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9
votes
1answer
2k views

Schrodinger equation with periodic boundary conditions

I have a couple of questions regarding the following: I am trying to solve the Schrodinger equation in 1D using the crank nicolson discretization followed by inverting the resulting tridiagonal ...
9
votes
1answer
646 views

Matrix Balancing Algorithm

I have been writing a control system toolbox from scratch and purely in Python3 (shameless plug : harold ). From my past research, I have always complaints about ...
9
votes
0answers
194 views

Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem Solving a non-linear system of equations. The number of variables is the same as the number of equations. When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
9
votes
1answer
669 views

What numerical quadrature to choose to integrate a function with singularities?

For example, I would like to numerically compute the $L^2$-norm of $\displaystyle u = \frac{1}{(x^2+y^2+z^2)^{1/3}}$ in some domain that includes zero, I tried Gauss quadrature and it fails, it is ...
8
votes
2answers
880 views

How to solve the stiff equation in this Restricted Three Body Problem numerically?

I've come across a stiff equation in solving the Circular Restricted Three Body Problem. [An object is moving considering the effect of the gravitational forces caused by two gravitational sources ...
8
votes
1answer
1k views

Which numerical methods preserve time reversal symmetry?

If I have a physical system which contains a time reversal symmetry (for example a Hamiltonian $H(x,p)=p^2/2m + V(x)$ with $V(x)$ real) and I want to solve the differential equations which describe ...
8
votes
4answers
532 views

Prolonging PBS job

It is quite painful to discover that a few-day long job is going to be prematurely killed due to an error in setting walltime limit for it. Is there a way to change it for a running PBS job?
8
votes
2answers
599 views

Applying matrix square root inverse in matrix-free regime

Let $A$ be a large symmetric positive definite matrix, and suppose that we can efficiently apply $A$ and have a fast solver to apply $A^{-1}$, but we do not have access to the matrix entries for ...
7
votes
5answers
305 views

External hardware resources for running long and computationally intensive simulations

I've written code for an obstructed random walker simulation and I want to run long simulations (6 hours or more on my computer). I don't want to run this simulation on my computer because I will want ...
7
votes
2answers
5k views

FEniCS: how to specify boundary conditions on a circle inside 2D mesh

I would like to numerically find a mutual capacitance of two stripes of metal on the opposites sides of a cylinder. The problem is obviously a 2D Laplace equation. I would like to find the potential ...
7
votes
1answer
663 views

full rank update to cholesky decomposition for multivariate normal distribution

This question is a specialization of full rank update to cholesky decomposition, to which I hope to get a more positive answer. When calculating the minus log of the multivariate normal distribution, ...
7
votes
4answers
2k views

Finite Difference Method Stability

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
7
votes
1answer
597 views

Which preconditioning for large linear elasticity problem?

The problem I want to solve is the displacement formulation of the linear elasticity : $$ \nabla \cdot \sigma = 0 \quad \text{in} \quad \Omega \\ \sigma = \lambda ( \nabla \cdot u ) I + \mu (\nabla \...
6
votes
1answer
198 views

Symmetric matrix which satisfies conditions of the form $v_i^T X v_i = 0$

I want to solve an underdetermined system of linear equations $A x = b$ with $A \in \mathbb{R}^{n \times r^2}, x \in \mathbb{R}^{r^2}, b \in \mathbb{R}^n$. The matrix $A$ has the following additional ...
6
votes
1answer
1k views

Algorithm for solving system of quadratic equations and linear equations

Let $x \in R^N$. From a Spectral Chebyshev collocation method, I have a system of quadratic and linear equations. Denote them, $$ x^T Q_i x + L_i^T x = 0 $$ and $$ A x = 0 $$ Furthermore, I know ...
6
votes
2answers
2k views

Sparse Incomplete Cholesky

I'm looking for an efficient, multicore, library to do incomplete cholesky (possibly modified). Many ILU code exists, but I can't find much about IC except in PETSC or Pastix. Could some of you drop ...
6
votes
3answers
520 views

