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

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18 views

In a floating number system, are there always as many numbers between 0 and 1 as between 1 and $\infty$.

The question is as the following, where $\beta$ is the base, $t$ is precision (length of decimals), $e_{\min}$ is the minimum exponent, and $e_{\max}$ is the maximum exponent. I am not sure, ...
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0answers
13 views

One-dimensional deblurring

I just begun studying image deblurring on my own, and I have a question. Most books I found say that I can see the images as arrays, and that I can "vectorize" the arrays of the images by stacking the ...
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0answers
29 views

Show that $|g'(x)|\le\frac{1}{2}$ whenever $x^2>2|c|$

Consider the fixed point iteration $$ x_{n+1}=-b-\frac{c}{x_n}=g(x_n)$$ How would I show that $|g'(x)|\le\frac{1}{2}$ whenever $x^2>2|c|$?
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1answer
23 views

Fixed Point Iteration Proof

Given the fixed point iteration $$x_{n+1}=\frac{-x_n^2-c}{2b}$$ where $b$ and $c$ are fixed, $x_n\longrightarrow x$, what does $x$ solve? Additionally, what is the region for $(b,c)$ values where our ...
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1answer
26 views

Trapezoidal rule - truncation error

I am trying to prove that when solving numerically diff. eq.: $$ y'(t)=f(t,y(t)), \hspace{0.5cm} y(t_{0})=y_{0} $$ using trapezoidal rule, namely: $$ y_{n+1}=y_{n} + \frac{h}{2} \left( f(t_{n},y_{n}) ...
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1answer
32 views

Inner product vs. vector triad form

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
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1answer
34 views

Conceptual Differences among Galerkin Methods

I have a conceptual question about numerical methods for second-order elliptic partial differential equations. What is the difference among finite element, continuous finite element, discontinuous ...
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1answer
22 views

Calculation of coefficients of a function with respect to Legendre polynomials.

$$f(x) = \begin{cases} -1 &x \in [-1,0],\\ +1 &x \in (0,1]\\ \end{cases}$$ the formula for calculation of coefficients in terms of Legendre polynomial $L_k(x)$ is: $f_k= ...
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1answer
20 views

Numerically integration with a an infinite upper limit and non-zero lower limit

I have seen lots of quadrature formulas where we have definite limits or one of the limits is infinity and the other is zero. But what about the following case $$f(x) = \int_a^\infty e^{\frac{x}{t}} ...
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12 views

Gauss-Green cubature in 2d

Hello friends of maths, I've given an arbitrary polygonal cross section (in cartesian coordinates $y$ and $z$). On this cross section, there acts an arbitrary stress-field $\sigma = f(y,z)$ as ...
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33 views

Calculating Fast Fourier Transform from given set of data

I am trying to calculate the Fast Fourier Transform numerically from the given data : Given: f0 f1 f2 f3 f4 f5 f6 f7 1 2 3 4 4 3 2 1 I have to find the ...
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1answer
21 views

Symplectic integration of harmonic oscillator

I try to get numerical solution of ordinary harmonic oscillator with symplectic integrator. The problem is that what I obtain doesn't conserve energy (but symplectic integration should do). I ...
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2answers
65 views

Proving that $a\le \text{fl}\left(\frac{a+b}{2}\right)\le b$

Suppose that $a$ and $b$ are some floating point numbers such that $a\lt b$. How can I show that $$a\le \text{fl}\left(\frac{a+b}{2}\right)\le b$$ specifically in IEEE standard floating point ...
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2answers
19 views

Proof of a result in numerical analysis, error bound.

I would like to proove the Lemma 3.1. in this book. My attempt... I want to split the lemma into several parts. Part 1: $$\prod_{j=1}^{n} (1 + \epsilon_j) = 1 + \sum_{j=1}^n \epsilon_j + O(|u|) = 1 ...
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1answer
29 views

Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

I am really stuck on a homework problem, which boils down to the following: We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions ...
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1answer
43 views

Condition number for computing $x$?

The question is: Consider the linear system $\left( {\begin{array}{*{20}{c}} 1&\alpha \\ a&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = ...
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1answer
44 views

The fixed point iteration and find the converge interval

I've finished part a, which is quite easy. Can someone gimme some hints on part b and c? Thanks!
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1answer
38 views

Matrix-vector product of a banded matrix

Suppose $A\in\mathbb{R}^{n\times n}$ is a banded matrix, i.e., a matrix with all of its nonzero elements on the main diagonal, i.e., $\alpha_{i,i}\neq 0$, the first superdiagonal, i.e., ...
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45 views

Solving a system of nonlinear second-order differential equations with initial/boundary conditions.

