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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3answers
355 views

Accurate Way to Calculate Matrix Powers and Matrix Exponential for Sparse Positive Semidefinite Matrices

I do need to numerically calculate the following forms for any $x\in\mathbb{R}^n$, possibly in python: $x^T M^k x$, where $M\in\mathbb{R^{n\times n}}$ is a PSD sparse matrix, $n$ can be quite large ...
-1
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0answers
80 views

Reformulate a maximization into a minimization problem

I have the following maximization problem: $$\max_{\mathbf{w}, t, \Theta} t$$ $$\text{s.t. } ||\mathbf{w}||^{2} \leq P$$ $$\mathbf{S} \succ 0$$ $$||\mathbf{u}^{(n)}||+2\sum_{i=1}^{2} \mathbf{u}^{(n)...
2
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0answers
38 views

How to solve this boundary value problem which has more unknown than equation on MATLAB

I need your helps about solving the problem below with MATLAB. I am trying to solve 2D Stress Wave Propagation problem by using FDTD(Finite difference time domain) method on the cylindrical coord. I ...
1
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2answers
105 views

Diagonalization using LAPACK

Say, we have a Hamiltonian which for simplicity does not mix particle hole sectors. It is just a simple Hamiltonian in real space as shown, $H=\sum_{ij,\sigma} A(i,j)(c_{i\sigma}^{\dagger}c_{j\sigma} +...
0
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2answers
108 views

Given a symmetric matrix, is it ok to apply Cholesky decomposition to see if it has negative eigenvalues?

I intend to check the diagonal of L, where A = L'L, for negative elements. However, I don't know if Cholesky is meaningful in theoretical / computational sense if there are some negative eigenvalues.
0
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1answer
63 views

Solve for large array of PD matrices

I have N matrices that are positive definite, and I have to solve for a M vectors. As M is large in my case, doing all solves simultaneously using np.linalg.solve ...
0
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1answer
44 views

Algebraic multigrid as solver and as preconditioner

My question is around the efficiency of AMG. In which case AMG can perform better,as solver or as a preconditioner(for example a Krylov space method as CG)? Assume the case of elliptic pdes.
2
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1answer
88 views

Efficiently compute a projection matrix from Householders reflectors

Let $A \in \mathbb{R}^{m \times n}$ where $m \geq n$. Let $B$ and $\tau$ be the result of applying LAPACK's dgeqrfp method (R on the upper right triangle, and the ...
3
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1answer
109 views

A notion of resolution in inverse problems

Suppose I have a linear inverse problem of the form: \begin{align} Ax=b \end{align} I would like to reconstruct $x$ from the measurement $b$ via the objective $$\min_x\{\vert\vert Ax-b\vert\vert^2_2+\...
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0answers
58 views

Upper bound on condition number in linear preconditioning

I'm studying iterative methods for solving linear system, and I find the following setting in Wikipedia: Consider a matrix splitting $A = M-N$, where $A,M,N$ are all symmetric and positive definite ...
6
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1answer
207 views

Efficiently computing $e^{tX}$ for many different values of $t$

Given an anti-Hermitian and sparse matrix $X$, I am using Python (NumPy and SciPy) to compute the matrix exponential $f(t) := e^{tX}$ for many values of $t$. The method I am currently using is to ...
2
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1answer
99 views

Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular

I am reading the Chapra and Canale book on numerical methods, and was working through the chapters on solving linear systems. Now the book goes through direct methods including Gaussian Elimination, ...
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0answers
62 views

Given an unpivoted form of Aasen's algorithm, how does one add pivoting?

I've implemented the version of Aasen's algorithm described in the book Matrix Computations 4th Edition. The version there doesn't have pivoting. The book's description of how to add pivoting is a bit ...
2
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1answer
56 views

How do I extract the output of Aasen's algorithm into a usable form?

I tried implementing the algorithm in Aasen's 1971 paper on factorizing symmetric indefinite matrices. I've translated the code verbatim from Algol into Python, and I used the test example given in ...
1
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1answer
87 views

Complexity of solving an image differential linear system

Define an "image differential linear system" as a linear system $A\mathbf{x}=\mathbf{b}$ wherein $\mathbf{x}$ contains the ($\mathbb{R}$) pixels of an image and each row of $A$ constrains ...
6
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1answer
166 views

Parallelize Scipy iterative methods for linear equation systems(bicgstab) in Python

I need to solve linear equations system Ax = b, where A is a sparse CSR matrix with size 500 000 x 500 000. I'am using scipy.bicgstab and it takes almost 10min to solve this system on my PC and I need ...
5
votes
1answer
189 views

Symmetric matrix which satisfies conditions of the form $v_i^T X v_i = 0$

I want to solve an underdetermined system of linear equations $A x = b$ with $A \in \mathbb{R}^{n \times r^2}, x \in \mathbb{R}^{r^2}, b \in \mathbb{R}^n$. The matrix $A$ has the following additional ...
0
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0answers
60 views

How to make a directed graph symmetric?

