# Tagged Questions

**1**

vote

**0**answers

29 views

### Lanczos algorithm with thick restart on a dynamic matrix

currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries from time to time. The problem should be solved using an ...

**1**

vote

**0**answers

91 views

### Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers.
To be precise, I want ...

**3**

votes

**0**answers

108 views

### Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task:
Let $\mathbb{F}$ be a finite field.
Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let
$G=\langle ...

**8**

votes

**1**answer

456 views

### How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient
algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that
$n = \sum_{i=1}^k a_i m_i$?
...

**3**

votes

**1**answer

116 views

### Is there an algorithm to compute group presentations of or find generators for the centralizer of a matrix in $GL(n, \mathbb{Z})$?

Let $M \in H \leq GL(n, \mathbb{Z})$. Is there an algorithm that computes either matrix generators or even a group presentation for $C_H(M)$ given generators or a presentation of $H$? Also is ...

**7**

votes

**1**answer

212 views

### Main problems on lattice-basis reduction algorithms (such as LLL)?

What are the main open problems on lattice-basis reduction algorithms (such as LLL)? I am looking for problems satisfying the following two conditions:
(a) their solution would likely be of some ...

**1**

vote

**0**answers

82 views

### Row subset selection of matrix to optimize condition number

Given a matrix $\mathbf{A} \in \mathbb{C}^{N\times M}$ with $N \gg M$. This matrix results from a linear equation system and has a certain structure (however, I do not think that details provide any ...

**4**

votes

**1**answer

261 views

### Checking if a binary vector lies in the affine span of given binary vectors

Let $x_1,\ldots,x_N \in \{0,1\}^D$ be $N$ binary vectors in ${\mathbb R}^D$, assumed affinely independent. Is there an efficient algorithm for determining whether a new binary vector $x_{N+1}$ is in ...

**5**

votes

**2**answers

759 views

### Solve for $A$ and $B$ in $AXB=Y$

Let $R = \mathbb{Z}[x_{1}, \dots, x_{r}]$.
Let $X$ be $n \times n$ matrix with entries in $R$.
Let $Y$ be $m \times m$ matrix with entries in $R$ formed from $\mathbb{Z}$-linear or $\mathbb{R}$-linear ...

**3**

votes

**0**answers

173 views

### (Co)limit computations for diagrams of Vector Spaces

Fix a field $K$ and consider a finite directed graph $\Gamma$ where multiple edges between a pair of vertices are allowed so long as the total number of edges is finite. Associate to each vertex $v$ a ...

**1**

vote

**0**answers

43 views

### Most efficient algorithm for computing norm of the residual for the least squares problem in the rank deficient case

I have a large $m\times n$ data matrix $A$, $m>n$, and response $m$-vector $b$. I need to calculate $E = ||Ax-b||_2$ as quickly as possible, where $x$ is the least squares solution. I don't need ...

**2**

votes

**1**answer

246 views

### Finding a subspace disjoint from a union of subspaces

Let $k$ be a finite field (I care about $\mathbb F_p$, especially $\mathbb F_2$) and let $V_1,...,V_N\subset k^n$ be subspaces.
I want to find a subspace $S\subset k^n$ such that $S\cap V_i=0$ for ...

**2**

votes

**2**answers

237 views

### Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...

**5**

votes

**0**answers

320 views

### Determine if a matrix is unimodular

Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?

**12**

votes

**2**answers

373 views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg n.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...

**1**

vote

**2**answers

170 views

### Simultaneous Smith Normalization of a Composable Matrix Sequence

Let $\mathsf{R}$ be a PID and consider a collection of free, finitely generated $\mathsf{R}$-modules $V_1,\ldots,V_n$ along with module maps $m_j:V_j \to V_{j+1}$. That is, we have the following ...