How to solve a small least-squares problem

This question is not very deep. Suppose I have a small rectangular matrix $A$, with number of rows and columns between $50$-$100$, respectively. Given a right-hand side $b$, I want to solve the least-...
6
votes
1answer
1k views

generalized eigenvalue problem

I need to solve a real generalized eigenvalue problem $Ax= \lambda Bx(*)$ A and B are calculated from equations below: $$A=\sum_{i,j=1}^{N}W_{ij}(K_{i}-K_{j})\beta\beta^{T}(K_{i}-K_{j})^{T}$$ $$B=\...
6
votes
1answer
392 views

Caveats of Hessian free method

Hessian free iterative optimization techniques like Newton-CG, do not explicitly compute the Hessian but instead approximate the product of the Hessian with a vector through finite difference. The ...
6
votes
4answers
2k views

Efficient eigen-decomposition of covariance matrix

I am looking for an C/C++/Python algorithm implementation that calculates eigenvalues and eigenvectors of a symmetric, positive semidefinite covariance matrix. A general-purpose eigen-decomposition ...
6
votes
2answers
292 views

Is it possible to ignore/discard part of a matrix when finding eigenvalues?

I have have multiple large matrices for which I need to find the largest absolute eigenvalue. I know that there is a large submatrix that does not vary. Is it possible to ignore/discard the submatrix? ...
6
votes
1answer
208 views

Does mean removal increase accuracy of numerical differentiation?

I wish to compute the derivative of a vector through numerical differentiation. Let's say, we use a standard 2nd order central difference scheme, to arrive at a differentiation matrix, and apply it on ...
6
votes
2answers
235 views

Continuation procedure to solve for a 2D curve that satisfies f(x,y) = 0

I have some function of $R^2$, that must be numerically computed. For instance, I might be interested in a real-valued contour integral that begins from (x,y) = 0. $$ f(x,y) = \Re\left[\int_0^{x + iy}...
6
votes
0answers
272 views

Integrating highly oscillatory functions

I have a logarithmic grid, upon which i have two functions that are similar to this one (this is only the last 100 points): These are essentially very similar to a Sin function at this point. I need ...
6
votes
1answer
663 views

Estimate extreme eigenvalues with CG

CG may be used to estimate the extremal eigenvalues of a SPD matrix (by computing eigenvalues of tridiagonal matrix associated with the Lanczos algorithm). After a few iterations the largest ...
6
votes
4answers
2k views

Need Fortran 77 compiler

Does anyone know a compiler for Fortran 77 available as a free download? I have pre-written 77 code from a source published in the early 90's that I just need to compile, build, and run. But I don't ...
6
votes
2answers
1k views

matlab eigs: wrong eigenvalues for tridiagonal matrix

I try to compute eigenvalues of the tridiagonal matrix coming from finite difference scheme. For small mesh size, eigs works well. But for large size it fails. Here is an example where eigs fails. Is ...
6
votes
1answer
170 views

Can heat distribution in an optical element irradiated by laser be oscillating?

I am modelling a heat distribution in optical element irradiated by laser. System is radially symmetric, and element is thin, i.e. heat value depends only on distance from center. Heat is received via ...
6
votes
1answer
816 views

Inverse iteration to find the null singular vector of a rank-deficient matrix

I have an $n \times n$ unsymmetric matrix $A$ that results from the discretization of an ill-posed Poisson problem, and thus is rank-deficient with null space of dimension one. I want to compute just ...
6
votes
2answers
464 views

Implementing the pressure correction method using finite elements

Ok so I am nearing the completion of my finite element Navier-Stokes solver that uses the $\theta$-method for time stepping and the pressure correction method for the pressure. I am following the ...
6
votes
2answers
943 views

Python package for (adaptive) function plotting

Are there any mature Python packages that can plot functions, and possibly use adaptive sampling? I am looking to pass only a function (can be a numerical black box) and a range, and expect a plot as ...
6
votes
1answer
489 views

Defining a condition number and termination criteria for Newton's method

The condition number of function evaluation $$ \mathrm{cond}(f,x) := \left| \frac{x f'(x)}{f(x)} \right| $$ is infinite at a root of $f$. Hence it is useless for rescaling a tolerance which defines an ...
6
votes
1answer
176 views

A simple PDE solution question

I need to ask a question about partial derivatives. I want to solve this equation (steady state, one dimensional continuity equation): $$\frac{\partial (\rho u)}{\partial z}=0$$ which is equivalent to:...
6
votes
1answer
2k views

Why does std::complex<> initialize its value to 0 upon default construction?

