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

learn more… | top users | synonyms (2)

2
votes
2answers
70 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
0answers
20 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
17 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?
2
votes
0answers
36 views

Anyone knows a Good Textbook in Numerical PDES

I am planning on taking a course on numerical PDEs next semester. The course covers the following topics listed below. I am looking for a good book that covers these topics (or at least most of them). ...
1
vote
1answer
49 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
24 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
1answer
18 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
13 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
121 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.
0
votes
0answers
15 views

order of convergence for approximations

Let $u \in L^{2}(0,1)$ and $0 < x_{1}< x_{2}<... < x_{n} = 1$, where x$_{k}$ = k$\cdot$h, n$\cdot$h = 1, a partition of the interval [0,1]. Define I$_{k}$(x) = 1 if x $\in$ [x$_{k}$, ...
5
votes
1answer
58 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
40 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 ...
1
vote
1answer
27 views

Am I doing this approximation correctly? (least squares method)

Here is the problem. Find the function $f$ of the type $f(x) = a\cos x + b\sin x$ which best approximates the function $g$ in the points : $$ \begin{array}{ c | c | c | c | c | c | c } x & ...
2
votes
0answers
43 views

Convergence of Newton method under some assumptions

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 508 here) that if $x^\ast$ is the unique root of equation $$f(x)=0$$on interval $[a,b]$ and if the function has ...
2
votes
0answers
38 views

Why does this nonlinear ODE solution not work?

I am relatively new to Python and trying to use it to solve a second order nonlinear differential equation, specifically the Poisson-Boltzmann equation in an electrolyte. $$\phi''(r) + \frac2 ...
1
vote
2answers
29 views

Domain for which this matrix is positive definite

What is the domain for which this matrix is positive definite? $$\left(\begin{array}{cc} 12x^2 & 1 \\ 1 & 2 \\ \end{array}\right)$$ I'm trying to figure this out. I know the ...
1
vote
0answers
24 views

Rayleigh quotient ($|r(q)-\lambda|=O(||q-x||^2)$ ?)

how to show $|r(q)-\lambda|=O(||q-x||^2)$ $r(q)=q^*Aq/(q^*q)$ and $\lambda$ is an eigenvalue, A is a symmetric matrix. x is the unit eigenvector corresponding to $\lambda$. and q is a unit vector. ...
-9
votes
0answers
34 views

fixed point iteration code [closed]

So as I was working out my math with the following equations using the fixed point iteration, I stumbled to wonder if one can ever write a C program, MATLAB (or the likes) for such math. So my ...
1
vote
1answer
22 views

Gauss quadrature on tetrahedron

I have the Weights ${\bf w}=[w_1\,w_2\, w_3\, w_4]=[0.25, 0.25, 0.25, 0.25]/6$ and the points ${\bf x_1}=[0.1381966011250105 , 0.1381966011250105 , 0.1381966011250105]$ ${\bf ...
0
votes
0answers
15 views

Determine the order of dissipation for a finite difference scheme

Consider the PDE $u_t + au_x = f$ and the finite difference scheme: $$ \frac{3v_m^{n+1} - 4v_m^n + v_m^{n-1}}{2k} + a\frac{v_{m+1}^{n+1} - v_{m-1}^{n+1}}{2h} = f_m^{n+1} $$ I need to determine the ...
0
votes
0answers
25 views

Guassian quadrature

Consider the polynomial $L \in P_5$ defined by $$L(x)=\frac{d}{dx^3}(1-x^2)^4$$ My question is to use a technique similar to the one used in the analysis of the Gaussian quadrature to show that Q has ...
0
votes
0answers
8 views

What is numerical flux function?