I have developed a set of $n$ equations, $n$ variables for my dynamic system. The derivatives are second and first order in terms of $\theta$ (angle) of different components of the system (basically a ...
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1answer
57 views

The fixed point iteration part b

I've solved part a. And for part b, I solved $g'\left(x\right)^2$ and when c=0 or $x=-b/c$, we have the minimum, but according to the problem, we can't reach it. So the minimum occurs when ...
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0answers
19 views

the fixed point iteration. Find the rate of convergence

I got part a, but any suggestion how to solve b, c and d? Thanks!
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1answer
24 views

Backward Stability Lemma

Lemma-Let $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$ with components, $\xi_i$ and $\eta_i$, $1\leq i\leq n$, respectively, that are floating point numbers. Computing the inner product $x^Ty$ on a ...
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0answers
14 views

Find Complete Elliptical Integral K(k) and E(k) [on hold]

Please, I am trying to find total elliptical integral K(k) and E(k). I am trying to solve last equation Elliptical Integral
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0answers
10 views

Wide stencil for Second Derivative in finite difference - stability in maximum norm

I am given the problem $-u'' + a*u = f$. I already derived a 5-point wide stencil for finite difference with fourth order convergence, and then the matrix $A$ for the problem has a stencil like this: ...
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11 views

Which numerical method gives the most accurate solutions of Helmholtz equation for arbitrary domains?

There are many numerical methods for the solutions of PDE's such as FDM, FEM, SEM, Meshfree methods etc. I'm wondering which method gives the most accurate Dirichlet eigenvalues (and corresponding ...
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1answer
22 views

Accelerating linear solve in MATLAB for a specific type of matrices

Inside a DG solver (so far 1D) I need to solve a linear system of equations multiple times. The order of the system is rather small ($N=10..20$). I need to solve the system $Ax=b$, where $A$ is the ...
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1answer
18 views

Relation between absolute error and iterative approximation

I'm working on a university project studying numerical analysis and have hit a small snag. I have several theorems that deal with convergence of iterative procedures with respect to the absolute error ...
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0answers
18 views

A problem on Newton-Raphson method

The function $f(x)=0$ has a simple root in the interval $(1,2)$. The function $f(x)$ is such that $|f(x)|>3$ and $|f''(x)|\leq 4$ for all $x\in (1,2)$. Assuming that the Newton-Raphson method ...
1
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1answer
71 views

Numerical Integration Error Bound

I would like to use numerical integration to approximate $\int_{0}^{1} f(x) dx$ where $f(x) = \frac{1}{\sqrt{x}}$. But I can't figure out how to get an error bound. For example, if I use trapezoidal ...
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1answer
39 views

Trapezoidal rule or similar for integration over sphere (spherical triangle).

I would like to calculate numerically the integral of the function defined on the sphere. Moreover, the sphere is completely covered by non-overlapping spherical triangles, I need the integral to be ...
0
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1answer
29 views

Which is more appropriate here: multiplicative or additive error?

I am a beginner in numerical analysis and i have the following question at hand, but I am not being able to draw a logical conclusion: please help.. For estimating numerical errors in the process of ...
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0answers
65 views

The fixed point iteration to solve x

For part a, can I just substitute both $x_n$ and $x_{n+1}$ with x and then solve the equation? And any thoughts on part b? Thank you guys!
3
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1answer
56 views

Maximum accuracy in IEEE standard floating point arithmetic using bisection method

Just wanna make sure that I didn't make any mistakes. I use the bisection method to calculate $P_n$ and find out a pattern, which is $P_n= \left(-1\right)^{n+1}2^{-n}$. So the largest number that n ...
0
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2answers
65 views

Floating point arithmetic in IEEE standard floating point arithmetic

If real numbers a and b satisfy $a<b$, is it necessary for $fl\left(a\right)<fl\left(b\right)$ to be true? I think it is true because neither rounding the numbers or chopping them would change ...
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0answers
54 views

Sketching Region of a Fixed Point Iteration [on hold]

Suppose we have the fixed point iteration of $$x_{n+1}=\frac{-x_n^2-c}{2b}$$ in which $b, c$ are some fixed real parameters. Now, when $x_n\to x$, then what does $x$ yield? How would I sketch the ...
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1answer
40 views

For which arguments in the range $0\le x\le \pi/4$ will $\cos x=(1-\sin^2x)^{1/2}$ fail to give good accuracy?