Doing so strikes me as a waste of time. Consider std::complex<double> *a = new std::complex<double>[1<<28]; This could be near-instantaneous ...
5
votes
1answer
2k views

FEM: which is the correct way to impose Dirichlet B.C

I know the Neumann B.C. is implicit in FEM language. However, I have seen at least two ways to impose Dirichlet B.C. e.g. for the following problem 1D, $$\nabla^2 u + \nabla u= 0, u_{left}= 1, u_{...
5
votes
1answer
978 views

How is rigid bodies implemented in finite element codes

I am writing a finite element code for structural analysis, and I want to implement rigid bodies. How is this usually done? Say that I have a square mesh, with one half of the mesh being defined rigid ...
5
votes
2answers
226 views

Is there a special algorithm for computing the convex hull ordering when the candidate points are on the hull?

I'm dealing with a set of points which are already placed on the 2D hull boundary: a convex polygon. I know this for sure. However, the point set is not ordered, and I need the polygon points to be ...
5
votes
2answers
1k 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 ...
5
votes
2answers
1k views

minimization problem: sum of Rayleigh quotients

I would like to find $x$ which minimizes the following equation: $\frac{x^HAx}{x^HBx} + \frac{x^HCx}{x^HDx}$ where A, B, C, D are positive-definite. $x$ is not a very large vector (<1000 elements ...
5
votes
3answers
218 views

Is there a way to reduce aberration in computations of planets' trajectories?

I don't think the title is very accurate , sorry for that. I simulate bodies in space using two timestep: the TIMESTEP is the Δt wich I use to make the calculation and XTIME is the number of times ...
5
votes
1answer
197 views

What is the correct way of performing numerical experiments on desktops?

Suppose I want to set up an experiment to measure the performance of some numerical code on a desktop machine running Linux/Windows/MacOS. What kind of environment should I arrange in order to get ...
5
votes
3answers
354 views

PRIMA gives an unstable result?

I am working with Modified Nodal Admittance representation of circuits. I am doing Model Order Reduction using PRIMA on MATLAB. I am considering these circuits as Descriptor State-Space systems. I ...
5
votes
2answers
185 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 ...
5
votes
1answer
527 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 ...
5
votes
0answers
88 views

Interface Formulation at Finite Volume Boundaries when using the Dual Mesh

When using the dual mesh (vertex-centered) for finite volume methods, you end up with a cell center at the boundaries between materials. It is possible that the equations being solved in each ...
5
votes
0answers
714 views

Finding roots of systems of equations with a Jacobian that is singular everywhere

Given $a\in\mathbb{R}^{mn\times n}$, find a $C\in\mathbb{R}^{n}$, $x\in\mathbb{R}^{m\times n}$ such that $$ 0 = f_{k}(\boldsymbol{C}, \boldsymbol{x}):=\sum_{i=1}^{m} C_{i} \left(\prod_{j=1}^{n} a_{kj}^...
5
votes
1answer
306 views

Convergence of Jacobi's method for a semilinear elliptic PDE

I have an iterative finite difference scheme for the Poisson equation $$ \nabla^2 u=-\rho $$ It's the Jacobi method, which has the form (for 1D systems) $$ u^{n+1}_{i} = \frac{1}{2}(u^n_{i+1} + u^n_{...
5
votes
2answers
3k views

Fast algorithms to find the eigenvalues of some matrix on intervals of interest

I am curious how to quickly compute the eigenvalues for arbitrary matrices, sparse or dense, restricted on some given interval of interest. Suppose we have an arbitrary $n\times n$ matrix $A$, ...
5
votes
1answer
3k views

Stability of numerical method for 1D Burger's equation

I am trying to solve 1D viscous Burger's equation numerically and I cannot apply von Neumann analysis because the equation is non-linear. How do I predict the stability criteria for my system? I also ...
5
votes
3answers
6k views

Algorithm for Principal Eigenvector of a Real Symmetric 3x3 Matrix

I have a 3x3 covariance matrix (so, real, symmetric, dense, 3x3), I would like it's principal eigenvector, and speed is a concern. Is there a fast algorithm for this specific problem? I've seen ...

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