? How they improve solutions and what are the various techniques for the iterative refinements? Any ...
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0answers
356 views

Multigrid Interpolation and Restriction operators

I have a question about the restriction and the interpolation operators of a Multigrid algorithm. Let those be given: The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ ...
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0answers
133 views

Simpson's rule characteristics

I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that: ...
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0answers
140 views

What is the numerical algorithm for polygamma function?

would like to write polygamma function as a function in c++, Java or c# i find this function in Matlab as a psi(x)
5
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0answers
348 views

Runge's phenomen: interpolation error using Chebyshev nodes oscillates

We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
0
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1answer
77 views

solve a “wave equation” with an extra term

I want to solve the following "wave equation" $$\nabla^2\psi(\vec{r},t) - \frac{1}{c(\vec{r})^2}\frac{\partial^2}{\partial t^2}\psi(\vec{r},t) = R(r)\psi(\vec{r},t)$$ subject to initial conditions ...
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0answers
132 views

Qualities of Projected Gradient Methods

Consider the following constrained minimization problem: $ min_{x \in X} \ f(x) $ where $ X \subset \Bbb{R}^{n} $ is a nonempty closed convex set and f is continuously diferentiable. I'm ...
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1answer
2k views

Prove that if $A$ is symmetric and has a LU-decomposition then $A=LDU' \Rightarrow U'=L^T$, where $L^T$

Suppose the matriz $A$ has a LU-decomposition, in other words, suppose there exists matrices $L$ and $U$ such that $A=LU$ where $L$ is lower triangular and $U$ is upper triangular. We can to prove ...
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1answer
87 views

Weight function and orthogonalization

If we have the weight function $w(x) = e^{-x}$, then what are the constant and linear polynomials $p_0(x)$, $p_1(x)$, that are orthogonal on $[0,\infty)$ w.r.t. $w(x)$? And if $f(x) =\cos(x)$, what ...
3
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2answers
68 views

Deciding which number is bigger

Suppose that $a$ and $b$ are two algebraic numbers with $0<|a-b|\approx 10^{-50}.$ Suppose further that a calculator can evaluate $a$ and $b$ up to say 12 digits. Are there some general ...
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0answers
652 views

Lagrange Interpolation Polynomial Code for coefficients

Can someone help me with developing a simple MATLAB code that inputs the n datasets $ \lbrace x_i : i=0,...,n \rbrace$ and $ \lbrace f^i : i=0,...,n \rbrace$ and produces the coefficients $ \lbrace ...
0
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1answer
46 views

surface approximation using least squares

I am studying the following problem. Soppose you have two Bezièr patches with a common curve; suppose that the control points of the two patches are given by some initial guess (they are all known). ...
2
votes
2answers
68 views

Chain rule special application

I am given the equation $$ u(x,t)_t = u(x,t)_x $$ and i have to apply coordinate transformation with $$ x=x(\xi,\theta), \quad t=\theta $$ to get an equation of the form $$ \alpha u_{\theta} +\beta ...
2
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3answers
181 views

Newton (Iterative) Vs. Babylonian (Direct) For Roots

Is Newton's iterative method for finding a square root more efficient then the Babylonian method? Considering most roots are irrational, which method would get me within, say 16 decimal places, the ...
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0answers
223 views

Universal Approximation Theorem — Neural Networks

Universal approximation theorem states that "the standard multilayer feed-forward network with a single hidden layer, which contains finite number of hidden neurons, is a universal approximator among ...
1
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0answers
387 views

Simpson's Rule derived from Trapezoidal Rule

I was just wondering if I could have some assistance in regards to the Trapezoidal Rule and Simpson's Rule. I have a question where it asks to generalize the Trapezoidal Rule to the case of ...
3
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0answers
80 views

System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)

I have a large system (N>100) of equations $\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$ where $\vec{P}$ is a vector of N functions of the variable t. What is the correct ...