The question is In floating point system, consider using the trigonometric identity $\sin^2x+\cos^2x=1$ to compute $\cos x=(1-\sin^2x)^{1/2}$. For which arguments in the range $0\le x\le \pi/4$ ...
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2answers
65 views

Why it is more accurate to evaluate $x^2-y^2$ as $(x+y)(x-y)$ in floating point system?

The expression $x^2-y^2$ exhibits catastrophic cancellation if $|x|\approx|y|$. Why it is more accurate to evaluate as $(x+y)(x-y)$ in floating point system (like IEEE 754)? I see this is intuitively ...
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1answer
20 views

Euler's method, non-existent $y(x)$ function

I'm trying to approximate the solution to this equation using the Euler's method: $$y'(x) = 3-\tan(x) \cdot y(x), y(2) = 4$$ When solving for the step of $0.2$, I don't know what to calculate when ...
4
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1answer
93 views

The accuracy from left to right and that from right to left of the floating point arithmetic sums

Question 1 Show that floating point arithmetic sums $$s_n=\sum_{k=1}^n\frac{1}{k^2} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\dotsb+\frac{1}{n^2}$$ with accuracy $\mathcal O(n)\epsilon$ from left to ...
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2answers
24 views

Step size in Euler's forward method

I came across the following question. Kindly let me know if there is any generic solution to this type of question. $$ \frac{d^2 y}{dt^2} + 3 \frac{dy}{dt} + 2y = f(t) $$ Where $f(t)$ is an impulse ...
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0answers
16 views

Runge-Kutta 4 in polar coordinates

How is the Runge-Kutta method implemented on this differential equation: $$ \frac{d^2 \theta}{dt} = -\frac{g}{l} \theta $$ (pendulum motion) which is in polar coordinates? Let: $c = \frac{g}{l}$ ...
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1answer
17 views

Numerically solving the equation of a simple pendulum with Runge-Kutta.

I am trying to solve the equation $\dfrac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \dfrac{g}{L} \sin{\theta} = 0$ using Runge-Kutta. I have alread split it into the following equations ...
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24 views

Determining the minimal number of terms to use in a sum to approximate a number given a tolerance

In page 33-34 of Numerical Analysis by Burden & Faires an algorithm was given to compute the minimal value of $N$ for which $$|\ln{1.5}-P_N(1.5)|<10^{-5}\tag{1}$$,where ...
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1answer
25 views

Velocity Verlet method: How to calculate acceleration

The velocity Verlet method algorithm is as follows: Calculate: $$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t)\, \Delta t+\tfrac12 \,\vec{a}(t)\,\Delta t^2$$ Derive: $\vec{a}(t + \Delta t)$ from ...
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1answer
45 views

In common tongue, what is the differences between sparse and dense matrices?

What are the differences with sparse and dense matrices in practice, so as to offer some insight to new learners on a more intuitive level. Obviously everyone knows about the dictionary definition of ...
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0answers
25 views

Numerical integration of product of two functions

My question is in the context of spectroscopy/determination of excited state lifetimes. I have to numerically integrate the following integral using MS Excel $$ \int_0^{\infty} F(\lambda) \epsilon ...
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4answers
223 views

Which is bigger: $(n!)^{n!}$ or $(n^{n})!$? [closed]

To be honest I haven't spent a whole lot of time thinking about this other than the drive back home, and I won't have much time to think about it for a while due to shit-happening. So i thought I'd ...
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2answers
15 views

Is there a simple formula for the volume of an oblique triangular pyramid?

I have the xyz coordinates for 4 points in space that are not co-planar. These points form a triangular pyramid. Taking any 3 points as the base, the 4'th point will practically never be over the ...
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0answers
31 views

Error analysis for Runge Kutta, how to take Big O of 2 variables?

For the standard 4th order Runge Kutta: where the system is assumed to be smooth (so that the RHS has no discontinuous points) $\mathbf{y'} = \mathbf{F}(t,\mathbf{y})$ $\mathbf{y(t_0)} = ...
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1answer
21 views

Understanding the definition of an Interface

Im starting to learn about modeling of moving interfaces and am feeling daft about the basic definition itself: Given an $n$ dimensional space $\Omega$ an interface $\Gamma$ is a co-dimension 1 ...