All Questions
11 questions
2
votes
1
answer
241
views
How to solve this set of equations as efficiently as possible (with "efficiently" measured in FLOPS)?
The system of equations is the following:
$$
\Gamma_i^{\ -1} = \sum_{i=1}^nA_{ij}\Gamma_j,
$$
where $\Gamma = (\Gamma_i)$ is a vector of size $n$ and $A$ is a matrix of size $n\times n$, with $n \gt ...
- 21
3
votes
1
answer
836
views
Solving multilinear equations
Let $N=\{1,2,\ldots,n\}$. Suppose we are given $n$ equations, with each equation taking the form $\sum_{A\subseteq N}\left(c_A \prod_{i\in A}x_i \right) = 0$, where each $c_A$ is a real number ...
- 239
4
votes
1
answer
396
views
Resolvent of a triangular matrix
Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$?
Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
- 333
2
votes
0
answers
146
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 ...
- 21
2
votes
0
answers
263
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 ...
- 21
13
votes
2
answers
1k
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 ...
- 379
8
votes
2
answers
2k
views
Algorithm for solving systems of linear Diophantine inequalities
So, I posted on StackOverflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem ...
- 3,079
22
votes
9
answers
17k
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 ...
- 1,654
5
votes
4
answers
2k
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 $n/2-k$ ...
- 9,114
2
votes
0
answers
187
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 ...
11
votes
1
answer
3k
views
Best way to find a closest vector in a lattice
Let $v_1,\dotsc,v_n$ be linearly independent vectors in $\mathbb{R}^n$, and let $\Lambda=\bigoplus_{i=1}^n \mathbb{Z}v_i$. The question is, given a vector $w$ in $\mathbb R^n$, find the element $v$ ...
- 351