Say I have a directed graph given as an adjacency matrix $A$ in CSR format represented by the arrays ia (row indexes) and ja (...
3
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2answers
108 views

Equations that are easier to verify than to solve?

Are there interesting examples of (systems) of equations where it is known to be harder to find a solution (in terms of scaling with respect to problem size) than verifying a provided solution for ...
1
vote
1answer
84 views

Range of a matrix from its complete orthogonal decomposition

In this StackOverflow answer, @Gokul has shown how to get a basis of the kernel of a matrix with the help of the 'Eigen' function CompleteOrthogonalDecomposition. ...
1
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1answer
101 views

Factorization of cubic spline interpolation matrix

In cubic spline interpolation, we use the set of knots and function values $(x_i,y_i),i=1,...,n$ to construct a (tridiagonal) system of equations for the unknowns $\sigma_i$: $$ h_{i-1}\sigma_{i-1} + ...
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0answers
47 views

Formula for overdetermined logical matrix pseudoinverse not requiring SVD?

In https://commons.wikimedia.org/wiki/File:YI_%3D_PI.png, you will find a formula-based solution for an overdetermined logical matrix pseudoinverse. This simple formula gives the same result as the ...
4
votes
1answer
73 views

Trace of inverse from LU decomposition

Given an LU decomposition of $A\in \mathbb{R}^{n\times n}$, is there a way to compute $\operatorname{trace}(A^{-1})$ with lower complexity than that of the inversion ($O(n^3)$ in practice)? This ...
4
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0answers
53 views

Efficient computation of marginalized multivariate normal likelihood

In general,if we know that the marginal Gaussian distribution for some variable $\textbf{x}$ and a conditional Gaussian distribution for some $\textbf{y}|\textbf{x}$ of the forms: $$p(\textbf{x}) = \...
3
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0answers
41 views

Choice of iterative solver for a sparse asymmetric matrix with symmetric structure

I have a sparse $nxn$ matrix A with pretty interesting structure. It has a block structure with symmetric structure but asymmetric blocks. Expressed mathematically $A_{jk} = A_{kj}$ but $A_{jk} \neq ...
2
votes
1answer
91 views

Diagonalization of Hermitian matrices vs Unitary matrices

What are the general algorithms used for diagonalization of large Hermitian matrices and Unitary matrices? ($>5000 \times 5000$) LAPACK seems to diagonalize Hermitian matrices almost 20 times as ...
2
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0answers
83 views

Algorithm for computing inner products multiple times

I am taking a computational linear algebra course and i got stuck during a homework problem concerning the computation of inner products. I am supposed to compute the inner product:$$\mathrm{a}_{\...
4
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0answers
63 views

Optimize linear equation using inner products and subject to L1 norm

I have a linear system of the form $A x = b$ where $A$ and $b$ are known, $A$ is "square", and $\lvert b \rvert_1 = \lvert x \rvert_1 = 1$. Unfortunately, I am working in a framework that ...
2
votes
1answer
83 views

Efficient change of basis real positive definite symmetric matrix

I need to optimize a code where the most performance critical part is doing a 'change of basis', in other words it is an unitary similarity transformation on a big real positive definite symmetric ...
0
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0answers
66 views

Ill-conditioned stiffness matrix

I am writting a Fem code in c++ for a 2d plane stress model. My question is regarding the assembly stiffness matrix.I noticed that some elements of the matrix are not exactly zero but insted a number ...
1
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1answer
84 views

Jacobi iterative method

I'm using Jacobi iterative method for finding eigenvalue and eigenvector for hermitian or symmetric matrix. Eigenvectors corresponding to eigenvalues are not exact. The third eigenvector is totally ...
1
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0answers
25 views

Multiplying by E[xy'] where only some statistics of xy' are known

(cross-posted on crossvalidated) For random variable $(x,y)$ in $\mathbb{R}^{d}\times \mathbb{R}^{d}$ and vector $v \in \mathbb{R}^d$, I need to perform the following matrix vector multiplication. $$T(...
5
votes
0answers
125 views

Inverse problem with uncertain forward operator

Suppose I want to solve a linear inverse problem. In this example we take a convolution with the kernel: $$\frac{1}{(y^2+z^2)^{3/2}}$$ We only take a fixed $z$ for the computation and convolve with ...
6
votes
2answers
206 views

When is it easy to invert a sparse matrix?

