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Can you explain the difference between these two computational methods ?
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Are there cases where Hartree-Fock is not a good approximation to compute equilibrium geometry when the molecule is in a non bond-breaking condition?
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What is counterpoise correction exactly ? Can you explain when it is needed and why ?
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How do I pick a basis set for an ab-initio Hartree-Fock evaluation ? In other words, what are the important characteristics of a basis set so that a proper choice can be made ?
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Some basis sets are said to be "correlation consistent". What does it mean in practice ?
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Given two computational programs, one using a Gaussian basis, and the other using Slater basis, what are the practical differences, advantages and disadvantages for each choice ?
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Our workgroup produces a desktop application that simulates building energy performance. It is a .NET application and when the user is running a lot of simulations, they can be quite time consuming. The simulations are totally parallelizable, and we have some very significant HPC computing resources at "the office".
O... |
My background is in genomics, but I have recently been working with problems related to protein structure. I wrote a few relevant programs in C, building my own PDB file parser from scratch in the process. I didn't worry about making a really robust parser, I just knew that building one myself would be the best way to ... |
I'm new to molecular dynamics (MD) simulations. What is the complexity of a molecular dynamics simulation in terms of simulation time? In other words, if I want increase the simulated time from 10 nanoseconds to 20 nanoseconds, what can I expect in terms of the increase in runtime?
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When VASP 5 was released, the performance was mostly slower than our make of VASP 4.6. I wrote it off as an optimization issue, and went on in my life. Then, in VASP 5.2, with the release notes, I realized that the it was all due to slower FFT, and is now fixed. Is there any way for me, as a user, to tell if the VASP m... |
Specifically I want to extend work involving B3LYP started with Gaussian 03 but continued with GAMESS-US. The energies provided by the default B3LYP methods are not the same. There is a discussion about this in the GAMESS-US manual (Further Information section):
Note that B3LYP in GAMESS is based in part on the VWN5
... |
I often use inverse interval graphs to represent biologically relevant features along a genomic sequence. For example, given a (relatively) small genomic region, the graph would contain a node for each gene in the region, and there would be an edge between two nodes if the corresponding genes do not overlap (the comple... |
I'm from the field of accelerator physics, specifically related to circular storage rings for synchrotron light sources. High energy electrons circulate around the ring, guided by magnetic fields. The electrons circulate billions of times and one wants to predict the stability. You can describe the motion of the ele... |
EEGLab doesn't work well due to the data being from a Neuromag MEG. Any thoughts for alternatives?
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Suppose I have a matrix equation recursively defined as
A[n] = inverse([1 - b[n]A[n+1]]) * a[n]
Then the equation for A[1] looks similar to a continued fraction, for which there are some highly efficient methods that avoid tedious recalculation (See "Numerical Recipes" for some examples).
However, I wonder if there ... |
I am using PETSc's nonlinear solver package SNES to solve a system of nonlinear equations obtained by discretizing a partial differential equation. How can I determine why the solver is not converging and what can I do to successfully solve my equations?
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Various important sets of polynomials (Legendre, Chebyshev, etc.) are orthogonal over some real interval with some weighting. Are there known families of polynomials that are orthogonal over other curves in the complex plane?
For instance, I would like a basis for the polynomials of degree n that is orthogonal over, s... |
I have a PETSc Mat and would like to estimate its condition number.
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Many numerical methods for hyperbolic PDEs are based on the use of Riemann solvers. Such solvers are essential for accurately capturing shock waves. There are a range of such solvers available for the most well-studied systems (e.g., exact solvers, Roe solvers, HLL solvers). How should I decide which to use?
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Many numerical approaches to CFD can be extended to arbitrarily high order (for instance, discontinuous Galerkin methods, WENO methods, spectral differencing, etc.). How should I choose an appropriate order of accuracy for a given problem?
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Numerical methods for solving PDEs (or ODEs) fall into two broad categories: explicit and implicit methods. Implicit methods allow larger stable timesteps but require more work per step. For hyperbolic PDEs, the common wisdom is that implicit methods usually do not pay off because the use of timesteps larger than tho... |
I have a relatively simple convex optimization problem that involves less than 100 variables but contains a terribly ill-conditioned matrix. I have tried CVX and CPLEX; even though both can typically solve the problem in about 1 second, both fail when the condition number of the matrix becomes very large. An arbitrar... |
Standard multigrid and domain decomposition methods do not work, but I have large 3D problems and direct solvers are not an option. What methods should I try?
How are my choices affected by the following considerations?
coefficients vary over several orders of magnitude, or
finite element versus finite different metho... |
Engineers often insist on using locally conservative methods such as finite volume, conservative finite difference, or discontinuous Galerkin methods for solving PDEs.
