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

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32 views

Find a symmetric matrix of minimal Frobenius norm

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, And let $$x\in \mathbb{R}^n$$ be such that $\lVert Ax-b\rVert_2 = \min_{z\in \mathbb{R}^n} \lVert Az-b\rVert_2$. Show how to calculate a ...
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1answer
15 views

Trouble understanding Product Notation

I'm trying to determine the quadrature formula when the interval is [−2, 2] and the nodes are −1, 0, and 1. However, I get stuck on this step because I don't understand how the attached image is ...
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1answer
20 views

Norm Product expression

Prove the product expression $$\left \| AB \right \|_{U\rightarrow W} \leq \left \| A \right \|_{V\rightarrow W}\left \| B \right \|_{U\rightarrow V}$$ Hint: consider $(AB)u = A(Bu)$ and apply $\left ...
2
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1answer
43 views

Prove Operator Norm is a Norm on linear space [duplicate]

Prove that the operator norm defined by $$\left \| A \right \| = \left \| A \right \|_{V\rightarrow W} = \sup_{0\neq v\in V} \frac{\left \| Av \right \|_{W}}{\left \| v \right \|_{V}}$$ (Given norms ...
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0answers
20 views

Estimate the error of interpolation/extrapolation

I'd like to know how we can estimate the error of interpolation. For example, let's consider Lagrange interpolation. We don't know anything about the real function (it could be an algebraic or a ...
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1answer
41 views

How to derive the error $e^{n}=M^{n}e^{0}$ from iteration $x^{n+1} = Mx^{n} + f$?

$x^{n+1} = Mx^{n} + f$ is fixed-point iteration for solving the equation $x = Mx + f$, i.e., $(I-M)x = f$. The error $e^{n} = x - x^{n}$ How does one get $e^{n}=M^{n}e^{0}$?
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1answer
32 views

Largange polynomial second order derivative

Well I came across a problem to find a generalized version ($n+1$ nodes) of first and second order derivatives for Lagrange interpolation polynomial. In some former post, I found an expression for ...
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1answer
33 views

QR Factorization for Inconsistent Linear System

I am trying to recreate the problem found here on finding the least squares solution to an inconsistent linear system via QR factorization. Can someone explain the part about adding on vectors so that ...
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0answers
14 views

Solving coupled PDEs numerically

I have the following system of PDEs for which I have given parameters $\gamma, \tau$ and $\mu$, $$\begin{align} T_t = &\ \gamma\,(L +\tau F-T)\\ F_t = & -F_x-(F-LT)\\ L_t = &\ \mu ...
0
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1answer
32 views

How to verify correctness of my numerical method for Allen-Cahn equation?

\begin{equation}\label{Parabolic} \frac{\partial \phi(\mathbf{x},t)}{\partial t} - \Delta\phi(\mathbf{x},t)+\frac{f(\phi(\mathbf{x},t))}{\epsilon^2}=0 \end{equation} \begin{equation}\label{boundary} ...
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0answers
20 views

MATLAB plotting $\log(h)$ vs. $\log(e(h))$

So I have the initial value problems below. From a previous part, I have already confirmed that $v(t)$ is a solution to the the problem above it. Now I am left to write a MATLAB code that plots ...
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0answers
80 views

Implementation of a simulation of an incompressible Newtonian fluid with uniform density

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I want to simulate an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. The evolution ...
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1answer
30 views

Formation of Runge-Kutta method

A Runge-Kutta method for numerically solving the initial value ordinary value differential equation $y'(x)=f(x,y)$ ; $y(x_0)=y_0$ is given by ( for $h$ small ...
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1answer
56 views

Software where I can numerically evaluate multivariable integrals over a region?

I need, for example, to evaluate: $$\iiint(x-1)\,dx\,dy\,dz$$ over the region: $$y=0,\, z=0,\, y+z=5,\, z=4-x^2$$ but I have no ways to verify if I did it right. I needed a form of numerical ...
0
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1answer
35 views

Reference on checking result for a third order numerical EDO

Consider the following differential equation \begin{equation} y^{\prime\prime\prime}=xy\qquad\qquad y(0)=1,\:\:y^{\prime}(0)=0,\:\:y^{\prime\prime}(0)=1. \end{equation} Using the third order ...
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0answers
9 views

convergence rate of forward backward operator splitting algorithms

I am looking for some latest material on convergence rate of the basic forward backward operator splitting algorithm. After googling, I found the following: ...
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0answers
36 views
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1answer
22 views

Two step Backward differentiation formula

Derive the two step BDF method, the final solution should be $$y_n = \frac{4}{3}y_{n-1} - \frac{1}{3}y_{n-2} + \frac{2}{3}h f_n$$ I am pretty sure we can use backward euler to derive it i.e. $$y_n - ...
2
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1answer
46 views

Does the following equation have a positive integer solutions for $p$?

