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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3
votes
1answer
140 views

Is there any way/any python function to calculate the condition number of the roots of a polynomial directly?

I know that NumPy has linalg.cond(A) to find the condition number of a matrix A. But, if I want to find the condition numbers of the roots of a large polynomial ...
1
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1answer
72 views

Can someone explain the equivalence between Oja's rule and PCA in a simple way?

I have to give a presentation on unsupervised learning in 2 days, and I have to explain/show the equivalence between Hebb's learning rule (or Oja's rule to be more specific) and PCA. The thing is that ...
2
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0answers
90 views

Regularisation of ill-conditioned matrix-vector problem

I have a linear* problem which arises from an integro-differential system, and writes: $$ (\mathbf{I}+\lambda \mathbf{A})x = b $$ where $\mathbf{A}$ is a real full matrix, size $n\times n$, but is not ...
4
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0answers
102 views

PETSc-like library for Julia

I want to build an application for Material Point Method (and probably other meshfree methods too) in Julia and I am looking for library for direct and iterative solvers that can help me with it. One ...
5
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2answers
469 views

Computational method to compute both the (log) determinant and inverse of a matrix

Suppose I have a square matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$ and a vector $\mathbf{b}\in\mathbb{R}^n$. In my application I need to accomplish two things. I need to find the solution of the ...
4
votes
1answer
102 views

Solving for a single element of a solution of a linear system

I wish to solve a linear system $A x =b$ in which $A$ is dense but not too large, say no larger than $10\times10$. However, I am not interested in the full solution vector $x = [x_0, x_1, \dots]$, ...
2
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0answers
61 views

Multigrid method: linear solver and modified residual

I am trying to better understand the FAS multigrid algorithm for Euler equation in FV discretization. The usage of the modified residual (the residual with forcing) inside the different cases: ...
3
votes
1answer
81 views

Find $x$ that satisfy $(I-A^*A)+x(\frac{A+A^*}{2})\prec0$ using LMI or SDP on Matlab

Given $A\in\mathbb{C}^{n\times n}$, I want to use LMI or SDP to find feasibility of $x>0$ in the following inequality: $$(I-A^*A)+x(\frac{A+A^*}{2})\prec0,$$ where $D\prec0$ means that $D$ is ...
3
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0answers
86 views

Stable iterative solver for complex symmetric linear systems

I am interested in the iterative solution (preferably Krylov-type solvers) of a problem Ax=b, with x, b ∈ ℂn x 1 and A ∈ ℂn x n. A is symmetric, invertible, and its real and imaginary parts are ...
0
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2answers
119 views

Implementation of $[X, \cdot]$ as an $n^2 \times n^2$ matrix, where $X$ is an $n \times n$ matrix

Let $M_n(\mathbb{R})$ denote the set of $n\times n$ matrices with real entries. I have an $n\times n$ matrix $X\in M_n(\mathbb{R})$, and I would like to implement the linear operator $[X, \cdot] : M_n(...
5
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2answers
105 views

Optimizing a quadratic form integral over unit sphere

I have an optimization problem, which is to maximize the following integral over the unit sphere: $$ \max_B \int d\Omega \mathbf{f}^{\dagger}(\theta,\phi) (B^{\dagger} + B) \mathbf{f}(\theta,\phi) $$ ...
0
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0answers
60 views

Comparing minimas of two different functions

The goal is to find vectors $x_u$ and $y_i$, both of the same length $f=64$, and to do this the following loss function is minimized: $$\sum_{u, i} (1 + \alpha \cdot r_{ui})(p_{ui} - x_{u}^{T}y_i)^2$$ ...
6
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1answer
192 views

Algorithm for solving systems which are nearly symmetric/adjoint?

I am familiar with Cholesky decomposition and LU factorization for solving systems of linear equations. I have a problem where I have large sparse matrices (say, 1000x1000 or larger) where only one or ...
18
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4answers
2k views

Why do we usually not want the eigenvalues of non-symmetric matrices?

I came across this line in a class note I am reading where it discusses finding eigenvalues of matrices. In reality we don't go all the way with Arnoldi. We stop at a decent value of 𝑘. Then the 𝑘 ...
5
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1answer
133 views

Solving absolute value systems

Let $z, b \in \mathbb R^n$, $A \in M_n (\mathbb R)$ and $|z| := (|z_1|, \dots, |z_n|)$. I am searching for an efficient algorithm to solve the absolute value system: \begin{equation} z - A |z| = b. \...
1
vote
1answer
95 views

Efficiency of scipy.sparse.linalg.expm_multiply with sparse vs unsparse vectors

From the package scipy.sparse.linalg in Python, calling expm_multiply(X, v) allows you to compute the vector ...
1
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0answers
85 views

Matlab - Equality between 2 Fisher matrices constructed in a different way

I want to know if, on a Fisher matrix, the projection operation (with a Jacobian matrix) commutes with a matricial inversion operation. The 2 ways to build these 2 matrices are: 1) First method: 1.1) ...
0
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0answers
51 views

PLASMA usage (Linear algebra routines that supports multithreading)

I have been looking for linear algebra libraries that support multithreading. I have found PLASMA which looks promising. It is from the same group that developed LAPACK. http://icl.cs.utk.edu/...
0
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0answers
136 views

Do the projection (with Jacobian) and marginalisation (inversion of matrix and remove a row/column and reinversion) commute?

