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

learn more… | top users | synonyms (2)

0
votes
1answer
20 views

Explicit Euler Method's Order

I am seeing everywhere that the order of the Explicit Euler Method is 2 but I can't prove it on my own. The textbook that I found the method says that the proof is very easy so it is up to the reader ...
0
votes
0answers
9 views

Stability of gradient flow $x'(t) = -\nabla E(x)$, $E(x)$ is convex function.

Let $E: R^d \to R$ be convex and continously differentiable. The IVP $x'(t) = -\nabla E(x), x(0) = x_0, t>0$ is called the gradient flow. Show that gradient flow has following properties a) ...
2
votes
7answers
139 views

Showing that $\sin(x) + x = 1$ has one, and only one, solution

Problem: Prove that the equation $$\sin(x) + x = 1$$ has one, and only one solution. Additionally, show that this solution exists on the interval $[0, \frac\pi2$]. Then solve the equation for x with ...
1
vote
2answers
72 views

What is the enclosed volume of an irregular cube given the x,y,z coordinates of the 8 corners?

I have the xyz coordinates of 8 points that forms an irregular-shaped cube. This is an animation, so the cube is undergoing periodic or cyclical shape-change. The co-planarism of any group or set ...
0
votes
1answer
26 views

Using Newton's Method to solve $f(x)=x^2-2bx+b^2-d^2=0$

What would be the Newton's method in the form $x_{k+1}=g(x_k)$ to solve the equation $$f(x)=x^2-2bx+b^2-d^2=0$$ in which both $b>0,d>0$ are parameters? I also need to show that $|g'(x)|\le 1/2$ ...
-1
votes
1answer
12 views

For what values of $a>0$ is the convergence faster/slower than first order? [on hold]

$g(x)=a+x-x^2$ and sequence $x_{n+1} = g(x_n)$ For what values of a does this converge faster/slower than first order. I have no idea where to start. Would someone be able to assist me please?
-1
votes
0answers
31 views

Solve equation $x^3=0$ using Newton's method [on hold]

Use Newton's method in Matlab or Octave to solve the equation $$f(x)=x^3=0$$ Characterize the convergence as linear or quadratic by tabulating the number of correct bits at each step of the iteration. ...
3
votes
1answer
22 views

Presicion check for the solution of equations in Numerical Analysis

In order to have precision of 5 decimal points in a Numerical Analysis method for the solution of an equation then: $$|x_{n+1}-\rho|\leq \frac{1}{2}\cdot 10^{5}$$ where $x_{n+1}$ the current ...
0
votes
0answers
7 views

Bounding the perturbation between eigenvectors

Can somebody explain this part of the proof of a deduction from the Davis-Kahan $\sin \theta$ theorem? I understand how to get from: $||P_{u_1} - P_{v_1}|| \le \epsilon$ to $||P_{u_1}v_1 - v_1|| \le ...
1
vote
2answers
3k views

Lagrange Interpolating Polynomials - Error Bound

Let $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$. Construct interpolation polynomials of degree at most one and at most two to approximate $f(1.4)$, and find an error bound for the ...
5
votes
2answers
113 views

Exact Expression for numerical Solution 0.9595767

I need you to do just what any math genuis in a shallow Hollywood movie does: looking at big tables of numbers and seeing exact structure! These $3 \times 3$ matrices are solutions to a well-posed ...
2
votes
2answers
76 views

Newton's method arctan

Why does it oscillate? I looked up the graph of it and I think it is convergent? And when the function is $0$, the solution is also 0. What is the difference of choosing diverse starting values? ...
1
vote
0answers
41 views

Deriving a New Iteration Method by Solving a Quadratic Equation

My Question: Derive a new iteration method for solving $f(x)=0$ by solving the quadratic equation $$f(x_k)+f'(x_k)(x-x_k)+\frac{1}{2}f''(x_k)(x-x_k)^2=0$$ Complete your algorithm by specifying ...
0
votes
1answer
81 views
1
vote
0answers
25 views

Bisection Method Numerical analysis Problem.