I am learning "Numerical Approximations of Hyperbolic Systems of Conservation Law". I can not find answers for the following questions: 1.What is the numerical flux function? 2.How can one find ...
1
vote
1answer
21 views

Tissue Deformation Simulation using FEM

I need to simulate tissue deformation using FEM. Is it advisable to represent the object as a triangle mesh or a ...
0
votes
0answers
14 views

Polynomial Interpolant [duplicate]

I need to find the polynomial using divided differences with respect to these conditions: $ p(1) = 2 $ $ p(2) = 4 $ $ p(3) = 8 $ $ p'(2) = 4ln(2) $ How do I deal with the condition on the first ...
0
votes
0answers
17 views

Constructing integration rules

Consider the polynomial $L \in P_5$ defined by $$L(x)=\frac{d}{dx^3}(1-x^2)^4$$ i. Find the roots $(ξ_i)i=0,...,4$ of $L$ and show that they are distinct and lie in $[−1, 1]$. (Hint: −1, 0, 1 are ...
0
votes
1answer
24 views

Show that the following iteration rule fulfills the condition of the Banach-fixed-point theorem

Given the following system of differential equations: $$\dot{\textbf{v}} = \left( \begin{array}{cc} -800.2 & -399.6 \\ -399.6 & -200.8 \\ \end{array} \right) \textbf{y} - \textbf{c}$$ Show ...
1
vote
2answers
68 views

How to prove that trigonometric functions form a Chebyshev system?

How can be proven that $$\{ \operatorname{cos}(kx)\}_{k = 0}^n \text{ and } \{ \operatorname{sin}(kx)\}_{k = 1}^n$$ are Chebyshev systems in the interval $(0, \pi)$? Any ideas will be appreciated. ...
0
votes
1answer
66 views

Interpolation of polynomials

let $f(x)=2^x$ and $x_0=1$, $x_1=2$, $x_2=3$. Use divided differences to compute the interpolation polynomial $P(x)$ satisfying $P(x_i)=f(x_i)$, i=0,1,2 and $P'(x_1)=f'(x_1)$ and estimate error ...
-1
votes
3answers
30 views

How to turn a decimal into a number to divide something by into it.

So here is what things will convert to: 0.5 = 2; 0.25 = 4; + MILLIONS MORE 1 = The whole of a number ( / 1 ) 0.5 = Half of the number ( / 2) But what is the math to convert decimals into only a ...
3
votes
1answer
32 views

Finite differences and conservation law

I am using a Finite Difference scheme to solve a simple PDE in conserved form: $$\partial_t u = \partial_x (\partial_x u +au\partial_x u) = (1+a)\partial_x^2u +a(\partial_x u)^2 $$ $$\frac{u_{n+1,j} ...
1
vote
0answers
18 views

Matlab Newton's Method Non-linear system

There is something wrong with this program and I cannot seem to find it. I am trying to calculate the solution of a non-linear system using Newton's method. Matlab keeps saying there is a problem with ...
0
votes
1answer
24 views

Solving numerically a non-linear equation.

How is the more appropriate numerical method to solve the equation $$\cos(2\pi x)+\cos \left(\frac{2\pi N}{x}\right)=2,$$ for a given $N$? Notice that if $N \in \mathbb{Z}$, then $x\mid N$.
3
votes
1answer
59 views

solution of multidimensional PDE

I'm looking for a way to find a solution 'f' to the following PDE. $$ y \frac{\partial f}{\partial r} + g_1(r)\left(z\frac{\partial f}{\partial y} - y\frac{\partial f}{\partial z}\right) + ...
0
votes
2answers
61 views

Is there a proper way to prove that $f:[a,b] \to[a,b]$

Is there any proper way to know whether a function has the same domain and range $[a,b]$ where $a,b<\infty$ i.e. $f:[a,b] \to [a,b]$ ? For example: $$ f(x) = e^{−x} ,\qquad [\ln(1.1), \ln(3)] $$
1
vote
2answers
48 views

prove that $x \mapsto \mathrm e^{-x}$ has a unique fixed point on R

Can anybody prove $x \mapsto \mathrm e^{-x}$ has a unique fixed point on R using the fixed point iteration theorem?
0
votes
1answer
19 views