**2**

votes

**1**answer

409 views

### Does a product of matrices have eigenvalue 1

Start by fixing invertible matrics $A_1, \ldots, A_m \in \mathbb{Z}^{n \times n}$.
For a sequence $i_1, \ldots, i_k$ we construct $A = A_{i_1} \cdots A_{i_k}$. We would like to know "Is 1 an ...

**1**

vote

**2**answers

1k views

### Checking consistency of a system of linear equations and inequalities

I have a lot of systems of equations and inequalities of the following form:
$$ a_{1,1}x+a_{1,2}y+a_{1,3}z+a_{1,4}w = 2 $$
$$ \ldots $$
$$ 0 < x < 2 $$
$$ 0 < y < 2 $$
$$ 0 < z < 2 ...

**8**

votes

**3**answers

432 views

### Equitable Allocation of Individuals to Positions

I'm not a mathematician but I working on a problem that feels like it an example of a more general kind of problem and I'm hoping that someone might be able to point me in the right direction.
The ...

**3**

votes

**0**answers

241 views

### 3-SAT and a matrix of linear forms representing a non-degenerate matrix

This is a follow-up to the previous question on the same topic. Thanks to Emil Jeřábek I can now ask a more specific question.
As before, let $k$ be a field with $p$ elements. Consider the ...

**4**

votes

**1**answer

408 views

### determining if a matrix of linear forms represents a non-degenerate matrix

Let $k$ be a field with $p$ elements. Consider the following computational problem
Input: a natural number $n$, $n^2$ linear forms $M_{ij}$, $i,j=1,\ldots n$ in $n^2$ variables $X_{11}, \ldots ...

**1**

vote

**1**answer

373 views

### Need help to find an efficient algorithm for the following problem!

Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$.
Given $A_{n\times n}$ is the covariance matrix of $x$.
$u$ is a given n-dimensional vector of real ...

**4**

votes

**2**answers

460 views

### Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...

**0**

votes

**1**answer

792 views

### True divide and conquer inversion of large matrices

In http://math.stackexchange.com/questions/2735/solving-very-large-matrices-in-pieces there is a way shown to solve matrix inversion in pieces. Is it possible to turn it into a true divide and conquer ...

**3**

votes

**1**answer

315 views

### Mathematical Programming with other Algebras than Linear

Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization.
What analogies are there for ...

**2**

votes

**1**answer

171 views

### Computing a generating set of the kernel of a module

Crossposted from math.stackexchange, since I'm not getting any answer and I think the question is suitable here.
Given a generating set of a $\mathbb{Z_k}$-module $M \subseteq {\mathbb{Z}_k}^n$, is ...

**13**

votes

**2**answers

906 views

### Seeking proof for linear algebra constraint problem.

Given a symmetric real matrix with a zero diagonal $M$, I am trying to find a diagonal matrix $D$, such that the matrix $M + D$ is positive definite, and $(M+D)^{-1}$ has a diagonal consisting of all ...

**1**

vote

**1**answer

2k views

### Detection of Redundant Constraints

Suppose I pose the following query to a constraint logic programming
system:
?- Y <= 6 - X, Y <= (- 4) + 4 * X, Y <= 4 + X / 3.
Are there systems that would recognize the last inequality as
...

**5**

votes

**6**answers

2k views

### Fast evaluation of polynomials

Hello everybody !
I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer ...

**2**

votes

**1**answer

242 views

### Doing column permutation under row overlap constraint

In coding theory, there are parity-check codes whose parity-check matrices $H$ are generated via column permutations. For instance, the binary LDPC codes constructed in Gallager's 1962 IRE Trans paper ...

**5**

votes

**4**answers

904 views

### Determining a recurrence relation

I would like to solve the general problem of determining a linear recurrence relation that fits a given integer sequence of length $n$, or stating that none exists (with fewer than $n/2-k$ ...

**2**

votes

**0**answers

158 views

### Recovering a linear map from a non-linear approximation

The problem described here is algorithmic. We are given "black box access" to a map $f:R^d\to R^d$. By this we mean that one may query the value of $f(v)$ for an arbitrary $v\in R^d$.
We assume that ...

