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

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46 views

Numerical presentation of a term

Hello I have the following task: Let $$ y = \sin(x + \delta) -\ sin (x)$$ and $$\delta > 0$$ is very small. Write down a mathematical equivalent representation of this term which is stable. I'm ...
3
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1answer
31 views

Show that assumptions of theorem hold, determine the solution

Consider the initial value problem $$\left\{\begin{matrix} y'(t)=\sqrt{|y|}, 0 \leq t \leq 2\\ y(0)=1 \end{matrix}\right. \tag 1$$ Show that for this problem the assumptions of the following ...
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1answer
175 views

the Chord Method does not appear to be converging under the same cond as Newton's Method?

The Chord Method is: $x^{(k+1)} = x^{(k)} - {g(x^{(k)}) \over g'(x^{(0)})}$ The question is to compute the cube root of 2, using the Chord Method. Carry out the first few iterations, using x(0) = 0....
2
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1answer
65 views

A Variant of Gradient Descent

Suppose I have some objective function $f(\beta)$ which I would like to minimize for $\beta$. A standard gradient descent would be $\beta^{(t+1)}=\beta^{(t)}-\alpha \nabla f(\beta^{(t)})$, where $\...
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1answer
58 views

Evaluating a cumulative distribution function from normal distribution

How one can prove by using only pencil and paper that $$0.99815<\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{2.92}e^{-x^2/2}dx<0.99825?$$ I think there is a mistake in my book which says that $$\frac{1}{...
2
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0answers
45 views

Existence and uniqueness of solution of the ODE

Consider the initial value problem $(1)\left\{\begin{matrix} y'(t)=y^2 &, 0 \leq t \leq 2 \\ y(0)=1 & \end{matrix}\right.$. Verify that the following theorem: "Let $c>0$ and $f \in C([a,...
2
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3answers
238 views

Find the approximations to within $10^{-4 }$ to all the real zeros of the following polynomial using Newton's method.

We have $P(x)=x^3-2x^2-5$. I know the formula of Newthon's method. That is given here. The problem is, how do I find the approximations to within $10^{-4}$ to all the real zeros of the following ...
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1answer
38 views

Function doesn't satisfy the local condition of Lipschitz at intervals that contain $0$

The local Lipschitz criterion is the following: Let $c>0$ and $f \in C([a,b] \times [y_0-c, y_0+c])$. If $f$ satisfies in $[a,b] \times [y_0-c,y_0+c]$ the Lipschitz criterion as for $y$, ...
7
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1answer
142 views

Prove or disprove - Newton's method convergence in higher dimensions

It's not an exercise for uni or anything like that, just something that's been bothering me a bit and I can't seem to find useful information on the web on the matter. When talking about real valued ...
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0answers
41 views

Creating FEM mesh for image region — what is the most suitable shape function?

I wish to create a FEM mesh to solve an inverse elasticity problem, for an irregular domain. This domain is given by a medical image, so it is discretised and each square on the grid has one scalar ...
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85 views

Gram-Schmidt to prove recurrence relation of orthonormal polynomials

I have been told to apply Gram-Schmidt to $x\pi_k(x)$ to show that $$ \pi_n(x)=(a_nx + b_n)\pi_{n-1}(x)+c_n\pi_{n-2}(x) $$ with $n≥2$, $a_n$, $b_n$, and $c_n$ all constants that depend on $w(x)$. ...
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2answers
81 views

How to compute orthonormal polynomials from weight function?

I have a weight function $w(x)=e^{-x}$ with $x$ from $0$ (inclusive) to infinity. How would I compute the first four orthonormal polynomials with respect to this weight function?
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1answer
31 views

Using weight function to construct a minimum

I have a continuous, bounded function $f(x)$ and a weight function $w(x)$ on the same interval. If $$\{\phi_i(x)\}^N_{i=0}$$ is a family of basis functions for a linear space and $N>0$, I need to ...
2
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1answer
23 views

LU Decopmostions with block

So both $A_{11}$ and $\hat{A_{22}}$ have $LU$ decompositions say $A_{11}=L_{1}U_{1}$ and $\hat{A_{22}}=L_{2}U_{2}$. Show that $ \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}...
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0answers
192 views

Finding Fourier series coefficients numerically

Given a known function $f$, I am wondering how fast (depending on $n$) we can numerically approximate the Fourier coefficients $\int_0^1 f(x) e^{2\pi i n x} \, \mathrm{d}x$, either for fixed $n$ or ...
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0answers
53 views

polynomials to Chebyshev polynomials

I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined in the image. Anyone know? The link to the paper is: The actual paper, I'm referring to
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2answers
62 views

Numerical solution of the Volterra equation with an exponential factor

Given : $$u(x)=x+2 \int_0^x e^{x-t}u(t)dt$$ Solve the Volterra Equation numerically using Trapezoidal Rule in $(0,5)$ choosing $n=8$ and compare with the exact values. The Exact Solution I ...
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1answer
86 views

How to obtain a convergent solution iteratively for a linear system of equations?