Q. Does the following equation have a positive integer solutions for $p$? If are unable to obtain an answer for $p$, explain why this is the case. If you are able to find one case where the ...
1
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1answer
30 views

Numerical approximation WITH step functions

I'm interested in approximating a continuous curve (actually a data trace) with step functions. I know this is very similar to approximating the area under a curve with rectangles, but there is a ...
1
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1answer
30 views

Name of Newton's methods

We all know the famous Newton-Raphson method in order to solve $\Psi(\xi)=0$: \begin{align} \xi^{(\ell+1)} &= \xi^{(\ell)} + \Delta\!\xi^{(\ell)}, \\ ...
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2answers
123 views

Practical drawing of geodesics

I want to use a computer to draw geodesics on a known parameterized surface of revolution, starting from a known point and at a known angle to the meridian. What would be the easiest way of doing ...
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1answer
54 views

What does it mean to say that “formula is exact for all polynomials of degree less $n$”? [closed]

What does it mean that "formula that is exact for all polynomials of degree less than or equal to 2?'
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1answer
43 views

Is it possible to estimate the sign of real part of eigenvalues of a 10 by 10 matrix only by observing all the entries?

I have a symbolic 10 by 10 matrix. It is not difficult to get the eigenvalue expressions by using Matlab. But the expressions of some eigenvalues are too long to be analyzed. I was wondering if there ...
2
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0answers
38 views

Numerical solution of the stationary Navier-Stokes equations

Let $d\le 3$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I'm considering an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity $\nu$. In this case, the stationary ...
0
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0answers
16 views

System of 2nd-order ODE equations using Runge-Kutta

Given the following system: $$u''_1(t)+u_1(t)+\frac{1}{3}u_2(t)=0, u''_2(t)-0.5u_1(t)+0.5u_2(t)-2\sin(1.5t)$$ with the initial conditions $u_1(0)=u_2(0)=0$ and $u'_1(0)=u'_2(0)=0$, determine the ...
0
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1answer
18 views

Newton's Method in unconstrained optimization fails to converge

In order to show that Newton's method can produce a sequence of iterates that diverges, an example given in my book is apply Newton's Method to minimize $f(x)={2\over 3}|x|^{3\over 2}$. starting at ...
2
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3answers
67 views

Looking for fractals which are computationally demanding and preferrably parallelizable.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long ...
0
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0answers
44 views

Approximation of nonlinear ODE

I have an equation $$D \cos(\omega t) = \frac{d^2z}{dt^2} + B \frac{dz}{dt} - A g(z) - C$$ where $$g(z) = \frac{\partial}{\partial ...
0
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1answer
13 views

ODE: Euler iterative method

I´m doing some examples on the euler method to solve differential equations, and I came across this one: $\begin{cases} y'(t) = e^{-ty(t)} \\ y(0) = 1 \end{cases}$ for $t \in [0,1]$ I want't ...
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2answers
54 views

How to determine if an equation is algebraically solvable?

Problem I was given the following equation to solve for $x$: $$35x^{9 / 5} + 180x^{7 / 5} + 252x = 50400 / \pi$$ But don't get hung up on it. It's only an example. My solution ...was simply to ...
0
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1answer
30 views

Number of significant digits in apporximate value

Actual value of a quantity is $1.354675$ and the numerical approximate is $1.354595$ then what is the number of significant digits? By using absolute error i.e. $\vert\text{true value} - ...
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5answers
83 views

$10^9 \times \sqrt{3}$ what are first two digits after the decimal point?

Because of floating point error, my computer basically says $10^9 \sqrt{3} \approx 1.73205 \times 10^9$ so that if we ignore the numbers before the decimal point, the fractional part is: $$\{ 10^9 ...
0
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1answer
35 views

Reduce a fraction by interpolation

I am trying to solve this problem: Use $ x^2+1$ (polynomial interpolation) to reduce $$ \frac{(x^2+1)}{x(x-1)(x-2)(x-3)}.$$ I don't know how I can reduce a fraction by interpolation method.
4
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1answer
59 views

Numerical Solutions of the Telegrapher's Equation

I'm currently working on a brief report on the Telegrapher's Equation for my fractional calculus course. I am still new to Telegrapher's Equations, but I do know they are used to describe electrical ...
2
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1answer
40 views

How would you reduce roundoff error in “mod” when implementing a periodic function?

