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4
votes
1answer
393 views

Why my linear congruential generator generate low quality random numbers?

I am implementing an arbitrary bits random number generator (in [$0,2^n$)) and also want to ensure that the generator always generate unique numbers until all possible numbers generated, so I ...
4
votes
1answer
432 views

Best platform for complex SDPs with n and m around 5-15K?

I am looking to solve a class of SDPs with complex entries, with the semi-definite cone $S^n$, $n$ around 5000 to 15000. Also, $m$, the number of equality/inequality constraints is close to $n$. I ...
4
votes
1answer
76 views

trilinear hex elements

Do the faces of tri-linear hex elements have to be planar? Three nodes define a plane. If the fourth node does not lie on the plane, then the nodes are not planar and the face is not plane. In general,...
4
votes
1answer
221 views

Solving a nonlinear equation with random variable

I would like to solve an equation that looks like this UPDATE $E[(R^{1-\gamma})(r_k+\theta-r_z)]=0$ , where $R=\phi r_z+(1-\phi)(r_k+\theta)$ and $\phi\in[0,1]$, $\theta$, is a random variable ...
4
votes
1answer
2k views

Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method

I want to numerically solve the non-linear diffusion equation: $$ \frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right) $$ I want to use ...
4
votes
2answers
1k views

Computational complexity and implementation of UDU Modified Cholesky Rank 1 Update

I am attempting to increase the performance of a legacy Kalman Filter implementation. The state covariance is factored in terms of UDU, i.e. $\mathbf{P} = \mathbf{U}\mathbf{D}\mathbf{U}^T$. Many ...
4
votes
3answers
134 views

Algorithms to generate spherical codes

A spherical code, specified by the parameters $(n,N,t)$, is a set of $N$ coordinates on the $n$-dimensional unit hypersphere such that the set of dot products between any two unit vectors from the ...
4
votes
2answers
5k views

Creating 3D Mesh from stl files with gmsh

After long hours of searching for an answer I thought it might be better to ask the community. The problem I have is that I need to convert STL files to mesh files. I know that I therefore need to ...
4
votes
1answer
259 views

Multigrid stops converging when more grid levels are used

I'm having a problem with multigrid code I wrote. If I solve Laplace's equation in 2D and use more than 5 grid levels, the V-cycles stop converging after a few cycles (see below, convergence factor > ...
4
votes
0answers
394 views

Iteratively finding both left and right eigenvectors for non-symmetric complex matrix

I have a complex, non-Hermitian matrix $\mathbf{A}$, for which I need to find a few eigenvalues and eigenvectors in the generalised eigenvalue problem: $$\mathbf{A}\cdot \mathbf{x} = \lambda \mathbf{...
4
votes
1answer
67 views

How to deal with pseudo-compressibility of lattice Boltzmann method when you are calculating mass flux?

In lattice Boltzmann method, we have a concept, which is called pseudo-compressibility and it is defined based on the fact that LBM simulates incompressible flows by having small Mach number to ensure ...
4
votes
4answers
179 views

Can the mesh generation methods in FVM and FEM be totally based on the knowledge of the mesh generation theory in computer graphics?

The main references of mesh generation methods in computer graphics (CG) I found are discrete Differential Geometry [1] and a famous book "Polygon Mesh Processing" [2], while the "...
4
votes
1answer
276 views

Points on the interface

We consider the problem $\left\{\begin{matrix} k(x)\Delta u(x)=f(x) & \text{ in } \Omega\\ u=0 & \text{ in } \Gamma \end{matrix}\right.$ where $\Omega \subset \mathbb{R}^2$ open and ...
4
votes
2answers
143 views

Term for the typical “linear in the larger dimension, quadratic in the smaller” cost for linear algebra

Many dense linear algebra decompositions (QR, SVD...) on an $m\times n$ matrix have cost $$ O(\max(m,n)\min(m,n)^2) $$ when implemented in practice on a computer. Is there a colloquial name or a more ...
4
votes
1answer
792 views

Thomas algorithm for 3D finite difference

For 1D finite difference, the resulting linear system is tri-diagonal and can be solved in O(n) using the Thomas algorithm. I am trying to solve a finite ...
4
votes
3answers
654 views

Backward stable projection and normalization of a vector

Given a machine precision unit vector $n$, and an arbitrary vector $v$, I want an unconditionally backward stable method to compute $$f(v) = \frac{v-nn'v}{\left|v-nn'v\right|}$$ In other words, ...
4
votes
2answers
154 views

