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

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Why do we get oscillations in Euler's method of integration and what is the period?

When using Euler's method of integration, applied on a stochastic differential eq. : For example - given $$\dot v = -\gamma v \Delta t + \sqrt{\epsilon \cdot \Delta t }\Gamma (t) $$ we loop over $$ ...
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180 views

implicit non-linear equations with complex variables

I am trying to understand a methodology for solving implicit non-linear equations with complex variables. I would like to solve for z1 below where z2 is known. Also both z1 and z2 are complex ...
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68 views

Numerical Computation for K smallest eigenvalues of a large Real Symmetric Matrix with restricted methods

I'm writing some code on a distributed platform, using some programming language like Hadoop, and now I need to calculate the K smallest eigenvalues for a Large Matrix. K is a small constant at most ...
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56 views

Binary search (bisection method) - is it worth checking continuity

I am implementing a rather simple matlab code, that gets a function $f$ and 3 real numbers $a,b, \epsilon$ where $\epsilon >0$ is a very small positive number (for instance, no larger than ...
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65 views

Fastest way of finding eigenvectors from eigenvalues

Given the eigenvalue of a matrix of large dimensions, I want to know if there is a fast way of finding the corresponded eigenvectors?
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17 views

Choleski's Algorithm Query

In Choleski's algorithm, I wonder how can one be sure that the diagonals elements of L (except for the first one) to be all positive?
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185 views

Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
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75 views

Inverse Iteration to Find Eigenvalues - Question about Method

So I'm doing Inverse Iteration in Excel to find the dominant eigenvalue and eigevector of a matrix. This particular method involves estimating an eigenvalue, multiplying the identity matrix by it, ...
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241 views

Implementation of Total Variation Regularization Algorithm (Lagged Diffusivity Algorithm)

I am trying to compute the derivative of an experimentally-measured quantity as a function of time. The data are fairly noisy, which causes problems. For instance, using finite differences (central ...
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34 views

Influence of preconditioning on degenerate eigenvectors

I'm using a hierarchical decomposition of a sparse matrix $A$ as suggested here. I find that the method essentially finds eigenvectors using the QR algorithm. $A$ has some eigenvalues with ...
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142 views

Newton's method for multidimensional functions

Can Newton's method be used to find the root of a function f : $\mathbb{R}^n\to\mathbb{R}^m$. Can anyone provide a proof for this? (I have checked the method of solving system of equations with ...
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34 views

Causality and viscous wave equation

I've seen several papers related to the causality condition concerning the viscous wave equation resolution but never understood how causality and stability are linked ? Conceptually, it seems hard to ...
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161 views

Change in Singular Value Decomposition of a matrix on addition of a single row

Given that I know the svd decomposition of a matrix, is there any way to compute the svd decomposition of the matrix obtained by adding a single row to the original matrix? Is there any relation ...
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68 views

heat equation with Interface Crank Nicolson

I am currently working on solving the heat equation with an interface numerically using Crank-Nicolson. There are jump discontinuities at the interface which are dealt with using fictitious values ...
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172 views

If symmetric matrix in a least-square deconvolution problem positive definite?

I want to apply Gauss-Seidel method in a least square deconvolution problem. The convolution of two vectors is written in: $h * x = z$. $$z(n) = \sum_{i=0}^{N-1}h(i)x(n-i)$$ It is a linear transform ...
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176 views

Calculating critical value for Student's t-distribution through numerical approximation, for arbitrary values of degrees of freedom

I'm trying to make a plot in LaTeX/tikz which is supposed to show the progressive critical value for a t distribution of ...
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76 views

Sequence and intermediate value theorem

For part (a), By intermediate value theorem, there exist c between 0 and 1 such that $f(c) = 0$ Now, I supposed that there also exist d between 0 and 1 such that $ f(d) = 0 $ and $ c \neq d $ I am ...
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35 views