What can go wrong when using a method that is not locally conservative?
Okay, so local conservation is important for hyperbolic PDEs, what about ellipti... |
High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this regime because the inf-sup constant degrades as the square root of the aspect ratio or worse. The inf-sup constant is im... |
Physical quantities like pressure, density, energy, temperature, and concentration should always be positive, but numerical methods sometimes compute negative values during the solution process. This is not okay because the equations will compute complex or infinite values (typically crashing the code). Which numerical... |
In evaluating the quality of a piece of software you are about to use (whether it's something you wrote or a canned package) in computational work, it is often a good idea to see how well it works on standard data sets or problems. Where might one obtain these tests for verifying computational routines?
(One website/bo... |
I've been using GSL as the foundation of many of my simulations, but it's a little bit overkill for my purposes and it defines its own complex type for legacy reasons. Rather than code my own Runge-Kutta ODE solver, which would probably not be very efficient, are there any open source ODE solvers that use the native C9... |
I solve multi-species, compressible Navier-Stokes equations on a 3D structured grid. I have obtained a solution on a given grid (let's say a relatively coarse one). I want now to refine my grid and interpolate my previous solution on my new grid before restarting my simulation. Currently, we have an interpolation tool ... |
What's the state-of-the-art in the approximation of highly oscillatory integrals in both one dimension and higher dimensions to arbitrary precision?
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Source terms, such as those due to bathymetry in the shallow water equations, need to be integrated in a special way in order to preserve physical steady states. Is there a general way to construct well-balanced methods, or does it require special techniques for each equation?
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Multigrid methods usually solve Dirichlet problems on levels (e.g. point Jacobi or Gauss-Seidel). When using continuous finite element methods, it is much less expensive to assemble small Neumann problems than to assemble small Dirichlet problems. Non-overlapping domain decomposition methods such as BDDC (like FETI-DP)... |
The idea behind Large Eddy Simulation is to solve explicitely the scales of the flow that are resolved by the computational grid and to model the sub-grid scales. Ideally, the cut-off between these scales is located in the inertial range of the local turbulent field. This is because Kolmogorov's hypotheses predict a un... |
Given an evolution PDE
$$u_t = Au + Bu$$
where $A,B$ are (possibly nonlinear) differential operators that don't commute, a common numerical approach is to alternate between solving
$$u_t = Au$$
and
$$u_t = Bu.$$
The simplest implementation of this is known as Godunov splitting and is 1st-order accurate. Another well-k... |
We know that $\mathbf A$ is symmetric and positive-definite. We know that $\mathbf B$ is orthogonal:
Question: is $\mathbf B \cdot\mathbf A \cdot\mathbf B^\top$ symmetric and positive-definite?
Answer: Yes.
Question: Could a computer have told us this?
Answer: Probably.
Are there any symbolic algebra systems (like Mat... |
Automatic differentiation allows us to numerically evaluate the derivative of a program on a particular input. There is a theorem that this computation can done at a cost less than five times the cost to run the original program. This factor of five is an upper bound.
In what situations can this cost be further reduce... |
Sparse linear systems turn up with increasing frequency in applications. One has a lot of routines to choose from for solving these systems. At the highest level, there is a watershed between direct (e.g. sparse Gaussian elimination or Cholesky decomposition, with special ordering algorithms, and multifrontal methods) ... |
I have several challenging non-convex global optimization problems to solve. Currently I use MATLAB's Optimization Toolbox (specifically, fmincon() with algorithm='sqp'), which is quite effective. However, most of my code is in Python, and I'd love to do the optimization in Python as well. Is there a NLP solver with... |
Are there any good papers and or codes that couple discontinuous galerkin finite element solvers with Riemann solvers?
I need to explore coupling elliptic and hyperbolic problems but most splitting methods are ad hoc at best. Since I have a large amount of FEniCS code, I would like to just couple the Riemann solver ... |
There are a number of different libraries out there that solve a sparse linear system of equations, however I'm finding it difficult to figure out what the differences are.
As far as I can tell there are three major packages: Trilinos, PETSc, and Intel MKL. They can all do sparse matrix solves, they are all fast (as ... |
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?
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In my numerical analysis courses, I learned to analyze the efficiency of algorithms by counting the number of floating-point operations (flops) they require, relative to the size of the problem. For instance, in Trefethen & Bau's text on Numerical Linear Algebra, there are even 3D-looking pictures of the flop counts.
