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

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Absolute error and accurate digits question? [duplicate]

We have x=0.2234 and Δx =0.2*10^-3.Find how many accurate digits does the number have?So we know that $\Delta x≤(1/2) 10^{-n}$ where $n$ is the number of the accurate digits. Now I just have to ...
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1answer
25 views

Accuracy of the result question short?

We have 1/(4,5) . When we do the divison,what accuracy does the result have?
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53 views

Euler-Maclaurin Formula

I need a simple proof for the identity $$\int_a^{b} f(x)dx=(b-a)\frac{f(a)+f(b)}{2} + (b-a)^2\frac{f'(a)-f'(b)}{12}+O((b-a)⁴)$$ I do not want to use Bernoulli's numbers, as this is not the general ...
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2answers
38 views

We have x=0.2234 and Δx =0.2*10^-3.Find how many accurate digits does the number have?

So we know that $Δx≤(1/2)*10^{-n}$ where n is the number of the accurate digits. Now I just have to replace $Δx =0.2*10^-3$=$(1/2)*10^{-n}$ and find n. But why have I been given x=0,2234 since I dont ...
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1answer
84 views

Trapezoid Rule to Simpson's Rule Extrapolation

I need to show that one extrapolation of the trapezoid rule leads to Simpson's rule. I've looked through the other posts on ME, specifically the post with the same title, and this for help, but I ...
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26 views

Growth factor for Gaussian Elimination WITHOUT partial pivoting

A = $\begin{bmatrix} 1 & 7 & -11 \\ 4 & 29 & -50 \\ 6 & 49 & -107 \end{bmatrix}$ , after computing the LU factorization without partial pivoting I have L = ...
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1answer
118 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
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1answer
55 views

Modified Composite Quadrature formula

I'm trying to produce a modified composite trapezoid quadrature formula for $\int_a^b f(x)\ dx$ based on the Euler-MacLaurin formula. I know that it should involve $f'(a), f'(b), f'''(a)$ and ...
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1answer
25 views

Confirming an $LDL^t$ factorization

Hoping someone can confirm my work here. I'm trying to find the $LDL^t$ factorization (http://www.mathworks.com/help/dsp/ref/ldlfactorization.html) of the matrix $$ A = \left(\begin{array}{ccc} 2 ...
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2answers
48 views

Use bisection method to find the root of $2x(1-x^{2}+x)\ln(x)=x^{2}-1$ on [0,1]

I have to use the bisection method to find the root of $2x(1-x^{2}+x)\ln(x)=x^{2}-1$ on $[0,1]$. However I simplify I get, $$x^{2x(1-x^{2}+x)}-e^{x^{2}-1}=0$$ Which is undefined at $0$. How would ...
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0answers
74 views

Numeric solution of third order ODE

I need to solve the following third order (non-linear) ODE by numerical methods: \begin{equation}\tag{1} h^{3} \dfrac{d^3 h}{d x^3} = h-1. \end{equation} By assumption, the solution should approach $ ...
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1answer
103 views

(A + D)x = b … efficiently!

let say we have $A$ to be symmetric positive-definite (SPD), moreover block tridiagonal Toeplitz matrix and $D$ is block diagonal SPD (both with full rank). Let say we know everything about $A$ and ...
2
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1answer
40 views

how to find the distinct eigenvectors from a repeated eigenvalue

As the title, how to find the distinct eigenvectors of an certain eigenvalue if the algebraic multiplicity of that eigenvalue is not 1? The target matrix is real and symmetric. I know these ...
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1answer
36 views
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1answer
31 views

Power Iteration method for eigenvalues - Show the error is bound

Let $A \in $Sym$_{n}(\mathbb R)$ with eigenvalues $\lambda_i$ such that $|\lambda_1| > |\lambda_2| \geq |\lambda_3 |\geq ... \geq |\lambda_n|$ We define the following process as "Power Iteration": ...
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2answers
182 views

$f(x)=1/(1+x^2)$. Lagrange polynomials do not always converge. why?

