I'm in desperate need of a high performance algorithm to reduce a matrix to its independent vectors (row echelon form), aka find the basis vectors. I've seen the Bareiss algorithm and Row Reduction but they are all too slow, if anyone could recommend a faster implementation I'd be grateful!!! Happy to use TBB parallelisation.
Thanks!
What are you trying to do with the reduced echelon form? Do you just need the basis vectors to have them or are you trying to solve a system of equation? If you're solving a system of equations you can do an LU factorization and probably get faster calculation times. Otherwise gaussian elimination with partial pivoting is your fastest option.
Also do you know if your matrix is of a special form? Like upper or lower triangular for example. If it is then you can rewrite some of these algorithms to be faster based on the type of matrix that you have.
I am trying to parallelize gaussian elimination on sparse linear equations. I could not find data to test any where on the internet. If you could provide links to such data set that will be great.
Also could someone please explain how are sparse linear equations produced, that is practically, what problems produce such equations.
Thanks in advance.
First:
If indeed interested in linear problem sparse matrices, you may enjoy datasets from almost any linear Finite Element Method problem formulation.
Their problem-formulation may still use a vast amount of just linear equations.
Their problem-space is principally sparse in 3D-sense, so even if the original data were not represented as sparse-matrices, you can easily convert any of their dense-matrix representation into any sparse-matrix representation of your choice.
Next:
There are further sources of further interest for dense & sparse array I/O using the Matrix Market format
I have a sparse, square, symmetric matrix with the following structure:
(Let's say the size of the matrix is N x N)
Here, the area under the blue stripes is the non-zero elements. Could someone tell me if there is a algorithm to invert this kind of matrix that is simple yet more efficient than Gaussian elimination and LU decomposition? Thank you in advance.
Cholesky factorization is faster, O(n²). Or some specialized multi-band solvers, if you know the number of non-zero off-diagonals.
You can also apply iterative methods, maybe with preconditioning, it depends on your purpose.
There are a lot of sparse solvers. This can easily be solved using libeigen. What solver you choose is really going to depend on the properties of the sparse matrix besides the structure. Hope this helps.
I am taking a parallel programming class and am really struggling with understanding some of the computational complexity calculations and algebraic simplifications.Specifically for the bitonic sort algorithm I am looking when each processor is given a block of elements.
I am looking at situations when either a hypercube or 2D mesh interconnection network is used. I am given the following definitions for the calculation of Speedup, Efficiency, Iso-Efficiency and determining if the solution is cost optimal. I can understand how Speedup is determined but am totally lost as to how to solve for Efficiency and Iso-Efficiency. I think I understand cost-optimality as well. Given below are the equations.
The text which I am using for the class is
Introduction to Parallel Computing, 2nd Edition by Ananth Grama, Anshul Gupta, George Karypis & Vipin Kumar
For my question regarding the algebra of this problem, please refer to this
Several of my peers have mentioned that "linear algebra" is very important when studying algorithms. I've studied a variety of algorithms and taken a few linear algebra courses and I don't see the connection. So how is linear algebra used in algorithms?
For example what interesting things can one with a connectivity matrix for a graph?
Three concrete examples:
Linear algebra is the fundament of modern 3d graphics. This is essentially the same thing that you've learned in school. The data is kept in a 3d space that is projected in a 2d surface, which is what you see on your screen.
Most search engines are based on linear algebra. The idea is to represent each document as a vector in a hyper space and see how the vector relates to each other in this space. This is used by the lucene project, amongst others. See VSM.
Some modern compression algorithms such as the one used by the ogg vorbis format is based on linear algebra, or more specifically a method called Vector Quantization.
Basically it comes down to the fact that linear algebra is a very powerful method when dealing with multiple variables, and there's enormous benefits for using this as a theoretical foundation when designing algorithms. In many cases this foundation isn't as appearent as you might think, but that doesn't mean that it isn't there. It's quite possible that you've already implemented algorithms which would have been incredibly hard to derive without linalg.
A cryptographer would probably tell you that a grasp of number theory is very important when studying algorithms. And he'd be right--for his particular field. Statistics has its uses too--skip lists, hash tables, etc. The usefulness of graph theory is even more obvious.
There's no inherent link between linear algebra and algorithms; there's an inherent link between mathematics and algorithms.
Linear algebra is a field with many applications, and the algorithms that draw on it therefore have many applications as well. You've not wasted your time studying it.
Ha, I can't resist putting this here (even though the other answers are good):
The $25 billion dollar eigenvector.
I'm not going to lie... I never even read the whole thing... maybe I will now :-).
I don't know if I'd phrase it as 'linear algebra is very important when studying algorithms". I'd almost put it the other way around. Many, many, many, real world problems end up requiring you to solve a set of linear equations. If you end up having to tackle one of those problems you are going to need to know about some of the many algorithms for dealing with linear equations. Many of those algorithms were developed when computers was a job title, not a machine. Consider gaussian elimination and the various matrix decomposition algorithms for example. There is a lot of very sophisticated theory on how to solve those problems for very large matrices for example.
Most common methods in machine learning end up having an optimization step which requires solving a set of simultaneous equations. If you don't know linear algebra you'll be completely lost.
Many signal processing algorithms are based on matrix operations, e.g. Fourier transform, Laplace transform, ...
Optimization problems can often be reduced to solving linear equation systems.
Linear algebra is also important in many algorithms in computer algebra, as you might have guessed. For example, if you can reduce a problem to saying that a polynomial is zero, where the coefficients of the polynomial are linear in the variables x1, …, xn, then you can solve for what values of x1, …, xn make the polynomial equal to 0 by equating the coefficient of each x^n term to 0 and solving the linear system. This is called the method of undetermined coefficients, and is used for example in computing partial fraction decompositions or in integrating rational functions.
For the graph theory, the coolest thing about an adjacency matrix is that if you take the nth power of an adjacency Matrix for an unweighted graph (each entry is either 0 or 1), M^n, then each entry i,j will be the number of paths from vertex i to vertex j of length n. And if that isn't just cool, then I don't know what is.
All of the answers here are good examples of linear algebra in algorithms.
As a meta answer, I will add that you might be using linear algebra in your algorithms without knowing it. Compilers that optimize with SSE(2) typically vectorize your code by having many data values manipulated in parallel. This is essentially elemental LA.
It depends what type of "algorithms".
Some examples:
Machine-Learning/Statistics algorithms: Linear Regressions (least-squares, ridge, lasso).
Lossy compression of signals and other processing (face recognition, etc). See Eigenfaces
For example what interesting things can one with a connectivity matrix for a graph?
A lot of algebraic properties of the matrix are invariant under permutations of vertices (for example abs(determinant)), so if two graphs are isomorphic, their values will be equal.
This is a source for good heuristics for determining whether two graphs
are not isomorphic, since of course equality does not guarantee existance of isomorphism.
Check algebraic graph theory for a lot of other interesting techniques.