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

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1answer
13 views

Numerical Integration by Undetermined coefficients

The important part of my question is after the bold "Now" The method of undetermined coefficients is defined so that the error of a function $f(x)$ to be integrated is zero. I.e. $E=\int_{a}^{b} ...
0
votes
1answer
12 views

order of convergence PDE

If I have the generic PDE \begin{equation} u_t + u_x = f, \end{equation} approximated with a first order in time and a second order in space numerical scheme, how can I show that the solution ...
1
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1answer
21 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
2
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0answers
34 views

Fourier transform of periodic function

Is it possible to Fourier transform a periodic function f(x) = f(x+L) with period L, numerically, only over the range x = 0 to L and use periodic boundary conditions to enforce the periodicity of the ...
1
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0answers
19 views

Adi-method for Diffusion-reaction equation in 2d

i'm trying to solve this pde using an adi-method (alternating-direction-implicit). $\frac{d f}{d t}=D\nabla^2_{x,y} f+Q(x,y)f+C$ After discretizing, the equation looks like this. Implicit in ...
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1answer
20 views

$O(\text{polylog}(1/\epsilon))$-time Algorithm for Numerical Integration to Within Additive $\epsilon$?

I'm trying to approximate a 1D definite integral to within an additive $\epsilon$ for a given $\epsilon$. I was wondering whether there is an $O(\text{polylog}(1/\epsilon))$-time algorithm for this. ...
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0answers
20 views

Name of numerical methods for second-order differential equation

Numerical methods that try to solve first-order differential equations of the form: $$ \frac{\partial}{\partial t} y = f(y,t) $$ are often Runge-Kutta methods, and there is a whole family of ...
2
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0answers
18 views

Fast Way to Compute DFT with index summation subject to a constraint

I really appreciate if anyone can help me with this problem. Problem: Let $W_n=e^{\frac{2\pi i}{N}}$ which is the $N$th root of unity. The backward Discrete Fourier Transform of a complex vector ...
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0answers
17 views

Ruling span derivation?

I have recently read a paper about the ruling span for electrical wires and they have an approximation that looks like it can be derived with mathematical intuition only. I'd like to find a derivation ...
-1
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1answer
19 views

Finding approximation of root

AS for newton approximation of the reciprocal of the square root of 5. Does the function f(x)=1/x - 5^1/2 apply for newton method
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0answers
17 views

numerical methods (Numerical Differentiation) [on hold]

How will we choose the value of "q" in Richardson Extrapolation. "q" can't take on the values "0", "1" and "-1" because of the presence of "q((q^2)-1)" in the denominator of the formula.
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3answers
61 views

How do i solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x$?

How do I solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x $? Note : I have tried using trigonometric transformation but it seems very complicated to get the result .. may ...
1
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0answers
17 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
0
votes
1answer
53 views

Determine the roots of equation if possible

How to determine the roots of equation using numerical methods? I have this particular equation: $$\arctan(e^x)=\ln \left(\sqrt{\frac{e^{2x}}{e^{2x}+1}}\right)$$ In my solution I have that this ...
1
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0answers
38 views

How to take derivative of $F(u)=\sum_{i=1}^{N} \int f^2(x) u_i^q(x) dx $

I have to find the derivative of a function. Could you help me to find it $$F(u)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$ where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership ...
1
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1answer
33 views

Underdetermined vs Overdetermined Problem

I'm trying to create a model which is of the form $$y = (a_0 + a_1l)[b_0+\sum_{m=1}^M b_m\cos(mx-\alpha_m)] [c_0 +\sum_{n=1}^N c_n\cos(nz-\beta_n)]$$ In the above system, $l$,$x$ and $z$ are ...
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0answers
16 views

Implicit numerical integration: error bound

Suppose I'm solving this equation numerically with a time step $h$: $$x''(t) = f(x)$$ Discretizing it and using implicit integration: $$x^{n+1} - 2x^n + x^{n-1} = h^2f( x^{n+1})$$ $x^{n-1}$ and ...
1
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1answer
24 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
2
votes
1answer
33 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
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5answers
58 views

How to determine if 2 line segments cross?

