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

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20 views

Discretisation of a product of two functions

Suppose I have two functions, $f(x,t)$ and $g(x,t)$, and for an upwind scheme I want to use the quantity $\partial_x (fg)$ to solve the advection equation $$ \frac{\partial f}{\partial t} + ...
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1answer
33 views

Machine limit analysis of $\sqrt {x^2-a^2}-(x-a)$

Let $L(x)=\sqrt {x^2-a^2}-(x-a)$. I've been messing around with this equation on the calculator and found out that for certain values of $x$, the equations behave as $x \gg a$. Considering only for $x ...
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0answers
21 views

numerical errors near at the borders

I use some kind of partitioning on my data and then I do some interpolation and some other mathematical operations using chebyshev points. I have noticed that in the borders of each partition, It ...
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0answers
20 views

BDF2 and TR-BDF2: what is better?

What method of numerical solving ODEs is better? BDF2 or TR-BDF2? Namely, what advantages has TR-BDF2 over BDF2? The BDF2 method requires the values of $y_{n-1}$ and $y_n$ for computing $y_{n+1}$ ...
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1answer
24 views

Solving boundary value problem, put up linear equation system

For $\Omega = (0,1)^2 \subseteq \mathbb R^2, f \in C(\Omega)$ consider the boundary value problem: $-\Delta u(x,y) + u(x,y) = f(x,y)~ \forall (x,y) \in \Omega \\u(0,y) = u(1,y)~ \forall y \in (0,1) ...
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2answers
41 views

How to solve $Ax=b$ via backward and forward substitution on Matlab

How can I solve $Ax=b$ in Matlab code via LU factorization. I know that the command [L,U]=LU(A) stores the ...
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28 views

Help on Lagrange Error Calculation [duplicate]

Here is an example in Burden's Numerical Analysis book. My problem is in bold In example 2 we found the second Lagrange polynomial for $f(x)=1/x$ on $[2,4]$ using the nodes $x_0=2$, $x_1=2.75$, ...
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14 views

Solving a sum-exponential equation

I was wondering if someone could point me to the right resource towards numerically solving an equation of the form: $c = \dfrac{\sum_i a_i^{2x}}{\left( \sum_i a_i^x \right)^2}$ $c$ and $a_i$ are ...
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0answers
11 views

choice of iterative linear system method

while implementing an unconstrained optimization problem, using Newton's method, I am faced with a Hessian matrix that is very large (10^8 by 10^8) but very, very sparse - Non zero elements along the ...
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1answer
52 views

Why easier to numerically minimize than to maximize a function

Is it easier, in terms of coputational complexity or speed, to numerically minimize a function $f$ than to maximize $-f$? Why is that so? I have noticed that most optimization algorithms in Matlab are ...
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2answers
58 views

The greatest eigenvalue

I am searching for any estimates of the greatest eigenvalue for non-symmetric 3(5)-diagonal matrix $A$, i.e. any information about estimates like $$|\lambda_n|<F(a_{ij}), $$ where ...
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0answers
53 views

Error evaluation of consecutive terms at fixed point iteration methods.

Q: Give an evaluation of error between two consecutive terms for methods of type $p_{n+1}=g(p_{n})$. I tried solving it, but I think that my solution is correct only when the method converges to a ...
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0answers
26 views

What's the point of 1-norm matrix estimation? Why not brute force?

Calculating (brute-force) 1-norm of a square matrix should take $O(n^2)$ operations, with a small factor involved. Apparently, there is an algorithm (link) for estimating 1-norm that takes $O(n^2 t)$ ...
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3answers
43 views

RK4 method applied to $\frac{dy}{dt}=-\frac{y-t}{2}$ with $y(0)=1$

I tried to solve this question, I did it by Huen's method also, but I'm getting very different results than the answers sheet. Use the fourth-order Runge-Kutta method (RK4) to solve the following ...
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0answers
18 views

Numeric Integration of a Surface Element in Spherical Coordinates

I know Area is related to spherical coordiantes by $dA = r^{2}sin(\theta) d\theta d\phi$ So numerical values should become $\Delta A = r^{2}sin(\theta)*\Delta\theta\Delta\Phi$ However, I'm unsure ...
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1answer
26 views

Discretize differential Equation numerical methods

I don't know how to approach this question by numerical methods, any help will be appreciated: Discretize the following differential equation using central finite difference formulation: ...
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2answers
88 views

Does averaging always provide faster converging numerical methods?

