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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votes
1answer
402 views

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 ...
5
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2answers
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?
3
votes
1answer
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, ...
13
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3answers
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 ...
6
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1answer
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 ...
4
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2answers
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 ...
4
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2answers
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 ...
11
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1answer
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 ...
3
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2answers
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 $P$, $PAP^T$ ...
11
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1answer
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 ...
9
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1answer
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 ...
3
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0answers
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 ...
3
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2answers
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: $$X_{0} = \beta AA^t$$ ...
14
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2answers
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, ...
6
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3answers
5k views

How to find QR decomposition of a rectangular matrix in overdetermined linear system solution?

While trying to find cell-centered gradients in finite volume method computation of incompressible fluid flow I get over-determined linear system. This is a well known "cell based least-square" ...
5
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1answer
1k views

Comparing algorithms for tridiagonal linear systems solution

Below there are two algorithms for solving tridiagonal linear systems of the form $$ \left[ \begin{array}{ccccc|c} b_1 & c_1 & & & &d_1\\ a_2 & b_2 & c_2 & & &...
4
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3answers
125 views

Whats wrong with my running time calculation?

I am running a linear algebra iterative method (PCG) for solving Ax=b, the dimension of the matrix is 10000x10000. So, I did 2 preliminary analyses: Memory Analysis The size of the matrix dominates ...
9
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2answers
142 views

Estimate Norm of a black-box functional

Let $V$ be a finite-dimensional vector space with norm $\|\cdot\|$ and let $F : V \rightarrow \mathbb R$ be a bounded linear functional. It is only given as black-box. I would like to estimate the ...
4
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1answer
131 views

Constructing the origin position by transforming distance information

Suppose a set of $n$ points, $n\in M$, is given in some $d-$dimensional space, $X\in\mathbb{R}^{n\times d}$. Among these $n$ points, some $k\in K$ are selected, so $k<n$, and the distances from ...
4
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1answer
176 views

Estimating time for running serial/parallel codes

Assume I am running an iterative method, I have a rough estimate of how many iterations it will need, How do best estimate the time it will run for in serial? For instance, If I have Conjugate ...
9
votes
2answers
510 views

Is there a generalization of the Sylvester Inertia Law for the symmetric generalized eigenvalue problem?

I know that in order to solve symmetric eigenvalue problem $Ax = \lambda x$, we can use the Sylvester Inertia Law, that is the number of eigenvalues of $A$ less than $a$ equals the number of negative ...
10
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1answer
525 views

How to find the interior eigenvalues by krylov subspace method?

I am wondering how to find the eigenvalues of some sparse matrix in given interval [a, b] by iterative method. To my personal understanding, it is more obvious to use Krylov subspace method to find ...
11
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4answers
2k views

Finding the square root of a Laplacian matrix

Suppose the following matrix $A$ is given $$ \left[\begin{array}{ccc} 0.500 & -0.333 & -0.167\\ -0.500 & 0.667 & -0.167\\ -0.500 & -0.333 & 0.833\end{array}\right]$$ with ...
5
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0answers
719 views

Perron-Frobenius theorem on general real symmetric matrices

From the Perron-Frobenius theorem, it might be concluded that the spectral radius is the largest eigenvalue for positive matrices, ie, matrices with strictly positive entries. In other words, the ...
16
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4answers
3k views

Why can't Householder reflections diagonalize a matrix?

When computing the QR factorization in practice, one uses Householder reflections to zero out the lower portion of a matrix. I know that for computing eigenvalues of symmetric matrices, the best you ...
3
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0answers
70 views

Understanding how Numpy does SVD [duplicate]

Possible Duplicate: Understanding how Numpy does SVD I have been using different methods to calculate both the rank of a matrix and the solution of a matrix system of equations. I came across the ...
13
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3answers
8k views

Understanding how Numpy does SVD

I have been using different methods to calculate both the rank of a matrix and the solution of a matrix system of equations. I came across the function linalg.svd. Comparing this to my own effort of ...
6
votes
2answers
294 views

Is it possible to ignore/discard part of a matrix when finding eigenvalues?

I have have multiple large matrices for which I need to find the largest absolute eigenvalue. I know that there is a large submatrix that does not vary. Is it possible to ignore/discard the submatrix? ...
2
votes
1answer
253 views

Proof continuation for rigid transformation on PCA solution

Suppose a matrix $X\in\mathbb{R}^{n\times 3}$ is given as a Principal Component Analysis (PCA) projection from some high dimensional space. The 2D PCA solution on X, say $Y\in\mathbb{R}^{n\times 2}$ ...
7
votes
3answers
9k views

Gershgorin Circle Theorem to estimate the eigenvalues

In order to estimate the eigenvalues of a real symmetric $n\times n$ matrix, I intend to use the Gershgorin Circle Theorem. Unfortunately, the examples one might find on the internet are a bit ...
9
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4answers
2k views

Generating Symmetric Positive Definite Matrices using indices

I was trying to run test cases for CG and I need to generate: symmetric positive definite matrices of size > 10,000 FULL DENSE Using only matrix indices and if necessary 1 vector (Like $A(i,j) = \...
1
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3answers
266 views

