# 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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### In Octave, how do I specify that the solution to a matrix equation should be over integers?

In Octave, how do I specify that the solution to a matrix equation should be over integers? I.e., Given matrix $A$, vectors $x$ and $b$; $Ax=b$. Find vector $x=A^{-1}b$ such that all its entries are ...
530 views

### How to estimate the maximum argument of eigenvalues?

How to estimate $$\max_i |\mathrm{Arg}(-\lambda_i)|,$$ where $\{\lambda_i\}$ are eigenvalues of a large sparse matrix $A$ all lying in the left complex half-plane?
247 views

### 3D to 2D projections, a generalization

Given some data points in 3D, $X\in\mathbb{R}^{n\times 3}$, could one say that $$Y=XP,$$ for some $P\in\mathbb{R}^{3\times 2}$ actually corresponds to a particular viewpoint on a 3D data? Basically, ...
1k views

### Blaze linear algebra library?

The paper "Expression Templates Revisited: A Performance Analysis of Current Methodologies" in SIAM Journal of Scientific Computing references the "Blaze" linear algebra library. I haven't heard of it ...
943 views

### Practical efficacy of parallel back substitution

The fact that the back substitution is not done in parallel is not important, because it uses a negligible amount of computer time when N is large, compared to the forward elimination. This is a ...
154 views

### Simplifying some operations on Gram matrices

Suppose two Gram matrices are given $A, B\in\mathbb{R}^{n\times n}$, such that $$A=XX^T,~~~~~~~~~~~~~B=YY^T,$$ for some $X, Y\in\mathbb{R}^{n\times k}$, $k\ll n$. Also, suppose a Gram matrix based on ...
214 views

### How do the properties of a matrix affect the linear system solving

For a general matrix A, there are many properties to describe it: symmetric positive definite or indefinite, condition number, spectrum and so on. I am curious about how these properties affect the ...
861 views

### Projecting out the null-space of $A$ from $b$ in $Ax=b$

Given the system $$Ax=b,$$ where $A\in\mathbb{R}^{n\times n}$, I read that, in case Jacobi iteration is used as a solver, the method will not converge if $b$ has a non-zero component in the null-space ...
926 views

### Solving sparse matrix systems which can be reordered to block diagonal form

I have a class of matrices $A$ which are created by a domain decomposition method. Each matrix represents several subproblems of equal size, and I know that for some permutation matrix $P$, $PAP^T$ ...
192 views

### How to detect the multiplicity for the eigenvalues?

Suppose A is a general sparse matrix, and I want to compute the eigenvalues. I do not know how to detect the multiplicity for the eigenvalues. As far as I know, for a special case, finding the ...
405 views

### Rank structure in the Schur complement

I am doing research on the structure in the Schur complements and find an interesting phenomenon: Suppose that A is from 5--pt laplacian. If I use nested dissection ordering and multifrontal method ...
107 views

### Efficient principal pivots

It was suggested I should try posting this question here from Mathematics Background I'm working on a numerical linear algebra package in C#. I'm trying to implement a variety of "principal ...
188 views

### How to establish that an iterative method can be applied to large matrices whose size may reach 10^3?

I have an iterative method for computing the Moore-Penrose generalized inverse of matrices, that is $$X_{k+1} = ((I-\beta X_{k}A)^t) + X_{k}$$ with initial approximation: $$X_{0} = \beta AA^t$$ ...
1k views

### Why would a computational scientist need to implement their own version of std::complex?

Many of the better-known C++ libraries in computational science such as Eigen, Trilinos, and deal.II use the standard C++ template header library object, ...