I am trying to solve the following $$(\frac{x}{2})^2-\sin x = 0$$ with initial starting points $a_o = 1.5, b_0 = 2$ and $n = 1(1)5$ using Bisection Method. From the little I have studied, I went ...
1
vote
1answer
52 views

RK4 wrongly predicts null solution

When solving the hydrogen radial Schroedinger equation (with $r > 0$ the radial coordinate) for angular momentum $L=1$ and the modified radial wave function $P(r)=rR(r)$, $P(r)$ satisfies: ...
1
vote
0answers
32 views

Newton's Method Solving an Equation

What would be the Newton's method in the form $x_{k+1}=g(x_k)$ to solve the equation $$f(x)=x^2-2bx+b^2-d^2=0$$ in which both $b>0,d>0$ are parameters? Additionally, I need to show that ...
2
votes
1answer
81 views

Differential equation: $\ddot{y}(x) + \alpha\dot{y}^2(x) + \beta y(x) = 0$

I am interested in finding an approximate solution for this differential equation, since the exact analytic solution seems to not exist. I tried with Mathematica and it spits out nothing. ...
0
votes
1answer
26 views

Simulate random variable with PDF $x^2+\frac4 3x$ on $[0,1]$

Consider $X$ a random variable with the following density function: $f(x) =$\begin{cases} 0, & \text{x ∉ [0,1]} \\ x^2+\frac4 3*x, & \text{x \in [0,1]} \end{cases} I need to write a ...
1
vote
1answer
49 views

Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
0
votes
2answers
86 views

Find to how many digits the value $\frac{355}{113}$ is an accurate approximation of $3.1415929204$.

Find to how many digits the value $\frac{355}{113}$ is an accurate approximation of $3.1415929204$. What i did was i computed it using a calculator and got the of $\frac{355}{113}$ to be ...
2
votes
1answer
52 views

Name and explanation of a Numerical Analysis method for solving systems of non-linear equations

In a non-english textbook of Numerical Analysis there is a method for solving systems of non-linear equations. But not only I can't understand how this method is used but I can't even found the name ...
1
vote
1answer
28 views

Stable algorithm for computation of $\Phi(20)$, when $\displaystyle \Phi(x)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{k!(2k+1)}$

Let $\displaystyle \Phi(x)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{k!(2k+1)}$, i.e $\Phi$ is the MacLaurin series of the function $\displaystyle ...
1
vote
0answers
22 views

Galerkin formulation of 1D linear Schrodinger equation

I am interested in the weak (Galerkin) formulation of the 1D Schrodinger equation: $i u_t−\beta u_{xx}=0$ and $u(x,0)=u_0(x)$. As usual, we do integration by parts which yields: $i (u_t,v)+\beta ...
0
votes
1answer
29 views

Runge-Kutta 4 failure?

Say we want to solve numerically $y'(x) = f(x) \cdot y$, with $y_0 = y(x=0) = 0$ and applying RK4 method with step $dx = h$: \begin{align} k_1 &= f(0) \cdot y(0) \cdot h = 0\\ k_2 &= f(0+h/2) ...
5
votes
3answers
3k views

rewriting to avoid catastrophic cancellation

why is rewriting $x^2 -y^2$ as $(x+y)(x-y)$ a way to avoid catastrophic cancellation? We are still doing $(x-y)$. Is it because the last operation in the second form is a multiplication?
1
vote
3answers
60 views

Gödel's Incompleteness Theorem and Numerical Analysis

I am no expert in Mathematical Logic so I can't really express my question formally but I do hope that it will make sense. As far as I know the implication of Gödel's incompleteness theorem for ...
1
vote
1answer
52 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
votes
1answer
28 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
-1
votes
0answers
60 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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
48 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, \\ ...
2
votes
1answer
840 views

Sum of Lagrange basis polynomials

Let $L_i(x)$ be Lagrange basis polynomials for $n+1$ points $(x_0,y_0),\ldots, (x_n,y_n)$. How do you prove that $\sum_{i=0}^n (x-x_i)^pL_i(x)=0$ for $p\leq n$?
0
votes
3answers
26 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 ...
0
votes
0answers
23 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 ...
1
vote
1answer
14 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
votes
3answers
115 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 ...
4
votes
2answers
465 views

Fixed-Point Iteration method unable to converge to any of a function's infinte roots

An equation is given to me which has to be solved by direct iteration method: $$\sin(x) = {x+1 \over x-1}$$ or $$f(x)=\sin(x)-{x+1 \over x-1} = 0$$ I follow the following procedure with reasons ...
0
votes
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 ...
4
votes
1answer
80 views

Is there a name for this piecewise cubic interpolation kernel

I went looking for a way to do piecewise cubic interpolation, like natural cubic splines, but: expressible as a convolution of data points with a piecewise cubic kernel; and still C2-continuous ...
0
votes
2answers
29 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, ...
2
votes
1answer
78 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 ...
2
votes
1answer
26 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
votes
0answers
78 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? [closed]

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
votes
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
votes
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 ...
1
vote
0answers
24 views

Relating to representation of real numbers. [closed]

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 ...
1
vote
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
votes
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 ...