So I am studying SICP (Structure and Interpretation of Computer Programs) and doing one of the excercises which is based on the fixed-point method for finding the fixed-point of $f(x)$. In a ...
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1answer
26 views

Estimated solution to system of equations with phase-shifted functions

Forgive my first attempt at MathJax. I have a system of $n$ equations of the form $$ v_j(t) = \sum_{i=0}^{m-1} \frac 1 {|\vec p_i - \vec q_j|} u_i \left(t - \frac {|\vec p_i - \vec q_j|} s \right) $$ ...
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0answers
15 views

How many binary digits for the fractinal part?

Suppose we want to convert $0.3$ to its binary form by this way $$ 0.3 \times 2 = 0.6 + 0 \\ 0.6 \times 2 = 0.2 + 1 \\ 0.2 \times 2 = 0.4 + 0 \\ 0.4 \times 2 = 0.8 + 0 \\ 0.8 \times 2 = 0.6 + 1 $$ ...
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0answers
26 views

$ \|u(x)+u(x)-x +o(\|x\|^p)\|<r $?

Set $B_r=\{ x\in \mathbb{R}^n : \|x-0\|< r \}$ for any $r>0$. Let $C^p_0(B_r,B_r)$ the set of all smooth functions $u:B_r \to B_r$ of class $C^p$ such that $u(0)=0$. I would like to prove the ...
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1answer
44 views

A big contradiction in interpolating point and number of it's

For calculating divided (fraction) difference table for interpolating $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, $n(n+1)/2$ difference fraction was used. I ...
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0answers
31 views

Is it better to compute $A^tA$ once and then $Ax$ several times or compute $y=Ax$ and then $A^ty$ every time?

So I have this algorithm which given a matrix $A$ it assigns $A=A^tA$ outside the loop and then on the algorithm loop it solves multiple instances of $Ax$ for different $x$s, (meaning that it's ...
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0answers
84 views

Problem regarding the speed of two points $A$ and $B$ moving with constant speed in the plane [duplicate]

Consider a Point A that moves linearly on the positive x-axis with the speed 1 m/s and another Point B at a distance L from A with position (L,0). With each forward motion of point A the Point B moves ...
2
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0answers
29 views

Formula for $s_n = \sum_{i = 1}^n i^3$ Newton's Forward Difference Interpolation

Use Newton's Forward Difference formula to find an expression for $$ S_n = \sum_{i = 1}^{n} i^3$$ This is from an Introductory Numerical Analysis paper. I cannot figure out the connection ...
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2answers
24 views

Find the curve of $ y = (Ax + B)^2$ for points $(k,2k^2+k)$,given$ (k=1,2…10)$

I would like to solve this and run the matlab code: since $y=(Ax+B)^2=A^2x^2+2BAx+B^2$ let $A^2=\alpha ,2AB=\beta,B^2=\gamma$ $\min{(\alpha x^2+\beta x+\gamma-f_m)^2}=\phi(\alpha,\beta,\gamma)$ ...
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2answers
30 views

Newton's method convergence criteria

To use Newton's method on interval $[a,b]$ we need to guarantee that $f(a)f(b)<0$ on the interval which is true for $[0,1]$. $f'(x)$ and $f''(x)$ are continuous on the interval $[a,b]$ (which ...
3
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2answers
62 views

Good Textbook in Numerical PDEs?

I am currently taking a course on Numerical PDE. The course covers the following topics listed below. Chapter 1: Solutions to Partial Dierential Equations: Chapter 2: Introduction to Finite ...
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0answers
29 views

Finite difference scheme for the continuity equation

I am currently trying to solve a system of PDE's numerically, one being the equation; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 $$ I have been reading up on ...
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0answers
49 views

Tikhonov regularization vs truncated SVD

To find $\mathbf{x}$ such that $$A\mathbf{x}=\mathbf{b}$$ we can use least squares when the problem is not well posed. Further, we can use Tikhonov regularization when $A$ is ill-conditioned. In ...
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2answers
46 views

construct $\mathcal{O}(h^2)$ finite difference scheme for $(a(x)\cdot u'(x))'$ operator

Obviously, Taylor expansion for $(a(x)\cdot u'(x))'$ is to be used somehow, but I'm not sure how to start at all... The scheme that I'm looking to derive is actually $\frac{(a\cdot ...
3
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0answers
121 views

Diophantine approximation with additional constraints

I am trying to compute best rational approximations to various transcendental numbers $c$, subject to the following constraints: $$\frac {i j} {2^k} = c + \epsilon, \space\space2^n \le i, j \lt ...
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0answers
20 views

Parallel Programming the 1-D dam breakage problem

I am to write a parallel program to simulate the 1D dam break problem by using the Galerkin Equations with WENO limiter. The equations are on domain [0,2000]. At the beginning a dam divides the ...
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7answers
153 views

Question about the Accuracy of $\pi$

I have always been confused regarding the accuracy of $\pi$. In the books which are written on this subject $\pi$ , there are references of people and their methods for finding the value of $\pi$. ...
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1answer
74 views

Differential problem solving with Euler and Heun methods

I have to write application which solves task presented below. I only know some c# so I will stick to it. It is some kind of homework but I am asking for help with understanding this and advice for ...
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4answers
234 views

How to compare products of prime factors efficiently?

