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3
votes
2answers
590 views

Unimodular Matrix calculation

I know for a given matrix $M$, there exists a matrix $U$ over the integers with determinant $+1$ or $-1$ such that $UM=E$. I know $E$, but $M$ is not a square matrix. Is there any easy way to get $...
15
votes
1answer
617 views

What is the current state of polynomial preconditioners?

I wonder what has happened to polynomial preconditioners. I am interested in them, because they appear to be comparatively elegant from a mathematical perspective, but as far as I have read in surveys ...
5
votes
2answers
308 views

Approximating and visualizing basins of attraction

I am working on estimating the position and orientation (pose) of a model (rigid object) from its silhouette in an image. For this, I have constructed an error measure between the model in its pose ...
17
votes
5answers
7k views

What is the best way to determine the number of non zeros in sparse matrix multiplication?

I was wondering whether there is a fast and efficient method to find the number of non zeros in advance for sparse matrix multiplication operation assuming both matrices are in CSC or CSR format. I ...
3
votes
1answer
917 views

generating a non-uniform grid with Chebyshev discretization

I often see that it is common to put "more points" in the region of interest in the computational domain of the numerical method, i.e. use non-uniform grid. The proofs are usually done for the uniform ...
2
votes
1answer
135 views

Is it necessary to project the initial condition onto the variational space in a fully discrete galerkin method?

I'm solving a simple 1D heat diffusion problem $$u_t=u_{xx},\quad \Omega\times[0,T]$$ $$u=0\quad, \partial\Omega\times [0,T]$$ $$u(x,0)=f$$using a fully discrete galerkin finite element method. This ...
5
votes
1answer
3k views

Logistic regression with Python

I am trying to code up logistic regression in Python using the SciPy fmin_bfgs function, but am running into some issues. I wrote functions for the logistic (...
4
votes
2answers
174 views

Efficiency of Repeated Sparse Matrix-Vector Products

I recently read Wolfgang's answer to the question found here and found myself wondering about a related followup question. Assume you have two sparse matrices $A$ and $B$. You need to do the ...
1
vote
0answers
165 views

Enforcing continuity conditions in pseudospectral domain decomposition methods for time dependent PDEs

I have a partial differential equation of the form $$ \frac{d}{dt}f(x,t) = \Theta(x) f(x,t) \qquad \Theta(x) \sim \left[\frac{d^2}{dx^2} + k^2(x)\right] $$ subject to $f(x,t=0) = f_0(x)$, and $f(x=0,t)...
5
votes
3answers
189 views

Where do I find data to start a graph data structure to practice on?

I am interested in learning firsthand about graph data structures but have no data to put into the database. Can anyone recommend a source for a beginner to find graph data? (Preferably a source that ...
5
votes
2answers
271 views

How to do upwinding in finite volume schemes for nonlinear equations?

In finite difference theory, you learn, that you have to use upwinding for equations with high convection, like Burgers' equation. What does the finite volume equivalent look like? What if the ...
3
votes
0answers
80 views

Conservative field mapping between two topologically disconnected surface meshes

Some background: the Front-Tracking method uses a triangular surface mesh to describe the boundary between two immiscible fluids. To deal with the breakup and coalescence of the fluid interface, ...
3
votes
3answers
205 views

Difference between Computational science and Computational X (X = some natural science or discipline)

Where is the divide between a computational scientist and a computational X (X = some natural science or discipline)? What questions could someone ask themself to decide which is the best route for ...
3
votes
0answers
169 views

How to accurately decompose positive semidefinite matrix and use the lower triangular part in linear equations

I have $n$ arbitrary $p\times 1$ vectors $x_i$, and $p\times k$ matrices $A_i$, and $n$ $p \times p$ positive semidefinite matrices $S_i$, where some (often most) of the $S_i$'s are same (for example ...
4
votes
1answer
692 views

On implementations of the complex QR decomposition

I have an mathematical theorem on the QR decomposition, which relies on the QR decomposition of an invertible square complex matrix always constructing a triangular matrix with real diagonal entries. ...
8
votes
2answers
144 views

Computing ratio of trigonometric functions

I have need to compute the functions: $$ f(x) = \frac{\sin^{-1}x}{x}$$ and $$ g(x) = \frac{\sin a x}{\sin x} $$ where $a\in[0,1]$ and $ x\in[0,\frac{\pi}{2}]$ and is often very small ($x\ll 1$). Are ...
2
votes
1answer
295 views

Partitioning of mesh with 20 noded hexahedral elements [closed]

Is there a way to partition a mesh consisting of 20 noded hexahedral elements for parallel processing? I used METIS for partitioning mesh with 8 noded hexahedron elmements, which works fine but i don'...
5
votes
1answer
196 views

What is the most efficient approach to interpolate values between two FEM meshes in 2D?