Let $f(x) = \frac{1}{1+x^2}$. Error of Interpolation with Lagrange polynomials for $n+1$ points is given by $$ e(x)=f(x)-P_n(x)=\frac{f^{(n+1)}(\eta_x)}{(n+1)!}\prod_{i=0}^n (x-x_i) $$ Carl Runge ...
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1answer
40 views

Quadrature Rule “is exact for polynomials of degree n”

Could someone kindly explain what "a quadrature rule is exact for polynomials of degree n" means? Here is what I understand about numerical (Newton-Cotes) quadrature rules: Suppose we want to ...
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2answers
56 views

Least Squares Problem - Show that $F(x) = (b - Ax)^T(b - Ax) + \alpha x^T x $…

Consider the function $$F(x) = (b - Ax)^T(b - Ax) + \alpha x^T x $$ where $A$ is a real $ m \times n$ matrix and $\alpha$ is a positive real number. We want the minimum point of $F$ for given $A$, ...
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0answers
38 views

Upper Hessenberg Form

I am given a matrix. I would like to reduce it to its upper Hessenberg Form. We are discussing eigenvalue computations in Numerical Analysis and the textbook just gives the algorithm for it without an ...
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0answers
28 views

Numerical Integration of Highly Oscillatory Integral with Misbehaving Derivatives

I'm attempting to numerically handle an equation of the following form: \begin{equation*}f: x \rightarrow \int_{0.00001}^{2} d\omega e^{i \omega x} f(\omega)\end{equation*} where $f(\omega) = ...
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1answer
12 views

What is the recomputation of residuals in the deepest descent method?

From Jonathan Richard Shewchuk: By using equation (13), we have lost track of $x_(i)$ in the process of iteration. How is it possible to recompute the correct residual using equation (10) ever ...
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1answer
48 views

Necessary and Sufficient conditions for convergence of matrix iterations

I need some help figuring out how to go about the iteration part of the problem...I don't really know where to start. If someone can please help take me through it that would be greatly appreciated. ...
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3answers
42 views

Using Newton's Method in Backward Euler Method

I'm currently looking at this example problem from my course notes (introductory numerical analysis): and am a bit confused about how to write the equation for Newton's method. From its general ...
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25 views

nonlinear question

For a function (1-Dimension) $u(x)=\sum_{i=1}^4H_i(x)u_i $ where $H_i$ is the interpolation function (Hermite interpolation basis functions), $u_i$ be the nodal value. now, for the nonlinear term ...
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1answer
25 views

Estimating rates of convergence

If I have a set of data points obtained from a numerical approximation say 15.3828 15.2458 15.2095 15.2003 how can I estimate the rate of convergence?
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1answer
51 views

Fourier Series trouble

"For $f(x) = x^2$ on the interval $[-1,1]$ with period $2$, determine the Fourier series. Show that $\pi^2 / 6 = \sum_{n=1}^{\infty}(1/n^2)$". How is the first part of this exercise related to the ...
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27 views

Finite Differences and Scale Invariance

The semilinear heat equations $u_t = u_{xx} + u^p$ is invariant under the one parameter family of scalings $ t \to \lambda t, \quad x \to \lambda^{1/2}x, \quad u \to \lambda^{-1/(p-1)}u$. When we ...
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0answers
43 views

How to draw a diagram for the following PDE

Subject: Partial Differential Equations. Here are the details of the question: $$ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0 $$ for $0 < x < 3, 0 < y < 1$ ...
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1answer
29 views

What does “Formulate the system of equations for a finite difference discretisation of the problem” mean?

Subject: Partial Differential Equations. Here are the details of the question: $$ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0 $$ for $0 < x < 3, 0 < y < 1$ ...
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1answer
27 views

Least Squares Method Confusion

I'm learning about the Least Squares method. An exercise I am doing is "Find the constant c that makes the expression $$\int_{0}^{1} (e^x - cx)^2 dx$$ a minimum " Though, i'm not sure how to ...
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31 views

Split Step Fourier Algorithm

Consider the NLSE (Nonlinear Schroedinger equation) that can be written as the following partial differential equation: $$ \frac{\partial{A}}{\partial{z}}=({\cal{L+N}})A\quad\quad(1) $$ where $A: ...
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0answers
23 views

numerically solve quadratic air drag in xy-plane

I am trying to find a reference on solving for the position of a point mass as a function time, subject to air drag( quadratic term only) in both the x and y directions. The equations that describe ...
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1answer
74 views

Approximating $\sqrt{101}$ using Taylor series methods

I'm trying to approximate $\sqrt{101}$ using the Taylor series for the function $f(x)=\sqrt{x}$ centered at the point $x=100$. I need to obtain an approximation that is within $0.01$ of the correct ...
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1answer
51 views

How does LU decomposition work?