Maximization variant of semidefinite programming (SDP)

Consider the following program: $$\max_{\pmb a} \sum_i z_i\\ u.c. \pmb a \pmb P_i\pmb a^\top\geq z_i$$ where $\pmb a \in\mathbb{R}^p$ and the $\pmb P_i$ are all symmetric positive semidefinite ...
4
votes
2answers
980 views

Maxwell vs Kepler for GPU computation

I am looking at the Nvidia GT-860M which comes in both the old Kepler and new Maxwell architecture. The old one (1152) seems to have almost twice the cores as the new one (640). The new one has a ...
4
votes
2answers
197 views

Finding dominant eigenvectors of an operator that is small but costly to evaluate

Suppose I have a symmetric linear operator $A:\mathbb{R}^k \rightarrow \mathbb{R}^k$ where $k$ is "small" (eg., $k=100$), and I want to find it's first few eigenvectors, (eg., $10$ eigenvectors). If ...
4
votes
2answers
405 views

LAPACK - singular matrices - what does the positive integer info mean?

please can you help me with my code - I use Lapack to solve complex matrix (quite biq) and do it in two steps: I call zgetrf (LU factorization) and then ...
4
votes
1answer
191 views

Numerical evaluation of the Exponential Integral Ei by rational Chebyshev approximations fails

I am trying to evaluate the Exponential Integral $Ei(x)=-\int^{\infty}_{-x}\frac{e^{-t}}{t}dt$ for $x>0$ (interpreted as the Cauchy principal value) by using rational Chebyshev approximations, ...
4
votes
1answer
126 views

Floating point and global error in Euler Method

Inspired by this answer, I tried to understand when floating point errors come into visibility and to check it also comparing the plot of the numerical solution with Explicit Euler with the analytical ...
4
votes
2answers
3k views

Testing 1D Poisson Solver

I'm trying to test a simple 1D Poisson solver to show that a finite difference method converges with $\mathcal{O}(h^2)$ and that using a deferred correction for the input function yields a convergence ...
4
votes
1answer
233 views

Choosing preconditioner for unsymmetric pressure-velocity coupled system

I'm working with pressure-velocity coupled systems. It means that instead of solving 4 different linear systems in segregated approach (1 for pressure and 3 for Ux, Uy, Uz), we can solve only one ...
4
votes
3answers
443 views

How to implement this trigonometric polynomial maximum finding semidefinite program

Hi All, I posted this on the math.se site, but this may be a better location. I need a method of finding the maximum of a real valued trigonometric polynomial where I can trade accuracy for speed. ...
4
votes
3answers
2k views

4th-order Runge-Kutta method for coupled harmonic oscillator

I’m attempting to write a C program to gather values from a coupled spring system: There is a wall, connected to a mass $m_1$ by a spring, then this mass is connected to a second mass $m_2$ by another ...
4
votes
0answers
79 views

Large residual when integrating 2nd order ode close to singularity with SciPy ode / ODEPACK

I am trying to integrate a 2nd order ODE with a singularity at close to the initial condition. Why do I get large residuals when I plug-in the result of my integration back into the ODE? The equation ...
4
votes
1answer
131 views

Constructing the origin position by transforming distance information

Suppose a set of $n$ points, $n\in M$, is given in some $d-$dimensional space, $X\in\mathbb{R}^{n\times d}$. Among these $n$ points, some $k\in K$ are selected, so $k<n$, and the distances from ...
4
votes
1answer
485 views

Efficient and stable computation of inverse CDF

What is the most efficient and numerically stable algorithm for computing the inverse CDF $F^{-1}(y)$ of a probability function, assuming that both the PDF $f(x)$ and the CDF $F(x)$ are known ...
4
votes
1answer
164 views

Factorization for reweighted least squares

I am solving a problem using an iteratively-reweighted least squares method: http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares Essentially this requires solving a number of least-...
4
votes
2answers
207 views

How do the properties of a matrix affect the linear system solving

For a general matrix A, there are many properties to describe it: symmetric positive definite or indefinite, condition number, spectrum and so on. I am curious about how these properties affect the ...
3
votes
2answers
388 views

Knapsack problem with fixed number of elements?

I am looking at an optimization problem that looks like this: $$ \text{minimize}\;\; \mathbf{x}^TQ\mathbf x \;\;, \; \mathbf x \in \mathbb R^n, x_i \in \lbrace 0, 1 \rbrace\\ \text{subject to}\;\; ||...
3
votes
1answer
56 views

Shall I use global, heap allocated array or local, stack allocated one if references to its elements are made too many times?

