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Questions tagged [factorization]

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3
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0answers
32 views

Reweighted least squares factorization

This is a continuation of the question asked here. I want to solve numerous least squares systems of the form $$ D_i A x \approx D_i b $$ where $D_i$ are $m \times m$ diagonal matrices with positive ...
1
vote
1answer
96 views

Supernodal vs multifrontal factorizations

What are the differences between supernodal and multifrontal matrix factorizations? Can you provide a few references or high-level points about the approaches?
0
votes
2answers
115 views

Factoring a quadratic function

I have a quadratic binary optimization problem of the form \begin{align} &\max x^TQx \cr &\text{subject to }x\in\mathcal{X}\subseteq\{0,1\}^n, \end{align} where $\mathcal{X}$ is the feasible ...
1
vote
2answers
169 views

numerically stable routines to compute $M = B A^{-1} B$

Rather than gesv -> solve $AX = B$ gemm -> compute $M = BX$, somehow I feel there are better ways to compute $M$ with lapack/mkl?
5
votes
0answers
114 views

Rank-one Update to a Rank Revealing QR (RRQR) Factorization?

Suppose we are given an RRQR factorization for some matrix $A \in \mathbb{R}^{m \times n}$, $A\Pi = QR$ where $m > n$. Is there a cheap way to update $A' = A + uv^{\top}$ given this factorization?...
1
vote
1answer
315 views

Clever ways to update LU factorization for ridge regression

Ridge regression can be posed as minimizing the following objective function (over $x$): $$\frac{1}{2} \lVert Ax - b \lVert_2^2 ~+ \frac{\lambda}{2} \lVert x \lVert_2^2 $$ Which has a closed form ...
4
votes
1answer
79 views

Does $\log(\det(A))$ equals sum of log of diagonal elements of D in LDLT decomposition?

For a large matrix $A$, I need to evaluate the $\log(\det(A))$. I already have it's LDLT decomposition. Is it possible to evaluate the $\log\det$ with the elements of the diagonal $D$ of the LDLT ...
3
votes
1answer
209 views

Symmetric nonnegative matrix factorization

Suppose $A\in\mathbb{R}_+^{n\times n}$ is symmetric. I would like to factorize $A\approx UU^\top$ by solving $$ \begin{array}{rl} \min_U & \sum_{ij} \left(A_{ij}\ln\frac{A_{ij}}{[UU^\top]_{ij}}+[...
1
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1answer
65 views

Big errors while calculating Complex Cholesky Factorization

I am using my own Routine to calculate the Cholesky-Factorization of a complex, positive definite symmetric Matrix. My Code Looks like this: ...
2
votes
0answers
354 views

Difference between fast and normal Givens Rotations?

would someone be so kind as to explain me the difference between the ordinary givens-rotation and the fast givens-rotation? I know that the fast givens Rotation reduces the Count of operations to ...
1
vote
0answers
98 views

C++ Library: What is the Most Efficient Library that Factorizes a Polynomial?

I need to know what library is the fastest one that can factorize a polynomial of large degree whose coefficients are big integers. I have tried NTL library to factorize a monic polynomial but it is ...
4
votes
1answer
76 views

Is there an efficient $O(n^2)$ way to get the eigen decomposition given a LDL factorization?

Let's say I have a LDL factorization of a matrix A. Is there an efficient $O(n^2)$ way to get the eigen decomposition of A given it's LDL factorization? Is there a more efficient way, in case L and ...
1
vote
1answer
90 views

What is the cost of factorization for one-dimensional sparse problems?

In Golub and Van Loan's book, Matrix Computations, page 606, it is stated that: With standard discretizations, 2-dimensional problems can be solved with $O(n^{3/2})$ work and $O(n \log{n})$ fill-...
1
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1answer
55 views

Is there an upper bound for fill-ins for indefinite triangular factorization?

For $A=LU$, or $A=LDL^T$ factorization, bandwidth is preserved when there is no pivoting. This is true even for indefinite A, see question. However, when there is pivoting band structure is destroyed, ...
5
votes
1answer
83 views

Is the bandwidth of indefinite A equal to its factor L in LDL^T?

In George, Liu, and Ng's book Computer Solutions of Sparse Linear Systems, it has been shown that bandwidth of $A$ is equal to bandwidth of its factors in $LL^T$.(section 4.3) However, I guess this is ...
-4
votes
1answer
188 views

Can anyone give me some suggestions about optimize my Matlab codes?

recently, I try to write a Matlab codes to implement a sparse approximation inverse factorization method proposed by M. Benzi in his paper http://www.mathcs.emory.edu/~benzi/Web_papers/ainv.pdf this ...
1
vote
1answer
115 views

Compiling HSL_ME57 and interfacing with C code

Has anyone here had any success (or at least tried) compiling the HSL_ME57 (or similar such as MA57, etc.) matrix factorization libraries? Do any C wrappers exist for the Fortran function calls? I'm ...
4
votes
1answer
153 views

Factorization for reweighted least squares

I am solving a problem using an iteratively-reweighted least squares method: http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares Essentially this requires solving a number of least-...
1
vote
1answer
165 views

Non-negative matrix factorization for sparse input

Looking for some software to deal with 50kx50k sparse matrix applying non-negative matrix factorization. Do you know any?
3
votes
1answer
81 views

Missing nonzeros in U matrix computed by SuperLU

On page 18 of SuperLU's user guide, the following sample input matrix A is given: ...
6
votes
2answers
1k views

Sparse Incomplete Cholesky

I'm looking for an efficient, multicore, library to do incomplete cholesky (possibly modified). Many ILU code exists, but I can't find much about IC except in PETSC or Pastix. Could some of you drop ...
6
votes
2answers
829 views

Computing smallest eigenvectors of a sparse matrix, having its inverse

I'm a bit confused by the vast amount of literature on solving eigenvalue problems. I have a sparse (large) matrix which I have already factored (by Cholesky or LDU). I would like to compute few ...
4
votes
2answers
325 views

Existence of incomplete cholesky factorization

What is the current state of research on the existence of incomplete cholesky factorizations (in the context of preconditioning) for symmetric positive definite matrices? I wonder in particular ...