I'm interested in the algorithm of LU decomposition in order to solve a LSE like $Ax=b$, where $A$ is a square matrix. My question is: When I compute $PA=LU$ do I also need to interchange rows in $L$ ...
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27 views

Generalized Logarithmic Integral

Euler's logarithmic integral (of particular application in the Prime Number Theorem, for instance) is of the form \begin{equation*} \text{li}(x) := \int_0^{x} \frac{dt}{\log t} \end{equation*} and it ...
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1answer
23 views

Backward Stability

For $x$ close to $0$ the computation $y=\log(1-x)/x$ is numerically unstable but computing via $y=\log(1-x)/(1-(1-x))$ is not. This question comes from ...
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1answer
18 views

Calculating the limit of an analytic function which gives a log answer.

I am trying to read through a paper and have gotten stuck at the following calculation several times. I've left if for a few days and came back to try it again four different times, but still no luck. ...
3
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1answer
112 views

Fibonacci Search Algorithm

Can someone show me an example of using this method for 'find the minimum of $$F(x) = x^2 - 6x + 2 \; \text{ on } [0,10] $$' ? I'm trying to follow the algorithm detailed above, but I don't ...
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1answer
16 views

Numerically optimising a sequence of matrix multiplications

I am trying to set up an optimisation problem and solving it numerically. I am still formalizing it and unsure what is the best way to solve it. It seems like a common problem, and im sure people have ...
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1answer
74 views

Power method for finding all eigenvectors

This is my homework. I was asked to find all eigenvectors of a symmetric and positive definite matrix by inverse power method with shifted. I encountered three problems: The eigenvalues to the ...
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0answers
172 views

Change MATLAB code from Lax-Wendroff to Leapfrog

I want to see how leapfrog would look using this code, but I'm having issues implementing it. I think my biggest problem is adding in the $ U_j^{n-1}$ term, I just don't get the logic. Here's what ...
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0answers
23 views

Solving ODE numerically - getting local truncation error

Well I have NO idea how to do this or even where to start Compute the order of magnitude of the local truncation error of the following time integration scheme: $$y_{n+1} = y_{n-1} + 2h f(y_n)$$ H ...
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1answer
70 views

Rate of exponential decay

Good day all I have this curve (it is a solution of a partial differential equation that am working on) and I want to calculate numerically the rate of exponential decay but I don't know how to go ...
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1answer
46 views

Laguerre's method and zero division

I'm trying to understand Laguerre's method for root finding and I have hit one road block. Suppose I have a polynomial $p(x) = x^4 + 1$ and an initial guess $x_0 = 0$. This results in division by ...
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2answers
43 views

Numerical Differentiation using Numerical Methods

I am currently studying Numerical Differentiation in MATLAB using Numerical Methods in Engineering with Matlab by Jaan Kiusalaas, and I am stuck at exercise 13 from Problem Set 5.1 from Chapter 5 ...
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54 views

The definition and meaning of “machine epsilon” in MATLAB

I am taking a introductory course in numerical mathematics, using MATLAB and a numerical math text that refers to MATLAB often. In the text, the machine precision is defined as: The distance ...
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1answer
30 views

What is wrong with this algorithm [closed]

Crout factorization: n=10 A = full(gallery('tridiag',n,-1,2,-1)) i = 2:(n-1); bmid = i.^2 / ((n+1).^4) b = [1+1/(n+1)^4, bmid, 6+(n^2)/(n+1)^4]' for i = 1:n L(i,1) = A(i,1) end for j = 1:n U(1,j) = ...
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1answer
39 views

Backward error for Crout factorization

Ok, can someone please tell me what is the formula for the max error in LU decomposition of Crout factorization?
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1answer
275 views

Gauss Seidel iteration in matlab

I've posted this question before for crout factorization. Now, I need help with Gauss-Seidel iteration. Write a program that takes a value for n and solves for x using the following method: ...
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1answer
23 views

Gaussian Quadrature with polynomial of order one

Let $w(x) > 0$ be a weight function on the interval $[0, 1]$ and let $P_n(x), n = 0,1,2\dots$ be a sequence of orthonormal polynomials of exact degree $n$ which satisfy $$\int_0^1P_n(x)P_m(x)w(x)dx ...