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} &...
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1answer
35 views

Theory of computation

For any language $A$, $B$ and $C$ such that $A\subseteq B \subseteq C$, if both $A$ and $C$ are decidable, then $B$ is decidable. True or False? How can I find this?
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0answers
65 views

Solution to the heat equation using a finite difference scheme

I have used a difference scheme and Fourier Analysis to find an expression for the solution to the heat Equation. My problem is plotting my solution. $w_{k,j,m}=(1-\Delta t\mu_k)^msin(k\pi x_j)$ $\...
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1answer
84 views

What is the logic behind Jacobi iterative method?

The book I follow and on net also, all that I can find is the algorithm to find the solution, but I don't quite understand the physical significance or logic behind the algorithm. Can someone please ...
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0answers
87 views

Irregular grid for finite differences PDE solution

I got a project in a class I'm taking. In this project, I need to solve fluid dynamics and heat equations. Up to here, the problem is not so complicated, however the chamber shape is the problem: as ...
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1answer
41 views

How to show the relative error of $ \frac{x_A}{y_A}$

First, this is how the relative error of $x_Ay_A$ (approximated errors) is computed as compared to $x_Ty_T$ (true errors) - $ \displaystyle Rel(x_Ay_A) = \frac{x_Ty_T - x_Ay_A}{x_Ty_T}$, Letting $...
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1answer
71 views

deriving an integral quadrature rule on a triangle

I'm trying to look for references on this but I've not found any. I'm probably using the wrong keywords ... Let's suppose that our domain of integration $\Omega$ is the triangle in $R^2$ with a ...
2
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0answers
129 views

Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
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1answer
41 views

Does the following limit exist as a result of the bisection method?

Does the following limit exist as a result of the bisection method? $$\lim_{n\rightarrow\infty}\dfrac{|r-c_{n+1}|}{|r-c_{n}|}$$ where $r$ is the root as a result of the method, and $c_n=\dfrac{a_n+...
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2answers
4k 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 ...
3
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1answer
147 views

Numerical computation of unlimited small Julia set details

I've read the claim of a fractal image application to be able to show infinite levels of zoom for Julia sets for the classic iteration formula $z_{i+1}:=z_i^2+c$. The application has a realtime ...
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1answer
45 views

Find a polynomial

I have to find a polynomial with the following characteristics for a problem. Find a polynomial $p(x)$ such that $$p(-1)=p'(-1)=p''(-1)=p(1)=p'(1)=p''(1)=0$$ I know and understand the process of ...
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0answers
23 views

stability assessment

I have been asked to assess the stability of my numerical solutions to two different sets of transient differential equations that govern the same phenomena. I am not sure how I can assess and compare ...
5
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1answer
213 views

Convergence of Conjugate Gradient Method for Positive Semi-Definite Matrix

Let $A\in\mathbb{R}^{N\times N}$ be a positive semi-definite matrix, given $b\in\mbox{Col}\left(A\right)$ we want to solve the equation system $Ax=b$ . To add some notation, we define $\left<u,...
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1answer
59 views

Approximation of Derivative operator using finite difference

The derivative operator $D$ is given by $\frac{\ln(\varepsilon)}{h}$, where $\varepsilon$ is the shift operator. Using the Taylor Expansion and the relation $\varepsilon = I + \Delta _{+}$, the ...
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1answer
125 views

Find Number of Iterations of Euler's Method in order to approximate $\pi$

I am given the function $x(t)=4 \arctan t$ and told that routine computations will show that $x(0)=0$ and $x(1)=\pi$. I must determine a differential equation for $x(t)$ of the form $x'(t)=f(t,x)$. ...
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0answers
47 views

Newton Interpolation

Let $K$ be a field. Let $x_0,\dots,x_m$ be destinct elements of $K$. Let $\omega_n$ be the Polynomial $(X-x_0)\cdots(X-x_{n-1})$. Given a function $f\colon K\longrightarrow K$, let $$a_n=\sum_{j=0}^n\...
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0answers
43 views

Estimate uncertainty of a function

I have a question about estimate the uncertainty of a function. I have a variable $x$. I assume that I can predict the variable $x$ is followed by a function $f(x)$ such as $$f(x)=\exp(-x^2)$$ Now, ...
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1answer
40 views

Approximation of a series containing Bessel functions

I have this series: $$\displaystyle S=\sum_{k=0}^N\left(J_k(x)-J_k(y)\right)$$ where: $J_k(\dot{})$ is the Bessel function of order $k$ with $x\in\mathbb{R}$ and $y\in\mathbb{R}$. I have to calculate ...
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0answers
65 views

What are some really cool problems that involve “least squares and Eigenvalue problems”?