Find the real roots

I have the following equation: \begin{equation} (y-1)^a - C~~ y~~ \exp(b x)=0 \end{equation} where $a, b$ are real constants, $C$ may be a complex number. I need to find the real solution of the ...
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33 views

Taking the limit of a derived function

I need to evaluate an expression of the form $$ \int_0 ^ a dx \left[\frac{\partial}{\partial \alpha} \left[ \frac{\partial^n}{\partial \beta^n} \left[\frac{\partial}{\partial \gamma} ...
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112 views

Optimal numerial method for optimization of “Rosenbrock Banana”-like function

Which numerical methods would be optimal to find an extremum of a function with an almost flat "valley" (but a single minimum in the middle of the valley)? In this context optimal means the least ...
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54 views

Stability conditions

Below is a problem about stability conditions that I have been struggling with it during an exam: Find the stability conditions for $$A\left ( \frac{\partial^2 u(x,\, y,\,t)}{\partial x^2} + ...
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71 views

Numerical algorithm to solve quadratic eigenvalue problem.

Given the equation $$-4 \left(a^2+a (n-1) (2 t^2-1)\right) \left(\sum _{i=0}^n \alpha_i t^{2 i}\right)^2 \\ +\frac12 \left(\sum _{i=0}^n \alpha_i t^{2 i}\right) \left(t \left(8 a (t^2-1)+1\right) \sum ...
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28 views

What is the derivative of $\frac{f^{(3)}(\xi(x))}{6}$ at $x=x_0$

The error of interpolating polynomial is $$ E_n(x)=\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x)) $$ The derivative of $E_n(x)$ is $$ ...
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32 views

How to establish a lower bound on this difference operator?

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
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30 views

characterising attractors for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
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56 views

Continuation fixed points of parameter dependent Newton

Suppose I have the iteration operator of the Newton method for some $\beta$-parameter dependent function $g_{\beta}: \mathbb{R} \rightarrow \mathbb{R}$. Let us assume that $g_\beta$ is in ...
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51 views

Scalable QR decomposition algorithm

Suppose one has a processor for QR decomposition of complex matrix of size 4 x 4. So if it is necessary to decompose M x M complex matrix, A, one can represent it as R x R block matrix [Cij] (block ...
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56 views

Condition number composite function

I have a composite function $h(t)=g(f(t))$ and have to evaluate the condition number for $h(t)$ through the condition numbers of $g$ and $f$. I know that the condition number formula is ...
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231 views

Autocorrelation Function and Power spectrum from ACF

In my assignment I am required to write or use a C code to find the autocorrelation function of a given function and then find the power spectrum from it. The function is as follows: $$f(t) = \cos(10 ...
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163 views

Initial approximation to inverse of beta distribution function / quantile of beta distribution

I'm interested in implementing an algorithm to find the quantile of the beta distribution, and I'm looking at this paper: Journal of the Royal Statistical Society Series C (Applied Statistics). 1973, ...
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28 views

computing length of a curve given as set of points

Given a set of points $(x_i,y_i)$ from a simple curve. How can the length of the curve be computed approximately?? I understand these points can be connected by line segments and the sum of the ...
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efficiently solve for values of a coefficient in a function, so for those values, the function intersects another function a specific number of times.

This is my summer assignment for my freshman "Intro to Numerical Methods with Matlab: Unit 2" course. The task: "Write an efficient Matlab code, which will take any closed $f(x)$ and $g(x)$ and ...
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98 views

numerically solve quadratic air drag in xy-plane

I am trying to find a reference on solving for the position of a point mass as a function time, subject to air drag( quadratic term only) in both the x and y directions. The equations that describe ...
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48 views

Generalized Logarithmic Integral

Euler's logarithmic integral (of particular application in the Prime Number Theorem, for instance) is of the form \begin{equation*} \text{li}(x) := \int_0^{x} \frac{dt}{\log t} \end{equation*} and it ...
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1k views