Unique coordinates (solutions) in a single Gauss-Seidel iteration

I managed to reduce certain computational problem to the Gauss-Seidel solution of the following linear system: $$Ax=Ly,$$ where $A, L\in\mathbb{R}^{n\times n}$ are weighted Laplacian matrices (...
0
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1answer
1k views

A Comparison between GMRES, QMR and LU for Dense Matrices

As I see it, there are 3 ways to solve unstructured dense system of equations: GMRES, QMR and LU. Has anyone done a comparison for these three? As far as I know, LU is the preferred choice and it is ...
11
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3answers
478 views

Parallel algorithm for eigensystem of a tridiagonal matrix

I'm doing a Lanczos diagonalization of a large sparse matrix (~2 million elements). Almost all of the steps in the Lanzcos algorithm are done in parallel on the GPU, except for diagonalizing the ...
3
votes
2answers
122 views

Time-stable SO(n) matrix synthesis algorithm

Consider an equation $S(t)b(t) = a$, where $a, b(t) \in S^{n-1}$ are given and the vector $b(t)$ is continuous, i.e. its endpoint traces a continuous curve on the unit sphere. The task is to find ...
3
votes
1answer
969 views

Spectral decomposition with eigenvalue shift

Suppose a square, real and symmetric matrix $G\in\mathbb{R}^{n\times n}$ is given, and it is known to have one zero eigenvalue associated with all ones eigenvector, $1_n$. I'm aware that the (possibly)...
33
votes
6answers
29k views

What is the fastest way to calculate the largest eigenvalue of a general matrix?

EDIT: I am testing if any eigenvalues have a magnitude of one or greater. I need to find the largest absolute eigenvalue of a large sparse, non-symmetric matrix. I have been using R's ...
12
votes
3answers
2k views

What's the current state of the art regarding algorithms for the singular value decomposition?

I'm working on a header-only matrix library to provide some reasonable degree of linear algebra capability in as simple a package as possible, and I'm trying to survey what the current state of the ...
12
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1answer
3k views

weighted SVD problem?

Given two matrices $A$ and $B$, I'd like to find vectors $x$ and $y$, such that, $$ \min \sum_{ij} (A_{ij} - x_i y_j B_{ij})^2. $$ In matrix form, I'm trying to minimize the Frobenius norm of $A - \...
7
votes
1answer
339 views

float128 in linear algebra

Is there any paper or research concerning float128 arithmetics applied to linear algebra problems(e.g. iterative solvers, decompositions etc.)? How much benefit is really there in comparison with ...
9
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3answers
531 views

Computing the characteristic polynomial of real sparse matrix

Given a generic sparse matrix $A \in \mathbb{R}^{n\times n}$ with m << n (correction: $m \ll n^2$) non-zero elements (typically $m \in {\cal O}(n)$). $A$ is generic in the sense that it has no ...
3
votes
1answer
71 views

2D Jacobi line maintenance?

Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
9
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4answers
3k views

Condition number of A'A and AA' formulations

It's shown (Yousef Saad, Iterative methods for sparse linear systems, p. 260) that $cond(A'A) \approx cond(A)^2$ Is this true for $AA'$ as well? In case $A$ is $N\times M$ with $N \ll M$, I observe ...
8
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2answers
141 views

Initial guesses for perturbed linear systems

Suppose you solve a linear system $Au = f$ by an iterative method, e.g. conjugate gradients or Richardson iteration. Then you try to solve a linear system that is slightly perturbed in the matrix and ...
9
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2answers
1k views

Safe application of iterative methods on diagonally dominant matrices

Suppose the following linear system is given $$Lx=c,\tag1$$ where $L$ is the weighted Laplacian known to be positive $semi-$definite with a one dimensional null space spanned by $1_n=(1,\dots,1)\in\...
10
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2answers
1k views

Which iterative linear solvers converge for positive semidefinite matrices?

I want to know which of the classic linear solvers (e.g Gauss-Seidel, Jacobi, SOR) are guaranteed to converge for the problem $Ax=b$ where $A$ is positive semi definite and of course $b \in im(A)$ (...
5
votes
2answers
3k views

Fast algorithms to find the eigenvalues of some matrix on intervals of interest

I am curious how to quickly compute the eigenvalues for arbitrary matrices, sparse or dense, restricted on some given interval of interest. Suppose we have an arbitrary $n\times n$ matrix $A$, ...
7
votes
1answer
678 views

Jacobi iteration to reduce the quadratic function

Given certain function $f(X)$ which is quadratic in $X\in\mathbb{R}^{n\times d}$, $$\frac{1}{2}tr(X^TAX) - tr(Y^TBX)$$ for positive definite weighted Laplacian matrices $A, B\in\mathbb{R}^{n\times n}...
5
votes
2answers
534 views

Recommendation for a good article/book for frontal methods?

Can someone provide an article or book that explains the principle used in frontal solvers? Some examples also may help understand the frontal methods better.Thanks in advance!
6
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1answer
301 views

Question about the smoothing operators in multigrid methods for nonlinear PDEs

Suppose we are dealing with a nonlinear problem, say $$ A u := L u + G(u) = f $$ the nonlinearity of the operator $A$ is the polynomial type, ie, $L$ is a linear operator, and $G(u) = u^k$, or ...