# Questions tagged [linear-algebra]

Questions on the algorithmic/computational aspects of linear algebra, including the solution of linear systems, least squares problems, eigenproblems, and other such matters.

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### Permute a matrix in-place in numpy

I want to modify a dense square transition matrix in-place by changing the order of several of its rows and columns, using python's numpy library. Mathematically this corresponds to pre-multiplying ...
271 views

### Algorithms for linear system of ODEs

I wonder: what is the best algorithm to solve $$\frac{du}{dt} = Au$$ Where $A$ is a real $n\times n$ matrix. A is not explicitly time-dependent, usually sparse but not ...
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### What are the fastest available implementations of BLAS/LAPACK or other linear algebra routines on GPU systems?

nVidia, for example, has CUBLAS, which promises 7-14x speedup. Naively, this is nowhere near the theoretical throughput of any of nVidia's GPU cards. What are the challenges in speeding up linear ...
18k 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 ...
210 views

6k views

### Recommendations for a usable, fast Java matrix library?

This complements an earlier question on usable, fast C++ matrix libraries. I've looked at the Java Matrix Benchmark, and it seems like the performance of java matrix libraries is all over the place. ...
693 views

### C library - iterative sparse complex linear equation solver?

Where can I find a library to solve a sparse complex matrix equation iteratively in C. So far I've only found libraries for direct solution to complex systems, and libraries for iterative solutions to ...
3k views

### What is the most efficient way to diagonalize small matrices?

I have a problem where I need to diagonalize a large number of small Hermitian matrices. Typically the matrices are between 4 and 64 in size (skewed towards the low end) and the number of matrices is ...
668 views

### Finding a permutation that makes a matrix lower triangular

I have a system of linear equations in form of $AX=b$ where $A_{m\times n}$, $X_{n\times 1}$ and $b_{m\times 1}$. Coefficient matrix $A$ is quite sparse. However, using a practical LP solver like ...
1k views

### Applications of Moore - Penrose generalized inverse of a matrix and associated projection?

I am seeking applications in the industry for the Moore-Penrose generalized inverse $A^\dagger$ of a matrix $A$. The Moore-Penrose Inverse of $A\in \mathbb{C}^{m\times n}$, denoted by $A^\dagger$, ...
522 views

### Exploring feasible points in a linearly defined space

What is the quickest way to find a point inside a linear feasible space? (Defined by the intersection of several hyperplanes and halfspaces). I want to be able to choose an initial point in the ...
1k views

### Approximate spectrum of a large matrix

I want to compute the spectrum (all the eigenvalues) of a large sparse matrix (hundreds of thousands of rows). This is hard. I am willing to settle for an approximation. Are there approximation ...
2k views

### eigenvalues (and determinant) of a hadamard product of matrices

I need to compute the determinant of a matrix that is calculated as $B \circ A$, with $B$ and $A$ being square matrices and $\circ$ representing their Hadamard product. One way of doing this is ...
1k views

### SVD for finding the largest eigenvalue of a 50x50 matrix -- am I wasting significant amounts of time?

I've got a program that computes the largest eigenvalue of many real symmetric 50x50 matrices by performing singular-value decompositions on all of them. The SVD is a bottleneck in the program. Are ...
315 views

### Computing Permanents of $64 \times 64$ Matrices

I need to compute the Matrix Permanents of several $64 \times 64$, zero-one matrices. I have tried using the built in functions in both Sage and Maple, but both programs return out of memory errors. I ...
78 views

### Computing a sequence of row interchanges that realizes a given permutation matrix?

This question is aimed at cleaning up an implementation detail of an in-house sparse direct solver. It uses METIS to reorder $A$ into $PAP^{T}$ for reduced fill-in. Inside the $Lx=b$ and $L^{T}x=b$ ...
9k 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 ...
I would like to predict runtimes for dense linear algebra operations on a specific architecture using a specific library. I would like to learn a model that approximates the function $F_{op} \;::\;$...