Change MATLAB code from Lax-Wendroff to Leapfrog

I want to see how leapfrog would look using this code, but I'm having issues implementing it. I think my biggest problem is adding in the $ U_j^{n-1}$ term, I just don't get the logic. Here's what ...
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51 views

Solving ODE numerically - getting local truncation error

Well I have NO idea how to do this or even where to start Compute the order of magnitude of the local truncation error of the following time integration scheme: $$y_{n+1} = y_{n-1} + 2h f(y_n)$$ H ...
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39 views

Simultaneous iteration of Symmetric Matrices

Given a Matrix $A$ we can use Simultaneous iteration(Using power iteration on all columns simultaneously) to compute the d biggest eigenvalues. Now this method will give you the biggest eigenvalues, ...
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46 views

Gerschgorin Theorem singularity proof

I know how to prove the Gerschgorin Theorem but how exactly would one show that there are no values of $\mu$ s.t. $\mu<0$ for which $A-\mu B$ is singular where $$ A= \begin{bmatrix} ...
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33 views

Gaussian Elimination theoretical question

You know how Gaussian Elimination can be broken up into a sequence of L-U premultiplications right? Suppose that there is a matrix $A=a_{i,j} : j=1,...,n$ is an $n × n$ real matrix such that ...
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234 views

Aitkens Extrapolation

The Aitken's extrapolation can be written as $$X^n = X_{n-2} + \dfrac{(X_{n-1}- X_{n-2})^2}{(X_{n-1}- X_{n-2})-(X_n- X_{n-1})}$$ Verify it? And $X^n$ can be viewed being defined recursively by ...
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34 views

floating-point operations do not satisfy the well-known laws for arithmetic operations

Introduction to Numerical Analysis, Stoer, Chapter: Error Analysis, Page 8 if $|y|<\frac{eps}{\beta}|x|$ where $eps = 0.5\times 10^{1-t}$ then $$fl(x+y)=x+^*y=x$$ where $fl(x)=$ normalized ...
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29 views

Computing area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions

I need to compute the area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions. As I am a non mathematics guy, it will be great if someone helps me out with the ...
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126 views

Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
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161 views

Fourth order Runge-Kutta method validity

I wonder whether the fourth-order Runge-Kutta method is suitable for a second-order linear ODE with dissipative terms modelling free fall of an object through a viscous medium under the act of ...
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By what factor do winning chances increase based on total value?

Say I am entering 24/7 in endless sweepstakes, contests, giveaways, drawings, etc. Assuming each one I enter has no less than 1 in 1,000 chances, but no more than 1 in 1 million (and I enter at least ...
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55 views

Numerical method for indefinite integral

let be the indefinite integral $$ F(x)= \int_{0}^{x} g(t)dt $$ the integral depends on the parameter 'x' i can use a linear transformation to turn this integral into $$ F(x)= ...
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66 views

Solve Poisson's equation

I want to solve Poissons equation $$ C=\nabla^2 v $$ where $C$ is a constant and v my variable. I want to solve over some 2D domain D with the boundary condition that v is zero on the edge. How does ...
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83 views

Piecewise Linear Rayleigh-Ritz

I am trying to get this equation $-x^2y'' - 2xy' + 2y = -4x^2$ into Piecewise Linear Rayleigh-Ritz format $-\frac{d}{dx} (p(x) \frac{dy}{dx}) + q(x)y = f(x)$ I pretty much needs to figure out what ...
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25 views

How does one prove that a linear multistep method of order p can recover all polynomials up to and including order p?

It is intuitive that all polynomials up to and including order $p$ can be fully recovered i.e. without error, but how can one rigorously prove this? In the book by Lambert, there is a similar ...
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91 views

Simpson's rule error rate for N-dimension

I'm doing a project that involves numerical method, but I'm not too familiar on calculus. I'm using Simpson's rule to integrate n-dimension gaussian, I was able to get the integration result for ...