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

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24 views

An Elementary Proof of Error Estimates for the Trapezoidal Rule

I do not see how equation (4) is equal to (3) on page 2 of 'An Elementary Proof of Error Estimates for the Trapezoidal Rule'. Here is the relevant portion of the paper: ...define $$L_i = ...
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1answer
27 views

Matrix and vector norms Inequality

I want to prove that $\frac{\left \| u - \tilde{u} \right \|}{\left \| u \right \|}\leq \frac{k(A)}{1-k(A)\frac{\left \| A - \tilde{A} \right \|}{\left \| A \right \|}} \frac{\left \| A - \tilde{A} ...
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1answer
41 views

How to find “unique” eigenvalues when computed numerically?

I have a large sparse matrix, $L$, which represents the laplacian of a weighted graph: $L = \text{diag}(\sum_{j=1}^{N} w_{ij})-W$, where $W$ is the weighted adjacency matrix with $w_{ij}$ giving the ...
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1answer
25 views

problem with solving a set of equations using ODE solvers in matlab

I have used matlab ode solvers to solve equations of the form: $$dy/dt=f(y,t)$$ in other words, where for instance say with an explicit Euler time stepping $$y^{n+1}=y^{n}+\Delta t f(y^n,t)$$ Is it ...
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0answers
17 views

Determining error for numerical integration of ODE's

When determining absolute/local error of numerical integration schemes for, say, the solution to a system of ODE's, what does one use as their standard of comparison? Is there one particular scheme ...
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0answers
45 views

Solving a simple Schrodinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Piaevskii equation I have found a problem that can be tracked down to something buggy occuring in the simplest Schrodinger equation possible: $\partial_t \psi ...
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0answers
19 views

Integral with Simpson's method not converging

I'm trying to use Simpson's rule to integrate the following function in a program: $$\int_{z_a}^{z_b}\frac{Cf(z)}{(C^2 - f(z)^2)^{3/2}}\,dz$$ where $C$ is a constant and $f(z)$ are interpolated ...
2
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1answer
51 views

DFT and the unitary matrix

Let $F_n\in\mathbb{C}^{n\times n}$ be the unitary matrix representing the discrete Fourier transform of length $n$ and so $F_n^{H}\in\mathbb{C}^{n\times n}$ is the inverse DFT of length $n$. For ...
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76 views

What variant of exponential smoothing is used in the VEGAS numerical integration algorithm?

The VEGAS numerical integration algorithm uses the following procedure to update a vector x of length n: ...
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0answers
15 views

Good books on numerically solving nonlinear PDEs

I had one course in PDEs and we weren't taught numerical methods in this course, and from the books I've read on the topic it seems very hard to impossible to solve with methods like F.D or F.E to the ...
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1answer
95 views

Unitary matrix representing Discrete Fourier Transform

Let $F_n\in\mathbb{C}^{n\times n}$ be the unitary matrix representing the discrete Fourier transform of length $n$ and so $F_n^{H}\in\mathbb{C}^{n\times n}$ is the inverse DFT of length $n$. For ...
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1answer
14 views

A posteriori measures of numerical dissipation and dispersion

In PDEs, it is typical to find out how dissipative or dispersive a numerical method is by writing down the modified PDE corresponding to the numerical method, and seeing if that modified PDE contains ...
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1answer
53 views

Meaning of “up to a scalar”

I have heard this used, within the context of results between the same up to a scalar, but I'm not sure of its meaning. Can anyone provide an explanation and example in as simple terms as possible? ...
6
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1answer
92 views

Why does this sequence converge to $\pi$?

Over at our friends at codegolf.SE, I asked a question about programs that seemed to converge to $\pi$, but didn't actually do that. One of the answers (by ...
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1answer
57 views

How to “separate” a matrix into two vectors?

I have a matrix $M$ and I would like to find two vectors $u$ and $v$, that minimize $$ \sum_{i,j} (M_{i,j}-u_iv_j)^2 $$ How can I do this (numerically)? Actually this is very simplified ...
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0answers
45 views

Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
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1answer
28 views

Verifying a quadrature formula is exact for polynomials of a certain degree

Say I have a quadrature used to approximate $\int_0^1 f(x)\,dx$ defined as $$I(f) = \frac{1}{90} [ 7f(0) + 32f(1/4) + 12f(1/2) + 32f(3/4) + 7f(1) ]$$ How would I verify that this formula is exact ...
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1answer
21 views

Determining upper bound on truncation error

Determine an upper bound on the truncation error resulting from estimating $\sin x$ by $T_5(x), x \in [-0.2,0.2]$ Workings: Take $x = -0.2$ $f(x) = \sin (x)$ $\sin(x) = x - \frac{x^3}{3!} + ...
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2answers
44 views

othogonality of chebychev polynomials

Let the chebyshev polynomials be defined as : with zeros : My goal is to show that the family of polynomials : are orthogonal with respect to where : To achieve this we show : ...
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0answers
30 views

Why is $f = f_{0} + \sum_{i}\alpha_{i}X_{i} + \frac{1}{2}\sum_{i}^{n} \sum_{j}^{n}A_{ij}X_{i}X_{j}$ the standard quadratic form in n dimensions?