... |
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 laser radiation and is removed from upper and lower sides of the element. A significant characteristic of the problem i... |
I am solving a physical problem using implicit numerical scheme. This leads me to solving a linear equation with tridiagonal matrix. I've coded this algorithm from Wikipedia. I wonder if there is an efficient library which allows to solve this type of equation in optimised way. An important note is that matrix itself c... |
Informally in our lab, we have developed 2 metrics to compare CFD solvers over
the range of machines we have access to. One is called COMP, which stands
for COde Machine Performance. This single number is supposed to represent the
absolute performance of a given code on a given machine. It is computed, for a
given run... |
I've been considering using a multigrid method to calculate the eigenvalues of a particular PDE. I know that multigrid is extremely good at finding the least eigenvalues and their associated eigenvectors, in part because these eigenvectors are very smooth. Is multigrid also a good method to calculate all the eigenval... |
You've managed to finally find out how the atoms are spatially arranged on your newly-discovered molecular entity. Through, say, spectroscopic means, you are now in possession of a bunch of atom coordinates, atom types, bond lengths, bond types, and whatnot for your molecule. You are now interested in determining the p... |
How can the gravitational n-body problem be solved numerically in parallel?
Is precision-complexity tradeoff possible?
How does precision influence the quality of the model?
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Whenever one uses a quasi-Monte Carlo method for cubature or optimization, it seems that there's a wide variety of low-discrepancy sequences to choose from, associated with the names of van der Corput, Halton, Hammersley, Faure, Niederreiter, Sobol', and other names I don't quite remember. Are there any good rules of t... |
It's pretty clear from a survey of the top500 that the industry is trending towards an exponential increase in processing cores. The largest supercomputers all use MPI for communication between nodes, though there does not appear to be a clear trend for on-node parallelism, with the simplest (but not necessarily the m... |
A lot of numerical algorithms (integration, differentiation, interpolation, special functions, etc.) are available in scientific computation libraries like GSL. But I often see code with "hand-rolled" implementations of these functions. For small programs which are not necessarily intended for public distribution, is i... |
A recent answer mentioned the use of Fortuna or Mersenne Twister Random Number Generators (RNGs) to seed a Monte Carlo simulation. I hadn't heard of Fortuna before so I looked it up - looks like it is mainly intended for cryptographic use.
I currently use a Mersenne Twister in production code to seed a K-Means algorith... |
Texts, articles, and papers on Maximum Entropy Classifiers tend to come in two varieties: the more popular "upper level", and the more technical.
The popular variety are good at explaining the Maximum Entropy concept and why such classifiers are considered to be better than Naive Bayes classifiers (and also why they ar... |
Multi-physics simulation involves coupling multiple "physics", often with different space and/or time scales. Additionally, the single-physics codes are often written by different teams. The most commonly used coupling technique is first-order operator splitting, but this has poor accuracy and stability properties. How... |
The Lax equivalence theorem states that consistency and stability of a numerical scheme for a linear initial value problem is a necessary and sufficient condition for convergence. But for nonlinear problems, numerical methods can converge very plausibly to incorrect results, despite being consistent and stable. For exa... |
I am looking for a method to estimate the contribution from Pauli repulsion to the interaction energy of a molecular dimer in an electronic structure computation (e.g. with Density Functional Theory). I'm aware of SAPT and SAPT(DFT) decomposition schemes, but as far as I understand, these estimate the contribution from... |
I formulate questions about ion densities in biological and materials problems using Classical Density Functional Theory, as in this paper which is also on the arXiv. The discretization in that paper for the Rosenfeld functional for hard spheres is just pointwise collocation on a regular grid.
I know of no convergence ... |
I was playing around with PETSc and noticed that when I run my program with more than one process via MPI it seems to run even slower! How can I check to see what is going on?
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I recently encountered a case where I needed an integer division operation on a chip that lacked one (ARM Cortex-A8). While trying to research why that must be, I found out that in general division takes many more cycles than addition, subtraction or multiplication on pretty much any integer (or fixed-point) architectu... |
Developing scientific algorithms is a highly iterative process often involving changing lots of parameters that I will want to vary either as part of my experimental design or as part of tweaking algorithm performance. What strategies can I take for structuring these parameters so that I can easily change them between ... |
I have an application that can be trivially parallelized but its performance is to a large extent I/O bound. The application reads a single input array stored in a file that is typically 2-5 GB in size (but I expect this number to grow in the future). A typical computation applies the same operation to each row or colu... |
Fortran has a special place in numerical programming. You can certainly make good and fast software in other languages, but Fortran keeps performing very well despite its age. Moreover, it's easier to make fast programs in Fortran. I've made fast programs in C++, but you have to be more careful about things like pointe... |
I am strongly convinced of the value of using tests that verify a complete program (e.g. convergence tests), including an automated set of regression tests. After reading some programming books, I've gotten the nagging feeling that I "ought to" write unit tests (i.e., tests that verify the correctness of a single func... |
What efforts have been made to quantify the costs of the common numeric workhorse operations such as those found in LAPACK, BLAS, sparse matrix operations, etc....
