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

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approximation of a complex valued real rational function

all. I am now struggling with a approximation problem. Suppose we have a matrix-valued measure $\mathrm{d}\Lambda(\omega)$, with compact support $[a,b]$, then its Cauchy transform is a well-defined ...
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24 views

Is there a numerical method to restore symmetrizability and row-stochasticity in a matrix?

I have a radiation-transfer matrix F that had been computed using some raytracing Monte Carlo method. From physics it is clear that it must be row-stochastic :$\sum_{j=1}^{N}F_{ij}=1$ because energy ...
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82 views

Numerical analysis- Runge Kutta

I have: $$y'(x)= \sin(y); y(0)=1$$ I need to calculate the function values with Runge-Kutta. The range is [0,1]. My problem is that I need to choose the h (=dx) such that the error will be in order ...
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41 views

Understanding what to do for relative error when p = 0 (bisection method)

I noticed this was mentioned in class, but the detail wasn't really given as to how to deal with it (outside of using another error method such as absolute error). Given the relative error of the ...
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414 views

Local truncation error of Euler method

Wikipedia and this book say the local truncation error of Euler method is $O(h^2)$. But this book and A friendly Introduction to Numerical Analysis say it's $O(h)$. Which is correct? I have a similar ...
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246 views

How to find a separating hyperplane?

I know about support vector machine, and it's quadratic programming approach which delivers the best separating hyperplane. My question is: is there a relatively simple algorithm to find a non-...
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62 views

Solving ${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x \le c_2$

Is there any way to solve $${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x\le c_2,$$ for $x>1$, $0<c_1<1$, and $0<c_2<<1$? Thanks
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201 views

stability of FTCS scheme for parabolic equation

Can you suggest any method for stability analysis of FTCS scheme for the the following parabolic equation ? D.E: $u_{t}=a(x,t)u_{xx}+f(x,t,u)$, $0<x<1$, $0<t<T$, $T>0$ BCs: $u_{x}(0,t)...
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78 views

Will Gauss quadrature numerical integration work with a variable dx

The question kind of says it all, but I'm reading about Gauss quadrature from here: http://www.damtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/integration.htm which gives an equation of this form: ...
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327 views

Finite Difference - Forward Difference with 2nd order Accuracy: What to do at the boundary?

I implemented a BVP using a first-order finite difference scheme after the shooting method did not work reliably. Its the first time I have worked with this. The code works but I would like to move to ...
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69 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int \limits_{...
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40 views

Numerical methods for computing exponential, if I have computed an exponential of a perturbated matrix

I need to compute the product $e^{H_1}\,e^{H_2}\,\ldots\,e^{H_n}$ for antihermitian matrices $H_j$ that do not commute and $H_i-H_{i+1}$ is small. Is there a numerically convenient way to compute $e^{...
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67 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where $X$...
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32 views

Blended surface

Partially blended surfaces are extensively used in the literature for shape preserving interpolation. Most of these shape preserving partially blended surface interpolation is based on the result that ...
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134 views

efficient least squares A = BX+CXD (solving for matrix X)

I am interested in solving a least-squares solution of the form $$ \operatorname{argmin}_X \| A - BX - CXD \|_F^2 $$ for large (rank in hundreds to thousands) matrices $A,B,C,D,X$ I know this is ...
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33 views

Partial integro-differential equation methods

Which method is best suited to solve an elliptic partial integro equation? Is finite difference for the derivatives and composite trapezoidal rule for the integral part stable?
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56 views

Approximations for finite n in limit-based definition of the exponential function

The exponential function can be defined via: $$ e^x = \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} = \lim_{n \rightarrow \infty} g(x; n) $$ In my problem, I actually have the right ...
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56 views

Which (approximative) methods are there to compute the inverse of a complicated function?

I have a complicated function $f(x)$ for which I want to compute the inverse $f^{-1}$ over a certain range $R(f): a \leq f(x) \leq b$. The only way to find the inverse I can think of is power series ...
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31 views

Integration respect the Lévy measure

How I can compute numerically $$\int_{a}^{b}f(z)\nu(dz)$$ where $\nu$ is a Lévy measure and $f$ is a continuous function? Is it equivalent to $$\int_{a}^{b}f(z)d\nu(z)$$ Thanks
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119 views

Error bound by the Simpson's rule

My lecture notes have a little exercise. Two functions are given: $$ f(x) = \cos(x) \ \text{and} \ g(x)=\sqrt{x+1} $$ And we're asked about the error bound of the Simpson's rule to estimate the ...
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66 views

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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188 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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70 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 $10^{-10}...
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66 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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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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191 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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76 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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243 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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144 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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163 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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182 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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180 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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80 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} g(x,\alpha,\...
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115 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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72 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 $$ E^{\prime}_n(x)=\frac{d}{dx}\left((x-x_0)(x-x_1)\...
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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}, D^-_xV_{i}=\dfrac{...
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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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57 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 $C^{\infty}...
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52 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 $\frac{xf'(x)...
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238 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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171 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, ...