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

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3
votes
1answer
137 views

Prove $1! + 2! + 3! + \ldots + n! =y^3$ has only one solution in the set of natural numbers?

I actually know that the above equation is true for $n=1$ and $y=1$ but am unable to prove it for the entire set of natural numbers. Can anyone please help me solve this in a simple way?
1
vote
1answer
140 views

How does Matlab, Maple, etc…solve algebraic and differential equations internally?

I would be very interested finding out how does Matlab, Maple, etc…solve algebraic and differential equations internally? Anyone know how they do it?
0
votes
0answers
48 views

Finite Difference Approach for the 1D Conservative Advection Equation with Spacially Varying Velocity

I am attempting to numerically solve the following conservative advection equation in 1D, using a finite difference method. $\frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial x}\left(v(x,t)...
0
votes
1answer
55 views

Solving for many points in a curve at the same time

Suppose there is a well-behaving monotonic function $f(x)$ where do not have analytical form of $f'(x)$, and we need to solve for many points on this function at once, that is, we need to know the set ...
0
votes
1answer
37 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 ...
0
votes
0answers
74 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 $y_{n+1}$ ...
0
votes
1answer
47 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) \\...
0
votes
2answers
2k 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 ...
1
vote
1answer
95 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 ...
1
vote
2answers
68 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 $A=[a_{ij}];i,j=\...
1
vote
0answers
73 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 ...
1
vote
0answers
47 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 $O(n^2 t)$ ...
0
votes
3answers
50 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 ...
0
votes
1answer
32 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: $$y''(x)+y'(...
6
votes
2answers
135 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 ...
1
vote
1answer
42 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 $$ v_j(t) = \sum_{i=0}^{m-1} \frac 1 {|\vec p_i - \vec q_j|} u_i \left(t - \frac {|\vec p_i - \vec q_j|} s \right) $$ ...
1
vote
0answers
30 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 ...
1
vote
1answer
49 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 ...
1
vote
0answers
38 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 ...
2
votes
0answers
86 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
votes
0answers
105 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 between ...
0
votes
2answers
25 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 $A^2=\alpha ,2AB=\beta,B^2=\gamma$ $\min{(\alpha x^2+\beta x+\gamma-f_m)^2}=\phi(\alpha,\beta,\gamma)$ ...
0
votes
2answers
396 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
votes
2answers
165 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 ...
4
votes
1answer
476 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 ...
3
votes
2answers
56 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 u_{\bar{x}})_{x}+(...
5
votes
0answers
179 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 2^{n+1}...
6
votes
8answers
269 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 $\pi$. ...
1
vote
1answer
200 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 ...
11
votes
4answers
292 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 ...
0
votes
1answer
61 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 ...
0
votes
1answer
39 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 ...
1
vote
1answer
305 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 ...
1
vote
0answers
115 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 ( \mathbf{K}...
2
votes
0answers
99 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 $...
1
vote
1answer
162 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 ...
6
votes
4answers
341 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
votes
1answer
165 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 ...
2
votes
0answers
71 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
votes
0answers
239 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 $...
2
votes
3answers
244 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: ...
0
votes
1answer
342 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 ...
0
votes
1answer
24 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?
1
vote
1answer
340 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
votes
0answers
62 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 ...
0
votes
2answers
51 views

How to express a system of differential equations in a form suitable for numerical methods?

I am modeling rocket thrust equations using some of the formulas and derivations on page 37 & 38 here. For my Rocket model, I have the following two equations: $$dv/dt = 383v^2$$ $$dA/dt = 635.14 ...
0
votes
1answer
27 views

Meaning of indices for cubic hermite splines

While digging through some code about Perlin noise, I noticed, that a Cubic Hermite Interpolation polynome is used at some point. At this point, I wanted to know, which of the Hermite basis ...
2
votes
5answers
153 views

Is there a proof that $\int \frac {dx}{x}=\ln |x|+c$?

Is there a proof that $$\int \frac {dx}{x}= \ln|x|+c$$ for $x\neq 0$ I would be interest for any replies or any comment.
5
votes
1answer
82 views

Using numerical methods to calculate integral

$$ \mbox{How can I go about calculating}\quad \int_{0}^{\infty}\,{\rm e}^{-100\,x^{2}}\,{\rm d}x\quad \mbox{to}\ {\sf\mbox{five}}\ \mbox{decimal places of accuracy ?.} $$ Do I use Simpson's Rule ?. ...
1
vote
0answers
56 views

Newton method for maps between Banach spaces

I am trying to understand the following theorem, which can be found in Kolmogorov and Fomin's (p. 509 here): Let map $F$ [$:X\to Y$ where $X,Y$ are Banach spaces] be strongly differentiable in a ...