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

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31 views

Can we reduce the number of operations for this case?

Suppose we want to compute $(A-B)c$ with smallest number of operations where $A=2x$, $B=y+b^{*}$, $b^{*}=y-z$ and we are given $c, x, y, z, b^{*}$ that are non-negative. From above $A,B$ are double ...
2
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1answer
35 views

Show that the error $\mid f(x)-P_2(x) \mid$ is bounded by $\frac{\sqrt2e}{3}$ on $[0,1]$

Let $f(x)=e^x\cos(x)$ and let $P_2(x)$ be the 2nd Taylor Polynomial for $f$ about $x_0=0$. (i) Show that the error $\mid f(x)-P_2(x) \mid$ is bounded by $\frac{\sqrt2e}{3}$ on $[0,1]$. (ii) Show ...
2
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1answer
34 views

Newton's method and error term analysis for f(x) = x - cos(x)

I'm taking a first course in Real Analysis and the instructor is using Arthur Mattuck's book. I'm struggling with a particular exercise: prove that the Newton's method of finding roots works for ...
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1answer
13 views

Newton method norm of error is proportional to norm of residual?

Let $F(x):\mathbb{R}^n\rightarrow \mathbb{R}^n$. Newton's method is: $x_{k+1} := x_k + d_k$, where $d_k$ is computed to satisfy $F'(x_k)d_k = -F(x_k)$. If the error at the current step is $e_k = x^* ...
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0answers
19 views

Discrete Fourier Transform of a vector

Find the Discrete Fourier Transform of the vector $[1,0,-1,0]^T$ We have not covered this in class. What is it that I'm trying to accomplish and how do I report an answer?
2
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1answer
40 views

How to use Runge-Kutta 4th order method without direct dependence between variables

Following equation shall be solved using Runge-Kutta method of 4th order: $$ \frac{\partial E(z,t)}{\partial z} = \frac{\partial P(t)}{\partial t} $$ $P(t)$ is given as an array, so that the ...
2
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1answer
35 views

Fitting a curve given points

My set-up is the following, I have two variables $N$ and $TTR$, and I have these points for each variable: $N$ = [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] $TTR$ = [0.818, ...
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0answers
13 views

Good reference for “solving equation $f(x)=0$ by homotopy and continuation methods”

I need a good reference for "solving equation $f(x)=0$ by homotopy and continuation methods". If $f:X\to Y$ is a continuous map between to linear space $X$ and $Y$, we want to find the roots of $f$. ...
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0answers
8 views

Banded symmetric Toeplitz linear system

What is the best (if such exists in terms of stability, efficiency etc.) matrix decomposition (or any method) for a banded, symmetric, indefinite Toeplitz linear system? Let's say, we have a linear ...
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0answers
29 views

Order of convergence, accuracy, and consistency multistep method

I am having trouble finding a clear cut definition for different order terminology for multistep methods. In particular, I see questions asking for order, order of consistency, order of convergence, ...
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0answers
27 views

Adequate software to plot grids, numerical stencils…

Do you know any software (online or not) easy to understand and appropriate to create this kind of plots? I would like the image to be vectorized, can write subindices, arrows, dots and so on to ...
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0answers
25 views

LTE for the Cahn Hilliard Equation

I'm trying to Calculate the LTE of a first order finite difference (central difference) scheme for the following 1D equation: $u_t = \frac{\partial^2}{\partial x^2}(u^3 - u - \frac{\partial^2 ...
0
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1answer
17 views

How to guess initial intervals for bisection method in order to reduce the no. of iterations?

SO, A function $f(x)$ is given to me and but the initial intervals are not given. I need to find the root of the equation using Bisection method. Sometimes when I randomly guess the initial interval ...
1
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1answer
23 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 = ...
0
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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} ...
3
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1answer
40 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 ...
1
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0answers
16 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 ...
1
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0answers
42 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
50 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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0answers
75 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 ...
2
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1answer
52 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
88 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 ...
2
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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 ...
2
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0answers
44 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
20 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
29 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$ ...
1
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0answers
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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0answers
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 ...
0
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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: $$ ...
1
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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 ...
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0answers
26 views

About horner's method

suppose $p(x)=a_0+a_1x+a_2x^2+...+a_nx^n$ (a polynomial with nth degree) how many multiplication is necessary to find $p''(x)$ with Horner's rule ? there are multiple choices in question : ...
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
28 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
73 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 ...
2
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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
20 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 ...