The claim that $$f = f_{0} + \sum_{i}\alpha_{i}X_{i} + \frac{1}{2}\sum_{i}^{n} \sum_{j}^{n}A_{ij}X_{i}X_{j}$$ is the standard quadratic form for $n$ dimensions, where $\alpha$ is some $ 1 \times n$ ...
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58 views

A question about recurrence in a Newton's method problem?

The equation $x^3-x=0$ has three roots,$ -1, 0, 1.$ We use the Newton's method to find the roots. And there are three cases (i) If $x_0>1/\sqrt{3}$, the Newton's method will converge to $1$. (ii) ...
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38 views

Newton's method of finding roots of an equation.

Consider Newton's method on finding the roots of $x^3-x=0$, how to show that $x_n$ converges to $1$ for any $x_0>1/\sqrt{3}$? My attempt: The Newton's method says ...
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1answer
21 views

How do I find optimal ω for SOR method?

Following is the example from this book. My question is, what value of λ did he put in? He did not explain that, can anybody explain how did he get 1.24?
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0answers
39 views

Finite difference discretisation of the heat equation

Here is the equation to be discretised: $$ k\left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = \dot{q} $$ Using the following discretisation scheme: $$ ...
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1answer
23 views

Inequality involving floating point numbers

Given $x_i^{*}$ ($i\geq 0$) be positive numbers on the computer, and $\delta$ is a unit round-off error, $x_i^{*} = fl(x_i) = x_i(1+\epsilon_{i})$ with $|\epsilon_{i}|\leq \delta$ ($x_i$ are positive ...
2
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1answer
33 views

Do $0$ terms count as a term in Maclaurin expansions?

I have a question which basically asks to find $$\int_a^b \ln(1+\sqrt x) \,dx$$ using the first five terms of the Maclaurin series of $\ln(1+x)$. There are limits to the question and my solutions ...
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1answer
42 views

Number of the form $2^i3^j5^k$ closest to a given number $n$

How do I find a number of the form $2^i3^j5^k$ closest to a given number $n$, with $i, j, k \in \mathbb{N}$ numerically? Of course, I could try $\lfloor \log_2{n}\rfloor \times ...
1
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1answer
29 views

Perturbation of roots in Wilkinson's polynomial

I am studying numerical analysis. When I read the online definition I found on this paragraph: Suppose that we perturb a polynomial $p(x) = Π (x−α_j)$ with roots $α_j$ by adding a small multiple ...
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0answers
19 views

L-stability of Semi-implicit Runge-Kutta Single-Step Method

So I would need assistance to provide a proof for stability of a Runge-Kutta SSM. Let the Semi-implicit Runge-Kutte SSM, used to discretize the following autonomous ODE : y = f(y), be defined as : ...
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2answers
49 views

Numerical methods - iteration equations

(a) show that $e^x - x = 4$ has a root between $1$ and $2$ (fine with this part) (b) show that the iterative formula $x_{n+1} = {e^{x_n}(x_n - 1) + 4 \over e^{x_n} -1}$ Leads to a solution of the ...
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0answers
29 views

QR-decomposition with Givens-rotations

I want to compute the QR-decomposition of the following matrix, with the additional requirement that the diagonal elements $R_{ii}$ be positive $$A = \begin{bmatrix} 3 & -2 & \sqrt ...
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1answer
74 views

Newton-Raphson method very slow convergence

When we use Newton's-Raphson method in the following equation $f(x)=x^{50}-1 =0$ for $x>0$ with $x_0=\frac{1}{2}$, there is very slow convergence for the $x=1$ root.
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0answers
7 views

Reference for finite difference scheme for elliptic PDEs

Is there somewhere a gentle presentation for the numerical analysis of the finite difference method for elliptic PDEs. For instance in $\mathbb{R}^2$, $\Omega = (-L_x,L_x) \times (-L_y,L_y)$ for some ...
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2answers
27 views

prove that $x_{n+1}=\frac{x_n(x_n^2+15)}{3x_n^2+5}$ is cubic order of convergence near $x_0=\sqrt{5}$

To solve the equation $$x^2-5=0$$ There exitsts a iteration method $$x_{n+1}=\frac{x_n(x_n^2+15)}{3x_n^2+5}$$ I know that it is cubic convergence but I don't know how to prove it. I have tried the ...
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0answers
28 views