I recently wrote How calculators do trigonometry where I wrote a simple program for computing $\sin(x)$. For completeness, I include a slightly modified version of the program here: ...
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1answer
33 views

How can I use Lagrange polynomial to approximate an integral over [0,1]?

$\int_0^{1}f(x)dx$ is approximated by $Af(\frac{1}{3})+Bf(\frac{2}{3})$. I would like to derive above formula using Lagrange interpolation polynomial. How should I start?
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0answers
53 views

Damped Iteration

For splitting $A = M-P$ a damped iteration with damping factor $\gamma <1$ and scalar $\omega$ is $$x^{k+1} = x^{k} +\gamma M^{-1}r^{k}$$ where $$r^{k} = b-Ax^{k}$$ $$M =\frac{1}{\omega }I $$ $$P = ...
1
vote
1answer
41 views

Number of points needed for linear interpolation of sine in $[0,\frac{\pi}{2}]$ with given error bound

I want to get a set of equispaced points in $[0,\pi/2]$ and use piecewise linear interpolation generated by those points to fit the sine function. And I want to determine how many points do I need to ...
0
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0answers
21 views

Preconditioning for Jacobi Method without effect

Show that the following scaling doesn't affect the spectral radius of the Jacobi method iteration matrix $T_{J} = -D^{-1}(L+U)$. $\tilde A=D^{-1 /2}AD^{-1 /2}$, where $D = ...
0
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1answer
14 views

How can I see the order of accuracy of an ODE method?

Let's say I have solved an ODE with Euler's forward method, and also solved it using RK4, in both cases for varying decreasing step sizes $h$. Is there any way to look at the graphs and "see" the ...
0
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1answer
51 views

Richardson's Methods

I need to prove Richardson's Method and the first part of the proof is: Consider the linear system $Ax = b$ where the eigenvalues of $A$ are real and positive. Let $G_{\omega } = I - \omega A$, ...
1
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1answer
34 views

lower bound for the min function

We all know the well known upper bound: $$ \min(a,b) \leq a^s b^{1-s}$$ for $$a,b \geq 0, 0 < s < 1$$ I am looking for a lower bound on $\min(a,b)$.
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2answers
50 views

Finite Element Method for the 1d wave equation

I'm solving the 1D wave equation \begin{equation} \frac{\partial^2 \eta}{\partial t ^2} - \frac{\partial^2 \eta}{\partial x ^2} = 0 \end{equation} with boundary conditions \begin{equation} ...
3
votes
3answers
63 views

Numerical method for approximating the standard Normal distribution cdf with mean 0 and variance 1

The standard Normal distribution probability density function is $$p(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2},\int_{-\infty}^{\infty}p(t)\,dt = 1$$ i.e., mean 0 and variance 1. The cumulative distribution ...
0
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0answers
24 views

Open Newton Cotes Problem

Consider numerically approximating the integral $$I = \int_{a}^{b}f(x)dx$$ using the open Newton-Cotes with $n = 2$,i.e., 3 points $$I_{2}^{(o)} = \frac{4}{3}h_2[2f(x_0) - f(x_1) + 2f(x_2)]$$ 4b.) ...
0
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0answers
17 views

Finite element boundary conditions

I have a boundary condition given by $\mathbf{n}\cdot \nabla m=\phi$, where $n$ is a vector normal to a surface, $m$ is a physical quantity (say mass) and $\phi$ is a constant. The boundary condition ...
0
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0answers
39 views

Nonlinear Schrödinger Equation

I have to find equation and starting condition to solve Nonlinear Schrödinger Equation with periodic edge condition. This method should control the propagation of fiber optical signal. In details I ...
6
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1answer
158 views

Please, help to identify this numerical constant

I'm trying to find an answer to this question. Let $K(k)$ be the elliptic integral of the first kind and $K'=K(\sqrt{1-k^2})$. According to Abel's theorem (see this link) we know that if ...
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1answer
20 views

(approximately) compute absolute largest eigenvalue of symmetrix 3x3 matrix

I need to compute (an approximation may be good enough) the largest (by absolute value) eigenvalue of a real symmetric 3x3 matrix many ($10^{6-12}$) times. Is there anything better than just computing ...