Give two line segments, each defined by $2$ points in $x,y$ space, such as $L_1 = (x_1,y_1)-(x_2,y_2)$ and $L_2 = (x_3-y_3)-(x_4,y_4)$, and that these points are the result of sampled data (they are ...
0
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0answers
29 views

The equivalent of least squares, but for vectors

Given a set of poins, one can use a fitting method such as least squares to find the straight (or the parabola, or the 3rd grade equivalent) that's closest to all points at the same time (via ...
1
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0answers
23 views

How can I solve this specific set of equations?

Here are the equations: $$\sum_{k = 1}^n i_k + Y_n u_n = J \quad \quad (1)$$ $$i_1 + Y(u_1 - u_2) = J \quad \quad (2)$$ $$i_k - Y(u_{k - 1} -2u_{k} + u_{k + 1}) = 0, \quad \quad k = 2, ..., n - 2 ...
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0answers
33 views

find a method for twin primes and with Golbach conjecture [closed]

There are infinitely many twin primes. Two primes (p, q) are called twin primes if their difference is 2. Let be the number of primes p such that p<= x and p + 2 is also a prime. a sample ...
1
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1answer
60 views

Runge-Kutta method for PDE

I consider certain partial differential equation (PDE). For example, let it be heat equation $$u_t = u_{xx}$$ I want to apply numerical Runge-Kutta method for solving it. As a first step I ...
1
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0answers
24 views

Numerically stable SVD

In this question regarding SVD, it is explained why eigen decomposition of $ A^tA $ is not numerically stable compared to "direct SVD algorithms". Since the former is the algorithm I'm most familiar ...
1
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0answers
33 views

Using Multipule Scale Analysis to solve a non-linear differential equation

I would like to know if there are other methods to solve equations such as this one below. I don't really understand the theory behind the multiple scale analysis and why it works I understand some of ...
-1
votes
1answer
8 views

Computing clock-wise and counter-clock-wise areas of closed X-Y plots [closed]

Given a discrete set of sampled data (a time series of 2 signals) that forms a closed path when plotted against each other (X-Y), and given that the path may cross itself (there may be 2 points on the ...
1
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1answer
27 views

Numerically find a potential field from gradient

I know that the gradient of a potential field/scalar field is a vector field, and given the function of the gradient I know how to integrate each component to get back the original scalar field. But ...
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0answers
16 views

Gauss-Legendre quadrature error

I'm trying to evaluate the error in Gauss-Legendre quadrature formulae on $[a,b]$. So far I have that the error is less or equal to $$ \frac{f^{(2n)}(\xi)}{(2n)!}\langle p_n,p_n \rangle, \enspace ...
0
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0answers
11 views

Composite trapezoid rule and trigonometric functions

I am trying to solve the problem talked about in: Trapezoid rule over trigonometric polynomials Show that the composite trapezoid rule over an equidistant partitioning with interval size ...
1
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0answers
23 views

How to select the number of nodes in a spline interpolation?

I am writing a program to test the precision of different methods for imputing missing data in a time series. One of the methods I am going to test is a natural cubic spline interpolation. I'll be ...
1
vote
1answer
21 views

Quadrature on segment

Is there a quadrature formula on the segment [0,1], such that on [0,1/2] the points and weights are symetric with respect to 1/4, on [1/2,1] they are symetric with respect to 3/4, and such that the ...
0
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0answers
23 views

How many tiles are Symmetrical? [closed]

We have a tape of type $1 * 2015$ had tile from tiles unit square in four different colors so as not exceed two tile of the same color (tile unit square, any tile from type $1*1$) How many tiles are ...
0
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0answers
39 views

Is there any practical use of this algorithm?

Example The exact solution of a DE $\frac{dp}{dv}=-1.4\frac{p}{v} $ with initial condition $(P_1,V_1)=(1,1)$ can be obtain by solving the integral ...
0
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1answer
14 views

Iteration Scheme Converging

I am in a class called Numerical Analysis and I have a quick question regarding iterative schemes. How would I go about finding out whether or not a certain iteration scheme converges to a unique ...
0
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0answers
22 views

Finding area by integration, increasing inaccuracies with complex functions?