(Crossposted on cstheory.SE) When is it easy to invert a sparse matrix? Specifically, I'm wondering about the cases in which matrix inversion has similar cost to sparse matrix multiplication, hence ...
2
votes
1answer
42 views

Solving MX=N where M is structured as a Gaussian 4th-moment tensor

I'm looking to solve numerically the following equation for $(d,d)$ variable $X$, in Einstein summation notation $$M_{ijkl}X_{kl}=N_{ij}$$ Where $M$ is a $(d,d,d,d)$ 4th-moment tensor of random ...
2
votes
1answer
123 views

Ill-condioned Linear System and Gaussian Elimination

Suppose that I have a linear system $Ax=b$ such that $A$ is ill-conditioned. Can I say that it is dangerous to find a solution with Gaussian Elimination for this system, or does there exist some class ...
3
votes
2answers
167 views

1D FEM for nonlinear diffusion coefficient

I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$ in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$. ...
3
votes
1answer
95 views

Efficient solution to a structured symmetric linear system with condition number estimation

I have a real-valued linear system $Hx = b$ where $H$ is symmetric matrix** (not necessarily positive/negative definite) with a very particular structure: $$ H = \begin{bmatrix} D && B \\ B^T &...
0
votes
0answers
46 views

Norm estimates if adjoints can't be computed

Assume that you have two linear maps $A$ and $V$. For a given $x$ (of appropriate dimension) you can compute $Ax$ numerically, and for any $y$ (of appropriate dimension) you can calculate $V^Ty$ ...
5
votes
1answer
659 views

Cheap recalculation of eigenvalues and eigenvectors for a low-rank update of the matrix

Suppose I have a correlation matrix, $A$, and I already have the eigenvalues and eigenvectors of this matrix. For a given vector, $\mathbf{\mathit{v}}$, I want to calculate the eigenvalues and ...
2
votes
1answer
95 views

Runtime of Gaussian elimination/row reduction on an $m \times n$ matrix

The runtime of Gaussian elimination on an $n \times n$ matrix is $O(n^3)$. What is the runtime on an $m \times n$ matrix? I am taking Gaussian elimination to mean putting the matrix in reduced row ...
1
vote
1answer
85 views

Frobenius norm of a binary matrix

In term of the mathematical distance measurement, What is the significance of a Frobenius norm for a binary matrix?
1
vote
1answer
54 views

How to optimize nuclear norm subject to positive semidefinite constraints?

For finite dimensional symmetric positive semidefinite matrices $A$ and $B$, I would like to solve \begin{align}&\min |X - A|_1 \\ &\text{subject to}\\ &X \preceq B \\ &0 \preceq X\...
0
votes
0answers
70 views

Derivative-free ill-conditioned non-linear least squares

I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ...
8
votes
5answers
508 views

Checking singularity of a matrix

Suppose that we don't know $n \times n$ matrix $A$ explicitly but we are only able to compute products $Ax$ where $x$ is a column vector with $n$ elements. Is there an algorithm to determine whether $...
0
votes
3answers
143 views

How to determine global stiffness matrix is constrained or not

Background In solid fem, we often solve $$\mathbf{Ku}=\mathbf{p}$$ where $\mathbf{K}$ is global stiffness matrix, $\mathbf{u}$ is displacement, $\mathbf{p}$ is global load vector. If displacement not ...
2
votes
1answer
99 views

Efficient projection of a vector onto matrix kernel

Given an $m \times n$ matrix $A$ and a vector $x\in\mathbb R^n$, with $m<n$, what's an efficient way of computing the projection of $x$ onto the kernel of $A$?
7
votes
2answers
290 views

Is there an iterative solver for dense matrices with possible zero diagonal entries?

Is there an iterative solver that can handle potentially zero entries on the central diagonal? I am implementing a polynomial fitting algorithm (up to $10^{th}$-order) and my matrix is a "...
2
votes
1answer
270 views

Solution of the linear system using Sherman-Morrison formula for 1000000x1000000 (7450.6GB) matrix using MATLAB

Let $n = 10^6.$ Let $A \in \mathbb{R}^{n\times n} $ be the lower triangular matrix having 1's on and below the main diagonal. We want to solve the following linear system: $$ (A + uv^T)x = b$$ by the ...
3
votes
0answers
110 views

Compute Nullspace of Sparse Matrix

I am computing the nullspace of a sparse rectangular $m$ x $n$ matrix $A$, where $m$ << $n$. I do this by computing the QR decomposition of $A^T$ and extract the $n-m$ right-most columns of the ...

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