Showing that an iterative method solves a particular system

I have $A=\begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$, $b=\begin{pmatrix}3 \\ 5\end{pmatrix}$. The system to be solved is $Ax=b$. We're also given: $$B_\theta=\frac 1 4 \begin{pmatrix} ...
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0answers
21 views

Why does step size in fourth order runge-kutta methods less than 1? [duplicate]

I am currently reading about fourth order Runge-Kutta methods, and I noticed that for first order the total error is a factor of $O(h^2)$, and for fourth order the error is in order of $O(h^5)$, where ...
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0answers
16 views

FEM with piecewise linear for inhomogeneous Dirichlet BC

Given $-u''(x) + \alpha u(x) = f(x)$ ($0\leq x\leq 1$, $\alpha > 0$) with $u'(0) = A$, $u'(1) = B$. (a) Find the finite element approximation using piecewise linear elements in $3$ equidistant ...
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2answers
102 views

Solving system to find a curve passing through 3 points

A family of curves, depending on parameters (A,B,C) has equation $$y(t) = A / (1+B*C*t)^{1/B}$$If B=0 or B=1 we have exponential and harmonic cases. I am looking for a curve of that family that ...
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3answers
43 views

Numerical Methods for Algebraic Equations - Non root finding

I'm researching a topic for solving general algebraic equations using numerical method. My numerical recipe knowledge is rather rusty with the Bisection to Newton's methods but I don't think those ...
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0answers
4 views

Testing numerical solvers for multidimensional SDEs with non-commutative noise

I am implementing the multidimensional Milstein scheme to solve SDEs. I am trying to test the solver on benchmark equations but I cannot find an analytic solution for the case with a 2-d state vector ...
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2answers
47 views

Convergence of Newton Iteration

For $a>0$, I want to compute $\frac{1}{a}$ using Newton's iteration by finding a zero of $f(x)=a-\frac{1}{x}$. Newton's iteration formula reads $$x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}=2x_k-ax_k^2$$ ...
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1answer
33 views

numerical solution of ode singularity

I have to solve a nonlinear ode problem: \begin{align} \dot\gamma&=-2(D+0.5A+0.5A\cos\gamma)\sin k \\ -\dot k&=(E-B-B\cos\gamma)+(2D+A+A\cos\gamma)\frac{\cos\gamma}{\sin\gamma}\cos k ...
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0answers
31 views

Inverted pendelum Matrix numerical derivative

Here I've written a dynamic function as : ...
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1answer
27 views

Mean squares aproximation constant finding

Measurments $(t_k, C_k), t=1..n ; t_k, C_k > 0$ suggest that $C(t) = \frac{1}{At + Bsin(t) + 2}$. Using mean square approximation find probable values for constants $A$ and $B$. Should I start from ...
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3answers
29 views

Find constant $a$ that minimizes expression $\sum_{k=0}^{r}\frac{(y_k - a\sin(x_k))^2}{\ln(1+x_k^2)}$

$ E(a) = \displaystyle\sum_{k=0}^{r}\frac{(y_k - a\sin(x_k))^2}{\ln(1+x_k^2)}$ I need to find constant $a$ that minimizes this expression $E(a)$. I'ts long time since I've done calculus so I need ...
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1answer
28 views

Proof that $ ||f|| = \sqrt{\sum_{k=0}^{n}p(x_k)f(x_k)^2} $ is normed vector space

I've to prof given $X = \{x_1, x_2, ..., x_n\}$ and function $p$ with property $p(x: X) > 0$ that equation $ ||f|| = \sqrt{\sum_{k=0}^{n}p(x_k)f(x_k)^2} $ is norm on discrete set $X$. This ...
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0answers
12 views

Numerical algorithm for largest Eigenvalue problem

I am dealing with calculating an eigenvalue problem for differential operator of order 4: $$ \alpha \cdot\Delta^2 u+\Delta u-\Delta(u\cdot u_p(x))=\lambda u $$ where $\alpha\in \mathbb{R}$, $\Delta$ ...
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0answers
18 views

Same result using chebyshev and equidistant (Lagrange interpolation)

I have a question regarding Lagrange interpolation. Are there any functions for which the lagrange interpolating polynomial with equidistant nodes and the lagrange interpolating polynomial with ...
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1answer
21 views

Finding the initial value that brings the system through two predefined values

I have two differential equations: $$\dot{\mathbf{x}} = f(\mathbf{x}, \mathbf{y}) \\ \dot{\mathbf{y}} = g(\mathbf{x}, \mathbf{y})$$ Given two states $\mathbf{x_{start}}$ and $\mathbf{x_{end}}$ which ...
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0answers
33 views

How to use the Riemann sum for a line integral

Let us say that we have the function $f(x,y) = xy$ and the curve $C: x=\cos(t), y=\sin(t)$ on the interval $[0, \pi/2]$. I want to use the Riemann sum to find out the integral of ...