I try to check the equality or the inequality between 2 Fisher matrices. The goal is too see if the projection (with Jacobian) and marginalisation (inversion of matrix and remove a row/column and ...
3
votes
1answer
132 views

How can I extract the banded or block diagonal part of a sparse matrix in MATLAB?

Given a large sparse (square) matrix in MATLAB, how can I extract the banded or the block-diagonal parts (of fixed size) of it efficiently? These are useful operations when prototyping and testing ...
0
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1answer
49 views

Efficient solution to linear system involving Kronecker sum in MATLAB

High dimensional finite difference problems often lead to linear systems of the form $$ A x = b, \qquad A = B_1 \oplus B_2 \oplus \cdots \oplus B_d, $$ where $\oplus$ denotes the Kronecker sum. $B_i \...
4
votes
0answers
81 views

Unstable Algorithms which become stable when hardware provides Kulisch exact dot product instruction

In John Gustaffson's book The End of Error, he discusses Ulrich Kulisch's exact dot product, which (in double precision) requires a 2100 bit fixed point register which rounds only once after the ...
1
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0answers
54 views

Number of words that a processor can handle for PMMM

The PMMM which stands for parallel matrix-matrix multiplication essentially accelerates the algorithm of the matrix-matrix multiplication of two matrices $A$ and $B$ both of size $n$ so that $C:= AB$. ...
3
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0answers
97 views

Jacobian Matrix of 2D element mapped to 3D

Note: I previously posted this question to MathStackExchange, but got no attention there. So I'm rewritting and trying over here. Problem summary Given a common¹ set of shape functions defined at ...
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0answers
75 views

Solution of an underdetermined system stemming from a PDE with Neumann BC

Consider the Poisson's equation in 1D with homogeneous b.c.'s $\mathrm{d} \phi/\mathrm{d} x=0$ with the seven point Laplacian (1 -54 783 -1460 783 -54 1 / 576 on a uniform grid). The resulting system ...
2
votes
4answers
183 views

Eigenvalue decomposition for a very huge matrix of medical images (such as the pixel physical coordinates of CT images)

I am trying to do eigenvalue decomposition for a huge matrix larger than 788000×788000 for medical image analysis. The matrix is not sparse and every element in the matrix has a real value. And, for ...
4
votes
1answer
115 views

Solving geodesics on triangular meshes gives negative distances

I have implemented the heat method for geodesics: https://www.cs.cmu.edu/~kmcrane/Projects/HeatMethod/paperCACM.pdf When I run it I am getting a solution that, visually, seems correct: In this image, ...
3
votes
0answers
66 views

Check if LinearOperator is symmetric

I have a scipy.sparse.linalg.LinearOperator object. I'd like to check if its associated matrix is symmetric without actually instantiating the matrix in the most ...
3
votes
1answer
108 views

Lanczos algorithm for finding top eigenvalues of a matrix sum

I am trying to find the top k leading eigenvalues of a NumPy matrix (using python dot product notation) L@L + a*X@X.T, where $L$ ...
0
votes
0answers
180 views

Numerical Range of a matrix in Python

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex $n\times n$ matrix $A$ is the set $$W(A)=\left\{{\frac {{\mathbf {x}}^{*}A{\...
2
votes
3answers
161 views

Linear solver recommendation(s) for small problems

I am interested in solving many linear systems $Ax = b$, where $A$ is symmetric positive definite and small (i.e. less than 25,000 rows) --- $b$ will be changing. We can assume that $A$ arises from ...
2
votes
0answers
60 views

Software for solving large systems of linear equations over gf(2)

What available solvers are there for linear equation solver over GF(2) (Boolean), capable of dealing with large sparse systems (in the 10k - 100k variables range)?
-1
votes
1answer
34 views

Interpretation of error between Hessian approximation and real Hessian - Quasi-Newton Method

$$ ||I- H_{k}^{BFGS}\nabla^{2}f(x_{k})||_{2}$$ , where $H_{k}$ is the inverse of hessian approximation at each iteration. I am given this expression to assess the error in Hessian approximation in ...
-2
votes
1answer
77 views

Forward Euler Adaptive Step Size Stability

Given with a generalization using adaptive times-stepping as then is it still reasonable to assume that to ensure stability of the Euler’s forward method we need the growth factor for all n to be ...
8
votes
3answers
607 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 ...
2
votes
0answers
43 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
vote
2answers
180 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
votes
2answers
134 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
73 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 ...
1
vote
1answer
64 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
votes
1answer
135 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
votes
1answer
137 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+\...
1
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0answers
77 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
votes
1answer
264 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
votes
1answer
144 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, ...
1
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0answers
65 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
votes
1answer
58 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
vote
1answer
122 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
votes
1answer
321 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 ...
6
votes
1answer
201 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 ...

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