I am required to find a research topic in this domain, so I'm really interested in finding out what kind of problems are covered in this domain, and how others are using these techniques to solve them....
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4answers
314 views

Calculating of least significant digit of an expression

I want to calculate te least significant digit (1s place) of following: $ 1+2^{1393} + 3^{1393}+4^{1393} $ How we can calculate this? It's very hard for me!
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1answer
89 views

How to solve this convex resource allocation problem numerically? CVX doesn't work.

I got a resource allocation problem as follows: \begin{eqnarray} \min &\sum_{i=1}^M \frac{1}{1 + \text{exp}(C_i + \frac{r_i}{1+r_i})} ;\\ &\sum_{i=1}^M r_i \le R;\\ &r_i \ge 0 \,\,\,\...
0
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1answer
104 views

Proposing a numerically sound algorithm for a function as it approaches 0

Suppose I have the function $f(x) = 1 + x - \sin(x)/(x*e^{x})$ I am tasked with proposing a numerically sound algorithm for evaluating f(x) as it approaches 0. My initial thought would be to take ...
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2answers
169 views

Using Newtons Method to approximate intersection of Graphs

Using Newtons Method, I am to try to approximate the point R where the graphs of $y = e^{-x}$ and $y = ln(1 + x)$ First I was supposed to propose a function, f(x) for which the method can be used, ...
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2answers
94 views

Numerical solution to ${d\over dt} \nabla^2 p = {dp\over dx}$

I need help numerically solving $$\frac{\partial \nabla^2 p} {\partial t} = \frac{\partial p} {\partial x}$$ I know that to solve \begin{equation} \frac{\partial p} {\partial t} = \frac{\...
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0answers
159 views

Composite trapezoidal rule and Simpson rule question?

We have the following integral $$\int_{1}^{3} \frac x{x^2+4}\; dx$$ and $n=6$. I have to approximate it using composite trapezoidal rule and then composite simpson rule. Using trapezoidal rule: $$\...
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1answer
231 views

Three point formula exercise question

Using the table in the figure and the three point formula find the approximate values of the derivative required f'(1.2).Also calculate Ea and Ev ( Actual error and error bound) We notice that h=0....
5
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1answer
145 views

Numerically stable method for angle between 3D vectors

I'm looking for a numerically stable method for computing the angle between two 3D vectors. Which of the following methods ought to be preferred? Method 1: $$ u\times v = ||u||~||v|| \sin(\theta) \...
0
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1answer
71 views

Harmonic Oscillation using Gaussian Quadrature [closed]

Assume that the potential is symmetric with respect to zero and the system has amplitude $a$ suppose that the potential $V(x)=x^4$ and the mass of the particle is $m=1.$ Write a java function that ...
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139 views

Approximate Laplace Operator with Central Difference in Polar Coordinates

I'm trying to approximate the Laplace operator in polar coordinates with the central difference quotient and I know how to do this in cartesian coordinates, but with polar coordinates I just feel ...
2
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0answers
36 views

Good method for finding roots that *usually* fall within an interval?

I've been using Brent's method to find the roots of a monotonic, nonlinear, non-differentiable function. The roots often fall within a known interval, but Brent's method fails if they occasionally ...
0
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1answer
82 views

Intro to Numerical Methods

What is the gap between $2$ and the next larger double-precision number? I understand to do this it is $2^{-52} \cdot 2^1 = 2^{-51}$ I'm having a little more difficulty with this one though. What ...
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2answers
51 views

Integral of $\frac{r^{2}}{(2 \pi \sigma^{2})^{\frac{3}{2}}}\exp\left(-\frac{1}{2}(\frac{r}{\sigma})^{2}\right)$

I'm trying to solve the integral of the following function in a sphere of radius $5\sigma$ the function is: $$f(r) = \frac{r^{2}}{(2 \pi \sigma^{2})^{\frac{3}{2}}}\exp\left(-\frac{1}{2}(\frac{r}{\...