# All Questions

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### Are there any heuristics for optimizing the successive over-relaxation (SOR) method?

As I understand it, successive over relaxation works by choosing a parameter $0\leq\omega\leq2$ and using a linear combination of a (quasi) Gauss-Seidel iteration and the value at the previous ...
8k views

### Null-space of a rectangular dense matrix

Given a dense matrix $$A \in R^{m \times n}, m >> n; max(m) \approx 100000$$ what is the best way to find its null-space basis within some tolerance $\epsilon$? Based on that basis can I then ...
797 views

### Computing accurate fluxes with FEM

I have solved Poisson equation on a 3d domain with neumann and dirichlet boundary condition. I get the potential, take the gradient for each element and integrate on a surface of an element, I do this ...
64k views

### How to determine the amount of FLOPs my computer is capable of

I would like to determine the theoretical number of FLOPs (Floating Point Operations) that my computer can do. Can someone please help me with this. (I would like to compare my computer to some ...
7k views

### Understanding how Numpy does SVD

I have been using different methods to calculate both the rank of a matrix and the solution of a matrix system of equations. I came across the function linalg.svd. Comparing this to my own effort of ...
3k views

### Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
2k views

### The meaning of conservative discretization in Galerkin FEM and Discontinuous Galerkin

I do understand the meanning of "conservative discretization" within the FVM/FDM framework, indeed it is well explained in this post. Now, according to the table in this slide (pp.8), it concludes: ...
1k views

### Does Computational Science involve programming?

I read about computational science on Wikipedia, but my understanding is not very clear. Does computational science involve programming? How different is computational science from computational ...
1k views

### Estimation of condition numbers for very large matrices

Which approaches are used in practice for estimating the condition number of large sparse matrices?
2k views

### PDE discretization with the method of rothe and the method of lines (Modular implementation)

The Heat equation is discretized in space with FV (or FEM), and a semi-discrete equation is obtained (system of ODEs). This approach, known as the method of lines, allows to easily switch from one ...
2k views

### Solving a linear equation system with pure Neumann condition

I am trying to solve a linear equation system $\textbf{A}\textbf{x}=\textbf{b}$, e.g. a Poisson equation discretized in strong form, using biCGstab method. Since there are only natural Neumann ...
2k views

### Why can't Householder reflections diagonalize a matrix?

When computing the QR factorization in practice, one uses Householder reflections to zero out the lower portion of a matrix. I know that for computing eigenvalues of symmetric matrices, the best you ...
416 views

### Eigenvectors of a small norm adjustment

I have a dataset that is slowly changing, and I need to keep track of eigenvectors/eigenvalues of its covariance matrix. I've been using scipy.linalg.eigh, but it'...
378 views

### How to directly compute the inverse of an ill-conditioned dense matrix

I know that it is generally a bad idea to compute the inverse matrix directly. However, if it is necessary to compute the inverse of an ill-conditioned invertible dense matrix, then what can I try? ...
142 views

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Consider a scalar field $u$ on an unstructured triangle mesh which is constant on each face. Let $A_i$ be the area of triangle $T_i$, $N(i)$ the set of triangles sharing an edge with $T_i$, and $L_{... 1answer 108 views ### Apply for a cluster for scientific computing from a developing country? I don't have access to a computer cluster in my university. Is there website that accepts applications for free access to a computer cluster for scientific computing? Further information: I am in ... 1answer 100 views ### Solving a non-linear heat equation with the galerkin method gives negative values I am trying to solve a non-linear time-dependent heat equation $$\partial_tT=\nabla \left(k_T(T)\nabla T\right) + f$$ using the galerkin method, with neumann boundary conditions. For linearization of ... 4answers 30k views ### How does the MATLAB backslash operator solve$Ax=b$for square matrices? I was comparing a few of my codes to "stock" MATLAB codes. I am surprised at the results. I ran a sample code (Sparse Matrix) ... 4answers 2k views ### Why is local conservation important when solving PDEs? 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 ... 7answers 2k views ### What programming paradigms should I be investing in if I want my code to run on petascale machines in the future? 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,... 17answers 3k views ### Is it common not to use libraries for standard numerical algorithms, and why? 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" ... 6answers 39k views ### Python vs FORTRAN Which one is better: FORTRAN or Python? And I guess that in both cases you need Gnuplot, am I right? I'm working on a Windows machine at the moment. I'd like to use it to get numerical solutions for ... 1answer 6k views ### What is the general idea of Nitsche's method in numerical analysis? I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ... 4answers 16k views ### Dealing with the inverse of a positive definite symmetric (covariance) matrix? In statistics and its various applications, we often calculate the covariance matrix, which is positive definite (in the cases considered) and symmetric, for various uses. Sometimes, we need the ... 1answer 11k views ### Why is Newton's method not converging? 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 ... 5answers 4k views ### Are there any famous problems/algorithms in scientific computing that cannot be sped up by parallelisation Are there any famous problems/algorithms in scientific computing that cannot be sped up by parallelisation? It seems to me whilst reading books on CUDA that most things can be. 7answers 22k views ### What is the fastest way to calculate the largest eigenvalue of a general matrix? EDIT: I am testing if any eigenvalues have a magnitude of one or greater. I need to find the largest absolute eigenvalue of a large sparse, non-symmetric matrix. I have been using R's ... 3answers 2k views ### multigrid method to solve PDE I need simple explanation of the Multigrid Method or some literature about this. I am familiar with iterational methods including BiCGStab,CG,GS,Jacobi and preconditioning, but I am a beginner with ... 2answers 4k views ### Stopping criteria for iterative linear solvers applied to nearly singular systems Consider$Ax=b$with$A$nearly singular which means there is an eigenvalue$\lambda_0$of$A$that is very small. The usual stop criterion of an iterative method is based on the residual$r_n:=b-Ax_n$... 3answers 789 views ### Strategies for unit testing and test-driven development I'm a huge advocate of test-driven development in scientific computing. It's utility in practice is just staggering, and really alleviates the classic troubles that code developers know. However, ... 8answers 1k views ### Is there any open-source or easy-to-access software that can simplify algebraic expressions like$x^{2}+2x+3, x=\sqrt{2}t-1\$?

I always calculate things by hand, but now my comrades are getting nasty and making a lot of repetitive exercises involving just plugging things in like the expression above. I am particularly ...
4k views

### When is Newton-Krylov not an appropriate solver?

Recently I have been comparing different non-linear solvers from scipy and was particularly impressed with the Newton-Krylov example in the Scipy Cookbook in which they solve a second order ...
1k views