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

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14 views

Indicate whether True or False

a) The derivative of the cubic spline interpolant at the nodes agree with those of the function. b) If $f$ is a continuous function with $f(a) f(b) > 0$, the $f$ has no roots in $(a,b)$. c) ...
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1answer
18 views

numerical solution of a transcendental equation

I need some help with a program in MATHEMATICA or MAPPLE, that solves the following transcendental equation: $$\alpha+2x+2\sqrt{\beta+\alpha ...
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1answer
35 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
0
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1answer
37 views

Singular solutions of a system of nonlinear 2nd order ODEs

I'm faced with the following nonlinear 2nd order system of ODEs: $$ \phi''(r)+\frac{4r^3-1}{r^4-r}\phi'(r)+\frac{r^2 h(r)^2+2r(r^3-1)}{(r^3-1)^2}\phi(r)=0, \\ ...
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3answers
24 views

How do we find more appropriate constants for expansions of functions?

We all knonw that the expansion of $e^x$ is $$1+x+x^2/2+...$$. But what if I want to find more approximate expansion of $e^x$. I try that $$e^x-1-c_0(x)+(c_0+c_1)(x^2/2)-(c_0+c_1+c_2)(x^3/3)=0$$ and ...
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0answers
18 views

Calculating B-Splines and dimension of spline space

I've got the following assignment: Let $S$ be the space of piecewise polynomials of degree $3$ on the intervall $[-1;1]$ with knots $x_i = -1+\frac{i}{2}, 0 \leq i \leq 4$. (a) Calculate a basis of ...
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0answers
29 views

Solving equation using Newton's Method

Use Newton's method to solve the equation $$f(x)=\frac{1}{x}+\ln{x}-2=0$$ for $x>0$. Characterize the convergence as linear or quadratic by tabulating the number of correct bits at each step of the ...
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1answer
12 views

Conditioning in regards to matrix vector product

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 ...
3
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3answers
104 views

How to find the root of a polynomial function closest to the initial guess?

I need some easy to implement and fast numerical method that finds the root of a nonlinear function (a polynomial in my case) closest to my initial guess. If I know that there is one root ...
0
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2answers
25 views

find the limit of the first one

To prove the first one, can I just say that $\lim_{n\to\infty} {a^n/n^p}$= $(\lim_{n\to\infty} a^n)(\lim_{n\to\infty} 1/n^p)$ and when the absolute value of $a$ is less than or equal to one, ...
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0answers
32 views

let $\alpha \in \Bbb{R} $ and $\cos(\alpha \pi) = \frac{1}{3}$, prove $\alpha $ is irrational [duplicate]

Let $\alpha \in \Bbb{R} $ and $\cos(\alpha \pi) = \dfrac{1}{3}$, prove $\alpha$ is irrational. (Proof by contradiction) If we consider $\cos \left(\dfrac{m\pi}{n} \right)=\cos \left(\dfrac{ m\pi ...
0
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0answers
77 views

I am a Math Hobbyist. I have made some simple discoveries in Math. How do I share it with the Math community out there? [on hold]

I am a Computer Engineering graduate and have taken many courses in Math of course. While I was in the University, I got myself lost in the world of mathematics and I discovered stuff that I felt ...
0
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0answers
13 views

Statistical calculation for neural firing rates with negative rate on numerical simulation

I am now working on a biological neural network simulation (NEST-Simulator) project with a problem of calculating firing rates. Background: The data set as result of simulation is a set of events in ...
0
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1answer
14 views

Conjugate gradient projection

Let $V$ be a collectino of the search direction for the conjugate gradient applied on a quadractic minimisation problem. As a proof of orthogonality in conjugate gradient: $$ V^T V = I $$ Now ...
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0answers
24 views

Relating to representation of real numbers. [on hold]

Can someone tell me which representation is better for representing real numbers: fixed point representation or floating point representation? If the answer is circumstance dependent, please specify ...
2
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1answer
72 views

Does a convex hull solution in 3 dimensions result in a minimum-area or maximum-volume solution?

The wikipedia entry for convex hull shows a 2-d example of a random set of points on x-y plane, and the "elastic band" solution that bounds the points with the convex hull solution. The definition of ...
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0answers
31 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, ...
0
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0answers
16 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 ...
0
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0answers
32 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 $x^2>2|c|$?
2
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1answer
27 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
28 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}) ...
1
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1answer
44 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 ...
2
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1answer
41 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 ...
0
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1answer
37 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= ...
1
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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}} ...
2
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0answers
13 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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0answers
34 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 ...
2
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1answer
25 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 ...
0
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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
22 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 ...
0
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1answer
31 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 ...
2
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1answer
44 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) = ...
0
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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
50 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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0answers
47 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 ...
1
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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!
0
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1answer
31 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
15 views

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

Please, I am trying to find total elliptical integral K(k) and E(k). I am trying to solve last equation Elliptical Integral
0
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0answers
11 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: ...
0
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0answers
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
19 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
73 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 ...
1
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1answer
40 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 ...
0
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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
57 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 ...