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.
Related
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
Given an invertible matrix M over the rationals Q, the inverse matrix M^(-1) is again a matrix over Q. Are their (efficient) libraries to compute the inverse precisely?
I am aware of high-performance linear algebra libraries such as BLAS/LAPACK, but these libraries are based on floating point arithmetic and are thus not suitable for computing precise (analytical) solutions.
Motivation: I want to compute the absorption probabilities of a large absorbing Markov chain using its fundamental matrix. I would like to do so precisely.
Details: By large, I mean a 1000x1000 matrix in the best case, and a several million dimensional matrix in the worst case. The further I can scale things the better. (I realize that the worst case is likely far out of reach.)
You can use the Eigen matrix library, which with little effort works on arbitrary scalar types. There is an example in the documentation how to use it with GMPs mpq_class: http://eigen.tuxfamily.org/dox/TopicCustomizing_CustomScalar.html
Of course, as #btilly noted, most of the time you should not calculate the inverse, but calculate a matrix decomposition and use that to solve equation systems. For rational numbers you can use any LU-decomposition, or if the matrix is symmetric, the LDLt decomposition. See here for a catalogue of decompositions.
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'm looking for a library that performs matrix operations on large sparse matrices w/o sacrificing numerical stability. Matrices will be 1000+ by 1000+ and values of the matrix will be between 0 and 1000. I will be performing the index calculus algorithm (en.wikipedia.org/wiki/Index_calculus_algorithm) so I will be generating (sparse) row vectors of the matrix serially. As I develop each row, I will need to test for linear independence. Once I fill my matrix with the desired number of linearly independent vectors, I will then need to transform the matrix into reduced row echelon form.
The problem now is that my implementation uses Gaussian elimination to determine linear independence (ensuring row echelon form once all my row vectors have been found). However, given the density and size of the matrix, this means the entries in each new row become exponentially larger over time, as the lcm of the leading entries must be found in order to perform cancellation. Finding the reduced form of the matrix further exacerbates the problem.
So my question is, is there an algorithm, or better yet an implementation, that can test linear independence and solve the reduced row echelon form while keeping the entries as small as possible? An efficient test for linear independence is especially important since in the index calculus algorithm it is performed by far the most.
Thanks in advance!
Usually if you are working with large matrices, people use LAPACK: this library contains all the basic matrix routines and support many different matrix types (sparse, ...). You can use this library to implement your algorithm, I think it will help you
I am looking for an efficient algorithm to find the largest eigenpair of a small, general (non-square, non-sparse, non-symmetric), complex matrix, A, of size m x n. By small I mean m and n is typically between 4 and 64 and usually around 16, but with m not equal to n.
This problem is straight forward to solve with the general LAPACK SVD algorithms, i.e. gesvd or gesdd. However, as I am solving millions of these problems and only require the largest eigenpair, I am looking for a more efficient algorithm. Additionally, in my application the eigenvectors will generally be similar for all cases. This lead me to investigate Arnoldi iteration based methods, but I have neither found a good library nor algorithm that applies to my small general complex matrix. Is there an appropriate algorithm and/or library?
Rayleigh iteration has cubic convergence. You may want to implement also the power method and see how they compare, since you need LU or QR decomposition of your matrix.
http://en.wikipedia.org/wiki/Rayleigh_quotient_iteration
Following #rchilton's comment, you can apply this to A* A.
The idea of looking for the largest eigenpair is analogous to finding a large power of the matrix, as the lower frequency modes get damped out during the iteration. The Lanczos algorithm, is one of a few such algorithms that rely on the so-called Ritz eigenvectors during the decomposition. From Wikipedia:
The Lanczos algorithm is an iterative algorithm ... that is an adaptation of power methods to find eigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very large sparse matrices. In latent semantic indexing, for instance, matrices relating millions of documents to hundreds of thousands of terms must be reduced to singular-value form.
The technique works even if the system is not sparse, but if it is large and dense it has the advantage that it doesn't all have to be stored in memory at the same time.
How does it work?
The power method for finding the largest eigenvalue of a matrix A can be summarized by noting that if x_{0} is a random vector and x_{n+1}=A x_{n}, then in the large n limit, x_{n} / ||x_{n}|| approaches the normed eigenvector corresponding to the largest eigenvalue.
Non-square matrices?
Noting that your system is not a square matrix, I'm pretty sure that the SVD problem can be decomposed into separate linear algebra problems where the Lanczos algorithm would apply. A good place to ask such questions would be over at https://math.stackexchange.com/.