We have several libraries which implement identical interfaces found in BLAS and LAPACK. I suspect these all have different strengths and weaknesses on vari... |
How can I compute the Schur complement:
$$ S = K_{bb} - K_{ba} K_{aa}^{-1} K_{ab} $$
where
$$ K=\begin{pmatrix} K_{aa} & K_{ab} \\ K_{ba} & K_{bb} \end{pmatrix} $$
(in some ordering) is a PETSc matrix (Mat)?
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I have a Cartesian mesh in $d$ dimensions, and I would like to enumerate all the subcells of a given hexahedral cell. If I am just enumerating the vertices of a cell (or cells that contain a vertex) I can use Gray code, here is my code doing that.
You can see that this works because the Gray code sequence traverses the... |
ATLAS is a free BLAS/LAPACK replacement that tunes itself to the machine when compiled. MKL is the commercial library shipped by Intel. Are these two libraries comparable when it comes to performance, or does MKL have the upper hand for some tasks? If so, which ones?
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I am using a basic singular value decomposition (via LAPACK) routine in FORTRAN to solve an overdetermined system in the form of $A\cdot X = B$ where $\mathrm{size}(A) = [m,n]$ with $m > n$.
My sample data points come from a noisy sine function and I am trying to use linear regression with $x^i$ as my basis functions. ... |
In a particular class of detectors, our data comes out as pairs of points in two dimensions, and we want to string these points into lines.
The data is noisy, and is binned in one direction but not in the other. We can't guarantee a hit in every bin even when each detector element is working, so there may be skips.
Our... |
I have a 50-base, artificially created RNA sequence and I would like to obtain its 3D structure.
What software should I use for this purpose?
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I am researching heat diffusion in an optical element irradiated by laser. This problem is described by the PDE which I wrote down in this question. I am using an implicit numerical scheme to model the heat distribution evolving in time.
After some time has passed the system comes to a stationary distribution (corresp... |
I work fairly heavily in mathematical biology/epidemiology, where most of the modeling/computational science work is still dominated by sets of ODEs, admittedly sometimes fairly elaborate sets of them. One of the plusses of these models is that they're rather easy to describe and replicate. A table of parameter values,... |
I have a series of data points $(x_i,y_i)$ which I expect to (approximately) follow a function $y(x)$ that asymptotes to a line at large $x$. Essentially, $f(x) \equiv y(x) - (ax + b)$ approaches zero as $x \to \infty$, and the same can probably be said of all the derivatives $f'(x)$, $f''(x)$, etc. But I don't know wh... |
As from Title. Where can I find a database of simple chemical structures in XYZ format, possibly with a simple CAS -> XYZ or InChi -> XYZ REST service ?
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As far as global optimal sequence alignment goes, is the Needleman-Wunsch and Hirschberg's algorithm still state of the art? Or have there been any improvements to these algorithms since they were published, or any newer algorithms?
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I will start with my personal experience in our lab. Back in the ifort 9 and 10 days, we used to be quite aggressive with the optimizations, compiling with -O3 and processor specific flags (-xW -xSSE4.2 for example). But starting with ifort 11, we started to notice:
1. some inconsistencies in the results (because sema... |
I am running into a problem with COMSOL 4.2.0.187 on Ubuntu (both 10.04 LTS and 11.04). When using the option -np x with x > 1, COMSOL crashes systematically after a short, yet random amount of time with the following error:
> [thread 140538491361024 also had an error] terminate called without an active exception
> Ab... |
Say I have a simulation that produces a single floating point number as a result, and a different number is produced each time the simulation runs. These numbers are randomly distributed according to some unknown distribution. I would like to approximate the underlying density distribution function based on a finite sa... |
First post. I'm working on this problem using Bayesian methods. In desperation I'm considering using p-values (shock horror), specifically posterior predictive p-values. So I need to simulate from the posterior predictive. I don't see a lot of statistics stuff here - it's mostly applied math at the moment. I could take... |
I'm in a field that doesn't necessarily do a great deal of HPC work, and when it does encounter it, it's often the result of researchers from other fields exploring new applications to their methods and the like. Primarily what this means is that it never really gets introduced in the course of study, or get brought up... |
I am annotating protein-coding genes in a region of the human genome and am looking for a set of assembled ESTs I can use as evidence for constructing gene models. A quick search of NCBI's SRA shows 4,318 human RNA-seq data sets, so I understand there is no shortage of data. I am hoping, however, to save the time that ... |
Our plasma dynamics simulations often produce too much information. During the simulations we record various physical properties on a grid (x,y,z,t) that is as large as (8192x1024x1024x1500), for at least 10 properties. This information is processed after the simulation is completed. With it we
make movies of properti... |
I would like to write my own solver for compressible Euler equations, and most importantly I want it to work robustly in all situations. I would like it to be FE based (DG is ok). What are the possible methods?