Let's say that $n$ and $m$ are two very large natural numbers, both expressed as product of prime factors, for example: $n = 3×5×43×367×4931×629281$ $m = 8219×138107×647099$ Now I'd like to know ...
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1answer
49 views

Why won't my conjugate gradient algorithm work?

I made this simple Conjugate Algorithm on Matlab n = length(b); r0 = b - A*x0; p0=r0; k=1; n0=(r0')*r0; while n0 >= eps && k <= n ...
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0answers
26 views

Finding the basis functions given the boundary values and values of derivatives at the boundary

Given an interval $I=[a,b]$ we define $$P_3(I):=\{v:I\rightarrow\mathbb{R}\mid v \text{ is a polynomial of degree} \leq 3 \text{ i.e } \\v=a_3x^3+a_2x^2+a_1x+a_0 \text{ for } a_i\in\mathbb{R}\}.$$ How ...
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1answer
50 views

Solving autonomous system of ODE numerically by Runge-Kutta method

I have an autonomous system $y''= \exp(y)$ with initial conditions $y(0) = 1, y'(0) = \sqrt{2 e}$, which I have to solve numerically by secod-order RG method. (Actuall I must solve BVP, but now i'm ...
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0answers
19 views

Discretization of the Anisotropic Diffusion Operator for Finite Difference Method

I have to derive and apply a Finite Difference scheme to solve a steady state, anisotropic, diffusion equation. So I have to find a discretization of the following equation $$\nabla \cdot ( ...
2
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0answers
74 views

Solving systems of polynomials with an oracle

I need to solve a system of polynomials. Let the variables be $x_1, \dots, x_n$, and let the polynomials be $f_1, \dots, f_n$ Let's say we have these conditions we can already assume: there are ...
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1answer
50 views

Detection of self intersection point of curve

What numerical procedure is be adopted to detect self-intersecting parametrized points $ [x(t), y(t) ] $ in $ \mathbb R^2 $ ? Observation : @ roots ( t= 2, t=-1 ) parabola has double value with ...
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4answers
226 views

A calculation that goes awfully wrong if we let $\pi=22/7$

Me and one of my friends had an argument and he said that using $22/7$ as value of $\pi$ is sufficient for any calculation. Can we always take it $22/7$, or is there some example of some calculation ...
2
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1answer
49 views

Can I use Runge-Kutta to solve these two equations?

I have derived the following two equations: $$69ru\frac{du}{dr}=8r^2\omega^2-16r\omega v-21u^2+48v^2-\frac{480\pi^2\nu r^3u^3}{Q^2}$$ $$48ru\frac{dv}{dr}=-21r(v-r\omega)\frac{du}{dr}-69uv+37r\omega ...
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0answers
34 views

System of linear diophantine modular inequalities

How can we best find a numerical solution to a system of $m\ge2$ linear diophantine modular inequalities $$\big((a^j x+b_j)\bmod n\big)<c\;\text{ for }1\le j\le m$$ where $x$ is the only unknown, ...
3
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0answers
80 views

Numerical scheme to 1D advection equation

I am trying to numerically solve a system of equations which model the early universe in 1D. The equations I am stuck on are; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 ...
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3answers
157 views

How to create a computationally cheap function passing through given points?

I am trying to develop a function which goes through the follow points. The function will be calculated on a microprocessor which has 20 mHz. List of given points: ...
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1answer
51 views

Solve Karush–Kuhn–Tucker conditions

solving a constrained optimizing problem with equality constraints can be done with the lagrangian multiplier. (http://en.wikipedia.org/wiki/Lagrange_multiplier) This approach leads to a system of ...
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1answer
20 views

Can $LL^T$ decomposition of a matrix be computed by the same algorithm as $LU$-one?

I know that's the silly question. But if I perform $LU$ decomposition on a symmetric positive definite matrix, will this decomposition be the same one as $LL^T$ one?
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1answer
97 views

The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho, $$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
0
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0answers
29 views

Lagrange Newton Method Singular Matrix

i implemented the lagrange-newton method in python to find the problem to nonlinear optimizing problem for learning purposes. But every guess i made a guess for the initial values the resulting ...