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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38 views

Scientific calculator device malfunctioning

https://en.wikipedia.org/wiki/Scientific_calculator Can there be scenario/s that a scientific calculator gives incorrect answers in calculations to the end user?
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21 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 (...
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2answers
75 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 ...
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1answer
58 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. ...
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1answer
80 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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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
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1answer
66 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 ...
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0answers
50 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}) = \...
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39 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 ...
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1answer
79 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 ...
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0answers
81 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}_{\...
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0answers
60 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 ...
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1answer
78 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 ...
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63 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 ...
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1answer
80 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 ...
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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(...
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0answers
95 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 ...
4
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1answer
119 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
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1answer
41 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
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1answer
93 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
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2answers
157 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
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1answer
92 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 &...
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0answers
45 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
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1answer
419 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
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1answer
65 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 ...
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1answer
76 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?
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1answer
51 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\...
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68 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 ...
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5answers
479 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 $...
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3answers
107 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
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1answer
90 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 $A$?
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2answers
269 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
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1answer
224 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 ...
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0answers
94 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 ...
3
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0answers
44 views

Numerical calculation of the Berry connection

I'm doing some numerical calculations involving Hermitian matrices, and derivatives of the eigenvectors. Essentially, I have an n x n, Hermitian matrix H(x), which is dependent on some continuous ...
3
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0answers
88 views

Is the matrix exponential and the Jordan canonical form actually useful for solving differential equations?

All of my yearlong graduate-level Linear Algebra course notes from my professor—an algebraist/representation theorist—shows his love for the exponential map $e^A$ and the Jordan canonical form—and one ...
5
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1answer
93 views

Accurately Computing a Positive Vector in the Nullspace of a Matrix

I'm sure this question has been asked before yet after many hours of searching I am unable to find a definitive answer. The problem at hand is solving the linear system: $$A \mathbf{x} = \mathbf{0}$$ ...
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0answers
79 views

A parallelized GMRES solver?

My application calls for solving a dense, 40,000 x 40,000, ill-conditioned linear system. The native SciPy GMRES solver with preconditioning has worked well for my application and solving a single ...
2
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0answers
71 views

Why is LAPACK (seemingly) suboptimal for packed and banded eigenvalue problems?

Based on this LAPACK routines list, it looks like there is no relatively robust representation (RRR) driver routine for either packed or banded symmetric eigenvalue problems. According to the relevant ...
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1answer
164 views

Is LAPACK behind the cutting edge of dense linear algebra?

I have been digging into some numerical linear algebra lately, and reading in particular about how LAPACK solves symmetric eigenvalue problems. I noticed that the ...
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2answers
194 views

Why OpenFOAM uses its own data structures and linear solvers?

I wonder why OpenFOAM code has its own data structures Lists, HashTables, ... etc. when there is the STL in C++? Another ...
2
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1answer
74 views

Parallel, matrix-free estimate of the trace

What would be the best way to estimate the trace of a large, distributed matrix, if one only know its action on a vector throug a parallel "matvec" routine? In the application I am interested in, the ...
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0answers
40 views

Triangle on top of diagonal least squares

I need to solve many least squares problems with the following matrices: $$ \pmatrix{ R \\ D_i } $$ where $R$ is upper triangular and $D_i$ is diagonal. $R$ is the same for all the problems, while $...
2
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1answer
148 views

Boundary conditions in a finite element eigenvalue problem

I've been reading multiple papers and related posts for a while now, but I can't seem to find a specific answer to the issues I'm having so I hope someone can clarify things here. I'll provide some ...
3
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1answer
149 views

Implementation of a $O(n \log(n))$ method to compute eigenvalues of real symmetric tridiagonal matrices

I just came upon this paper, which details the implementation of a fast method to get eigenvalues of tridiagonal symmetric matrices : Coakley, Ed S.; Rokhlin, Vladimir, A fast divide-and-conquer ...
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1answer
72 views

Solving triangular matrix equations on a GPU

Suppose I have these two $N\times N$ lower triangular banded matrices: $A = \begin{bmatrix} a_0 & & \\ a_1 & a_0 & \\ a_2 & a_1 & a_0 \\ a_3 & a_2 & a_1 & a_0 \\ &...
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1answer
32 views

Norm of operator in finite element discretization of Heat equation

I am solving the heat equation discretized spatially via FEM and temporally via backward Euler. I get the system $$M \dot{u} = K u +f$$ where $u$ is a vector representing the solution at spatial ...
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0answers
33 views

Choosing the pivot for the rotation matrix in similarity transformation

I have arrived at an equation in the similarity transformation - $M_r$ = $T. M_{r-1}. T^t$ ,where $T$ is the rotation matrix and $M_r$ ,$M_{r-1}$ are similar matrices. My aim is to find the rotation ...
4
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0answers
106 views

Fastest matrix library for Android (with GPU is possible)

I was working on an Android app that requires some linear algebra with matrices. The matrices will be somewhat medium-sized as they are not too small or too big. I was originally using jBlas because ...
2
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0answers
38 views

Is there a published RQ decomposition column-major algorthm?

I am refactoring an existing algorithm where where a RQ decomposition (as opposed to the more common QR) would be rather useful. Most common books on the subject (e.g. Golub and Van Loan) discuss QR ...

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