Show that$ |e_n| \leq 2^{-(n+1)}(b_0 - a_0)$

I would like to know if someone can shed some light on it.I'm not sure but I think Lipschitz or contraction mapping theorem is involved. Let $x_n = \frac{a_n + b_n}{2} , r=\lim_{n \to \infty }x_n$ ...
2
votes
1answer
36 views

matlab program help

Wanting to write a matlab program to solve the following iteration: $x^{(k+1)}=b+\alpha\begin{bmatrix}2&1\\1&2\end{bmatrix}x^k,k=0,1,2,\cdots$ where alpha is a real constant. Find the values ...
1
vote
1answer
40 views

How do I solve for the zeros of a Chebyshev polynomical? (on a computer)

I am working on a computer program and have a method that returns a number for a given $x$, $y$. So $f(x, y) = z$, where $f$ is my method. if I know $y$ and $z$, can I find what $x$ will be, without ...
0
votes
0answers
31 views

How to solve one differential equation with two independent variables in heat transfer.

$$A\frac{ \partial T_a}{\partial t}=B(T_p-T_a)+C(D-T_a)-E\frac{\partial T_a}{\partial x}$$ Where $A, B, C, D, E$ are constants, $t$ is time and $x$ is $x$-axis of the box in which heat transfer is ...
0
votes
0answers
12 views

Whitney element lowest order (Nedelec, finite element)

where can I find an explication about Whitney element lowest order? The usual finite element in Nedelec spaces $H(curl,\Omega)$. Thanks!
2
votes
1answer
35 views

condition number for a matrix with a variable

How would I go about calculating $cond(A)$ for A= $\begin{bmatrix}1 & c\\c & 1\end{bmatrix}$, $|c|\neq 1$
0
votes
0answers
32 views

Applying Central Difference (Finite Difference Method) in MATLAB

I was given a rather complicated few problems to solve in MATLAB using the central difference method, and I'd like some help figuring out how to translate this into code. The goal is to discretize ...
4
votes
1answer
177 views

Approximating an integral with another integral with finite limits

I came across the following integral in my work $$\int_{-\infty}^{\infty} \frac{\frac{1}{(1- \ \ 2 \pi j s \theta)^{m}}-1}{2\pi j s }\ e^{-2\pi j s\sigma^2}\ ds $$ Assuming $\theta,m,\sigma^2$ are ...
0
votes
0answers
27 views

polymonial of $n+1$ degree inequality

Let: $$p(x) = x^{n+1} + a_{n}x_{n} + a_{n-1}x^{n-1} + \cdots + a_0 $$ Show $\lVert p \rVert_{\infty}^{[-1,1]} \geq \left(\frac{1}{2}\right)^{n}$ Obviously my first naive approach is induction. ...
0
votes
0answers
12 views

Multiple integral of iterated kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel $$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$ should be calculated. However, it is not ...
0
votes
0answers
30 views

Show that Newton’s Method is well-defined for all k and converges to 0 for $x_0>0$

Let $f : R → R$ with $f$ twice continuously differentiable, $\gamma > f''(x)>\delta, f(0)=0,f'(x)>\rho $ for $x ≥ 0$. Show that for any $x_0 > 0$ that Newton’s Method is well-defined for ...
0
votes
1answer
16 views

Incomplete Cholesky decomposition conjugate gradient method in Matlab

I have a problem in finding the numerical material that describing in detail for incomplete Cholesky combined with conjugate gradient method by using Matlab. Someone can help me? Many thank in ...
2
votes
0answers
28 views

Calculating gradient from finite difference results

I am solving the steady-state heat equation in two dimensions: $$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial ...
1
vote
1answer
21 views

Functions for interpolation

Do we always need to be given points to do interpolation? Or can we be given only a function? For lagrangian interpolation we require points, and does it apply for others also?
4
votes
1answer
85 views

Need some facts about Newton-Schulz iterative method and its application to sparse matrices

I am studying Newton-Schulz iterative method for obtaining an approximate inverse , which is given by $V_{k+1}=V_{k}(2I-AV_{k})$, wherein $I$ is the identity matrix and it converges, when the ...