I am looking for efficient algorithm to interpolate values from one unstructured 2D mesh grid to another. Both grids are constructed using the same type of elements (triangles or quadrilaterals). Both ...
16
votes
2answers
1k views

What are the efficient, accurate algorithms for evaluation of hypergeometric functions?

I'm curious to know what good numerical algorithms exist for evaluation of the generalized hypergeometric function (or series), defined as $${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{k=0}^{\...
5
votes
1answer
942 views

What is the runtime complexity of MATLAB operation A*B where A and B are general sparse matrices?

I tried to search the answer and I found that method cs_multiply from this book has been adopted for the purpose of multiplication of two general sparse matrices in MATLAB. In the book it says that ...
4
votes
0answers
174 views

Find maximum distance between elements given constraints on some

I have a list of numbered elements 1 to N that fit into positions on a number line starting with 1. I also have constraints for these elements: The element 1 is in position 1, and element N must be ...
3
votes
4answers
2k views

Heat equation (steady state) with boundary conditions at domain edges and inside the domain

I'm trying to solve the steady state of a heat equation problem in 2D $\Delta u = 0$ (3D also), with the method of solving the huge system of equations that arises from the discretization of the ...
6
votes
1answer
217 views

interpolation combined with methods of characteristics can cause oscillations for the transport equation?

I would like to know about the effect of using a higher order interpolator for the methods of characteristics. I am solving $$u_t+a(x,t)u_x=0$$ with some nonsmooth initial data $u_0(x)$ by the method ...
8
votes
2answers
1k views

Max of a convex combination over a convex hull of real variables

I have the following linear program: $$ \begin{array}{cc} \text{Maximize} & a^T x \\ \text{Subject to} & x_{\min} \leq x \leq x_{\max} \\ & \mathbf{1}^T x = 1 \end{...
12
votes
2answers
4k views

Strategies for Newton's Method when the Jacobian at the solution is singular

I'm trying to solve the following system of equations for the variables $P,x_1$ and $x_2$ (all else are constants): $$\frac{A(1-P)}{2}-k_1x_1=0 \\ \frac{AP}{2}-k_2x_2=0 \\ \frac{(1-P)(r_1+x_1)^4}{L_1}...
1
vote
1answer
323 views

Clustering data based on the radius of a sphere

I have a data-set that has four columns [X Y Z C]. I would like to find all the C values that are in a given sphere centered at [X, Y, Z] with a radius r. What is the best approach to address this ...
4
votes
1answer
176 views

Is there a nonlinear solver similar to CGNR evaluating only $A^TAx$?

First of all, I am quite new to this field and I excuse myself in advance for any stupid content in this question. In the field of compressed sensing or deblurring I have a nonlinear optimization ...
12
votes
2answers
971 views

What are some good data-types for unstructured cell-centered FVM CFD code?

I'm interested in an advice for efficient data structures for cell browsing in unstructured cell-based finite volume CFD. One example that I encountered (in dolfyn cfd code) goes like this (I'll show ...
2
votes
1answer
270 views

Reshape and Index (State) Products in Numpy

Consider the following: I start with a $2\times 2$ matrix $W_{ij}$. I then take this $W$ matrix and make a new tensor, $T$, by doing the following: $$ T_{ijkl}=\sum_{a}W_{ai}W_{aj}W_{ak}W_{al} $$ ...
1
vote
1answer
164 views

Understanding Inner blocking size in lapack/plasma

I could not get accurate description that aids to my understanding of inner blocking size parameter that is used in many LAPACK routines(such as DGEQRT, DLARFB). Thanks
4
votes
1answer
2k views

Creating spheres In Matlab that change surface color depending data

I am trying to create a series of randomly distributed spheres in Matlab, three will be three sub-sets. I would like each subset to have a different color. I am using a structure because each sphere ...
2
votes
2answers
102 views

How does an automaton actually “compute”?