I am looking for an explanation as to why the method of integration to find the area of function using limits provides a greater % difference between other methods (In this example Simpsons) with ...
0
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0answers
20 views

Error in Gauss-Legendre quadrature

I've tried Googleing this, but so far I haven't succeeded. Can someone point to me a webpage or book in which I can find the error estimate (detailed, not just the final formula) of the Gauss-Legendre ...
2
votes
1answer
20 views

Name and explanation of a Numerical Analysis method for solving systems of non-linear equations

In a non-english textbook of Numerical Analysis there is a method for solving systems of non-linear equations. But not only I can't understand how this method is used but I can't even found the name ...
2
votes
4answers
28 views

Curve fitting of a set of data

Suppose you have a set of data $\{x_i\}$ and $\{y_i\}$ with $i=0,\dots,N$. In order to find two parameters $a,b$ such that the line $$ y=ax+b, $$ give the best linear fit, one proceed minimizing the ...
2
votes
1answer
37 views

Classifying peak and valley *regions* of a histogram

I've been playing with a few ways of classifying contiguous regions of a histogram as: 1) peak, 2) valley, or 3) in-between bit. Global thresholding has worked minimally well for me so far, but I'm ...
1
vote
1answer
53 views

Finite Difference for Hamilton-Jacobi-Bellman without boundary conditions

Let $t\in\mathbb{R}_+$ denote time, $x \in X$ is the state and $u \in U$ the control. The objective function is $F:X \times U \to\mathbb{R}$ and $f:X \times U \to\mathbb{R}$ is the law of motion for ...
1
vote
1answer
22 views

Show that spectral radius is lower than 1

Consider the following matrix: $I - \frac{1}{h^2}\mu \Delta t A$. Where $A$ is an NxN matrix. The eigenvalues of A $\lambda_j$ are given by $4sin^2(\frac{j\pi}{2(N+1)})$ for $j=1,...,N$. And $\mu , ...
2
votes
1answer
13 views

finite volume methods: what do I have to do with the cell averages after each step?

I'm having a hard time understanding finite volume methods. If I take for example the scalar advection equation $$\partial{u}_{t}+a\partial{u}_{x}=0, a>0$$ with suitable initial and bondary ...
2
votes
0answers
22 views

What is a stochastic differential equation of the form $dZ = f(Z_{prev}, X_{prev})dt + CdW_t$ called?

At every time step I can approximate the change in $Z$ using the following equation: $$ dZ = f(Z_{prev}, X_{prev})dt + CdW_t, \quad(1)$$ $$dW_t = r\sqrt{dt}$$ where $C$ is some constant, and $r$ is ...
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0answers
8 views

Thresholding in spectra of partial traces of random symmetric matrices

I found an interesting behavior while looking at partial traces of random matrices. This is something I was studying numerically, and I haven't completely ruled out the possibility of numerical ...
2
votes
0answers
15 views

Reference request for finite difference method

I am trying to use finite difference method to solve the minimizing problem $$ J[u]=\min_{u\in BV(Q)}\{\|u-f\|_{L^1(Q)}+|u|_{BV(Q)}\} $$ where $Q=(0,1)\times (0,1)$ is a uint square and ...
2
votes
1answer
21 views

Approximate recursively defined error in fixed point iteration

Problem: With an initial guess of $x_0$, the fixed point iteration is given by $$x_{k+1} = e^{-x_k}, \mbox{ for } k=0,1,2,...$$ If $x^*$ is the exact solution, then the approximation error is $$e_n = ...
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votes
2answers
17 views

Solving ODE numerically, using derivative steps

Assume I have the ODE $\dot{p}(t) = f(t,p)$, with $p(0) = p_0$ and assume $f(t,p)$ (for simplicity) is only a function of $p$. I want to solve the ODE numerically, using derivative steps, kind of like ...
2
votes
1answer
41 views

How can I solve: $u_{xx} + u_{yy} = g(x,y)$ numerically?

If $u(x,y)$ is defined in $\mathcal{R} = {(x,y): 0 \leq x \leq a, 0 \leq y \leq b}$ $$ u(x,0) = 10 \mbox{ and } u(x,b) = 90, \mbox{ for } 0 \leq x \leq a \\ u(0,y) = 70 \mbox{ and } u(a,y) = ...
1
vote
1answer
88 views

Numerical precision of arctan function

I'm trying to convert points into spherical coordinates, do some filtering/manipulation of the points and convert them back into the Cartesian coordinate frame. These are my transformation equations ...