I am aware of doing 0th order DG (finite volumes) and that should work very robustly. I have implemented a ba... |
Here is a typical usage of function pointers in C. I would like to do something similar in Fortran. I have some ideas, but I would like to know if there is some canonical way to do so.
The function pointers and contexts passed in by the user are stored, then called later.
typedef PetscErrorCode (*TSIFunction)(TS,PetscR... |
do you know good introductions into or surveys $hp$-adaptive finite elements?
In particular I do not know how the heuristics for choosing spatial refinement or increased polynomial degree are constructed.
Thanks.
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I am used to thinking of finite-differences as a special case of finite-elements, on a very constrained grid. So what are the conditions on how to choose between Finite Difference Method (FDM) and Finite Element Method (FEM) as a numerical method?
On the side of Finite Difference Method (FDM), one may count that they a... |
In both domain decomposition (DD) and multigrid (MG) methods, one may compose the application of the block updates or coarse corrections as either additive or multiplicative. For pointwise solvers, this is the difference between the Jacobi and Gauss-Seidel iterations. The multiplicative smoother for $Ax = b$ acting as ... |
In my computational science PhD program, we are working almost exclusively in C++ and Fortran. It seems like some professors prefer one over the other. I am wondering which one is 'better' or if one is better than the other in a certain circumstance.
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I am using the Weka workbench to train a protein fold classifier. I imported my training data into Weka and performed PCA-based feature selection. This seems to have worked fine, but now I cannot evaluate my trained classifier on the test data because the test data contains all the original attributes. Of course, if I ... |
What efficient implementations of a 'drizzle' algorithm are available? The problem is, given a timestream of data in which each element is associated with a pixel in a map, how do you create that map? Each pixel may have many data points associated with it. Each data point may need to be weighted.
For example, in py... |
I use an example from finite element theory, but anybody who maintains a large datastructure and successively extends it will find something similar.
Suppose I have an unstructured mesh of points and triangles, where the points are given by coordinates (say $x$ and $y$) and the triangles each consist of three point ind... |
I want to calculate
$\binom{100}{i}a^i(1-a)^{(100-i)}$ for different $i$ with $a=0.001$ using
GMP-GNU.
How can this be done?
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Many times, when I've inherited or encountered scientific code written by other people (or occasionally, even my own work), I've noticed that documentation is either sparse or nonexistent. If I'm lucky, I see informative comments. If I'm very lucky, there's even Doxygen comments and a Doxyfile so that I have function i... |
My question is probably going to be too general to answer it with a couple words. Could you please suggest a good reading in that case. Projection methods are used to reduce size of the solution space for the problems. And there are at least two very interesting applications (from my point of view). The first is the so... |
I'm looking for a decent implementation of Gillespie's Direct Method in Python, as if I code the algorithm myself I'm nigh positive I'll do it inefficiently.
Anyone have a favorite?
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I've heard people say that plots produced by ORIGIN tend to look polished and "professional," whereas plots produced by Mathematica do not. However, most plot-creation programs are quite configurable and it stands to reason that with the right settings for things like tick location and labeling, font and color choices,... |
I am solving a system using singular value decomposition. The singular values (before scaling) are:
1.82277e+29
1.95011e+27
1.15033e+23
1.45291e+21
4.79336e+17
7.48116e+15
8.31087e+12
1.71838e+11
5.63232e+08
2.17863e+08
9.02783e+07
1.72345e+07
1.73889e+05
8.09382e+02
2.16644e+00
I have found that accepting all the sin... |
Does anyone have recommendations on a usable, fast C++ matrix library?
What I mean by usable is the following:
Matrix objects have an intuitive interface (ex.: I can use rows and columns while indexing)
I can do anything with the matrix class that I can do with LAPACK and BLAS
Easy to learn and use API
Relatively pain... |
Which approaches are used in practice for estimating the condition number of large sparse matrices?
|
The OpenCL programming paradigm promises to be a royalty free opens standard for heterogenous computing. Should we invest our time in developing software based on OpenCL? Pros/cons?
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