In my studies in computability I have come across the notion of the "machine", an abstract representation of a device that essentially computes. I have read about Turing Machines and Wolfram's binary ...
13
votes
3answers
9k views

Understanding the “rate of convergence” for iterative methods

According to Wikipedia the rate of convergence is expressed as a specific ratio of vector norms. I'm trying to understand the difference between "linear" and "quadratic" rates, at different points of ...
3
votes
1answer
85 views

Improve optimization over 'mapping' of indices

I have two tables at my disposal, one work dataset and one reference dataset. Each dataset has got two columns, lets say these are fields A and B. I would like to associate the rows in the reference ...
5
votes
1answer
278 views

Recommendation for a python book for data processing

I've been doing FORTRAN programming for 10 years and I've started using python for a few years now, mostly for data processing. I've been lucky enough to work with people that are knowledgeable ...
4
votes
2answers
643 views

Orthonormalized Bernstein polynomials using Gram-Schmidt

I was wondering, before trying to do that myself, has anyone attempted to do orthonormalization of Bernstein polynomials using Gram-Schmidt? I discussed this with several people and have been told ...
1
vote
2answers
370 views

Checking for error in conjugate gradient algorithm

What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
5
votes
1answer
286 views

How to design good finite difference schemes? [closed]

In principle, I know finite differences. In university, we discussed it and derived consistency and boundary conditions. But I am still left with a big question. How to design a good finite ...
1
vote
1answer
487 views

How to apply a Galerkin finite element method to a linear, one-dimensional boundary value problem

I have the following boundary value problem: $$-(\alpha u')' + \gamma u = f $$ in $\Omega = (a,b)$ with b.c. $u(a) = u(b) = 0$ and $\alpha > 0, \gamma ≥ 0$ and $f:(a,b) \to \Re $ The weak ...
5
votes
1answer
2k views

Magma vs. Plasma

I'm having a difficult time understanding the difference between the linear algebra packages MAGMA and PLASMA from just a quick glance. It looks like MAGMA is oriented towards GPU's and vector ...
6
votes
1answer
2k views

Flexible Mesh Framework

I am looking for flexible and easy to learn mesh framework which provides data structure for representing and manipulating meshes in 2D i 3D. I've already found a few: MSTK https://software.lanl.gov/...
2
votes
2answers
3k views

Implementing PageRank using the Power Method

I am trying to implement the PageRank algorithm described in this paper (Fig. 1). Here is the breakdown of the steps: http://www.louismullie.com/algo.png where: ...
12
votes
2answers
853 views

Numerical Integration - handling NaNs (C / Fortran)

I am dealing with a tricky integral that exhibits NaNs at certain values near zero and at the moment I am dealing with them quite crudely using an ISNAN statement which sets the integrand to zero when ...
1
vote
2answers
120 views

how to approach time zero when the equation is not defined at that point

Some background: define a process $Y_t=\frac{1}{t}\int_0^tX_sds$, where $X_t$ is a standard Brownian motion. Then I define a function $u(t,X_t,Y_t)$ and require it to be a martingale. Thus, by ...
5
votes
1answer
537 views

Poisson solver on unstructured mesh

For the 2D Poisson equation, there exist on finite difference mesh, some code taking $O(n \log(n))$ operations to solve it on a mesh with $n$ nodes. They rely on Fast Fourier Transform or Block Cyclic ...
3
votes
2answers
115 views

Are matlab C library versions backwards compatible?

I have some C++ code that links to matlab2008b. Are matlab 2012a and 2012b backwards compatible with 2008b? If it's not trivially compatible, are there some simple steps to make it compatible?
5
votes
2answers
148 views

What is the algorithm for computing block reflectors in xDLARFB

The theory behind computing a single Householder reflector to zero out part of a column of a matrix is pretty well described in Matrix Computations by Golub and Van Loan. However, the blocked ...
10
votes
1answer
514 views

How to find the interior eigenvalues by krylov subspace method?

I am wondering how to find the eigenvalues of some sparse matrix in given interval [a, b] by iterative method. To my personal understanding, it is more obvious to use Krylov subspace method to find ...
13
votes
5answers
2k views

C++ or Python for a development of CFD library

What would you say would be the advantages/disadvantages of two approaches to coding a general (finite volume, fem, dg) library for Computational Continuum Mechanics? This is how I see things right ...
40
votes
4answers
1k views

Scientific standards for numerical errors

In my field of research the specification of experimental errors is commonly accepted and publications which fail to provide them are highly criticized. At the same time I often find that results of ...

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