I actually have this data locality as a possible problem for why my fortran program runs somewhat slow. In one part of this program, I have nested loops and throughout these loops, a given section of ...
3
votes
3answers
241 views

What numerical methods are used to model deformations in elastic physics?

What numerical methods are used to model deformations in elastic physics? For example, here's an example of a hyperelastic deformation in Ansys: Perhaps more simply than hyperelasticity, for linear ...
3
votes
1answer
125 views

Quick way to find a common basis of eigenvectors between 2 matrices : valid or not?

Following the advise of @Federico Polonion a previous post, one suggested, to find a basis of common eigen vectors between 2 matrices, to simply do : Generate 2 ...
3
votes
3answers
301 views

$H^1$-convergence rate of finite element method for Poisson equation, depending on element order

I wanted to verify my FEM-program by applying the method of manufactured solutions, while solving the Poisson equation in two dimensions using the continuous Galerkin method $$-\nabla^2u=f$$ with $$u=...
3
votes
1answer
519 views

Solving the heat diffusion equation with source term

I am trying to solve the 1-D heat equation numerically with a variable source term. The system is basically a tank containing styrene in which it polymerizes to liberate heat. I have assumed that the ...
3
votes
0answers
112 views

Backward stable algorithm to get orthogonal projection onto the column space of a matrix

I have to find the orthogonal projection of a vector $b$ onto the matrix $A$ of size $m \times n$. In my application, I don't have the luxury of calculating the QR factorization. All I have are ...
3
votes
2answers
958 views

Graphing electric potential of a ring of charge using MATLAB help

Here is a summary of what I am trying to do: Use MATLAB to compute the potential $V$ at any point $(x, y, z)$ in space due to a uniform ring of charge. Use a Riemann sum to compute the integral ...
3
votes
1answer
426 views

Setting up optimization problem in GEKKO

I have the following dynamical system, $\frac{d \phi}{dt} = -M^TDM\phi \tag{1}\label{1}$ $\frac{d \hat\phi}{dt} = -M^T\tilde{D}M\hat \phi \tag{2} \label{2}$ $\eqref{1}$ represents the exact ...
3
votes
1answer
162 views

Derivatives of Approximate Matrix inverses

I am cross posting this question to the mathermatics stack exchange. please find it either at this link, https://math.stackexchange.com/q/2952989/430980, or below: I have a question concerning the ...
3
votes
0answers
146 views

Automatically generate constraints for trajectory optimization

This is a follow up to my previous post here I'm interested in performing trajectory optimization from the problem mentioned in abov link. I want to supply the following as dynamical constraints to ...
3
votes
1answer
839 views

Finite difference method basic implementation on Octave

Trying to study the error of FDM for a second order derivative versus step size I calculated the coefficients and validated them, but the output has errors for small step sizes. The function in ...
3
votes
2answers
200 views

1D FEM for nonlinear diffusion coefficient

I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$ in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$. ...
3
votes
1answer
3k views

Solving Advection (Convection) - Diffusion - Reaction Partial Differential Equation in Python

I am looking for library written in Python which will enable me to solve the coupled nonlinear equations which looks like: I need the library which will enable me to couple this solver to other ...
3
votes
2answers
3k views

C++ Library: What is the common libraries that do polynomial arithmetic?

I need to know what libraries (in C++) support polynomial arithmetic specially over a field. So I can give to it an array of coefficients of polynomial over a field and it returns the roots of ...
3
votes
2answers
2k views

What is the difference between “Newton-type” and “Newton-like” iteration?

Is there any clear classification between different iterative methods? What is the difference between Newton-type and ...
3
votes
2answers
186 views

Optimization of known function with respect to two unknown function arguments

I have a data set, composed of points $(x_i, y_i)$ for $i=1,N$. I also have a known function $F$, which maps these points $x_i$ to $y_i$ as such $F(x_i, a(x_i),b(x_i)) = y_i$, where $a(x_i)$ and $b(...
3
votes
1answer
356 views

singular value decomposition of a 2 x 2 complex matrix

This should be easy, but... I would like to express the singular value decomposition of a 2 x 2 complex matrix $A$ as function of its coefficients $A_{ij}$. In "closed form", no intermediate values, ...
3
votes
3answers
517 views

Numerical approximation for a known exact solution of advection-dispersion equation

My goal is to create a numerical solution of 1D-solute transport (Convective-dispersion equation, CDE) to match it's analytical solution based on experimental data. The CDE can be written as (where, C=...

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