**0**

votes

**1**answer

249 views

### [Matrices over Z] - An algorithm for calculating the diagonal with elementary operations

Dear mathoverflow,
Let
$
\left(
\begin{array}{cc}
a & b \newline
c & d
\end{array}
\right)
$
be a matrix with $a, b, c, d \in \mathbb{Z}$, $\gcd(a,b,c,d) = 1$ and $ad - bc = \pm N$, with $N ...

**0**

votes

**2**answers

618 views

### Fast algorithms for computing nullspace of a positive semidefinite matrix over Z

Let $A \in \mathbb{Z}^{n \times n}$ be a positive semidefinite sparse matrix. I am looking for asymptotically fast algorithms for computing the nullspace basis of $A$ (or just random elements in the ...

**2**

votes

**0**answers

228 views

### When is polytope compatible with network flow?

A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...

**7**

votes

**1**answer

607 views

### Best way to find a closest vector in a lattice

Let $v_1,\ldots,v_n$ be linearly independent vector in $\mathbb{R}^n$, and let $\Lambda=\oplus_i^n \mathbb{Z}v_i$. The question is, given a vector $w$ find the element $v$ of the lattice $\Lambda$ ...

**-3**

votes

**1**answer

528 views

### Eliminating redundant linear constraints? [closed]

I have an NxN matrix of linear constraints that is not of full rank. In other words, some of the constraints are linear combinations of other constraints. The "standard" linear algebra tools ...

**1**

vote

**1**answer

1k views

### Bidiagonalization and SVD of matrix

I can't find a single solid explanation of how to implement this -- whitepapers too detailed/confusing. Closest I came to an answer was this:
...

**3**

votes

**1**answer

1k views

### SVD complexity for structured sparse matrices

Hello,
For an $n \times n$ real matrix, the SVD (Singular Value Decomposition) algorithm is $O(n^3)$.
I have large matrices (say $10,000 \times 10,000$) that only have elements on few diagonals, i.e. ...

**2**

votes

**0**answers

252 views

### Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...

**6**

votes

**2**answers

525 views

### To what extent can algorithms in undergraduate linear algebra be made continuous/polynomial/etc.?

I feel like many of the algorithms that I learned — indeed, that I have taught — in undergraduate linear algebra classes depend sensitively on whether certain numbers are $0$. For ...

**5**

votes

**2**answers

392 views

### Other norms for Lattice reduction techniques (LLL, PSLQ)?

LLL and other lattice reduction techniques (such as PSLQ) try to find a short basis vector relative to the 2-norm, i.e. for a given basis that has $ \varepsilon $ as its shortest vector, $ \varepsilon ...

**21**

votes

**8**answers

8k views

### Fast Matrix Multiplication

Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in ...

**6**

votes

**2**answers

3k views

### Solving a system of linear inequalities — what is the dimension of the solution set?

It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$?
For the applications I have in mind the ...

**10**

votes

**0**answers

271 views

### Matroids with prescribed independent sets

Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...

**2**

votes

**1**answer

1k views

### Matrix approximation

Let A be an $m\times n$ matrix and $k$ be an integer. Assume that $A$ is non-negative. We want to find a scalar $\epsilon$ and an $n\times n$ matrix $B$ such that $A\leq A(\epsilon I + B)$ (where ...

**3**

votes

**3**answers

193 views

### Rank(A) and other algorithms as a polynomial

If $A = (\alpha_{ij}) \in \mathbb{C}^{nxm}$ we have simple algorithms by which to determine $\mathrm{rank}(A)$. However, is there a polynomial $f \in \mathbb{C}[\alpha_{ij}]$ where $f \colon ...

**11**

votes

**2**answers

2k views

### How to compute the rank of a matrix?

Okay, that's a misleading title. This is a somewhat subtler problem than undergraduate linear algebra, although I suspect there's still an easy answer. But I couldn't resist :D.
Here's the actual ...