# 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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### 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 ...
84 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 ...
465 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 ...
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 $x = [x_0, x_1, \dots]$, ...
58 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: ...
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### 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 ...
84 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 ...
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### 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 ...
53 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 $C:= AB$. ...
85 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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### 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 ...
163 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 ...
112 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, ...
64 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 ...
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### 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 $L$ ...
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### 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 ...
257 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 ...
131 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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### 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 ...
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### 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 ...
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### 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 ...
283 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 ...
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### 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 ...
Say I have a directed graph given as an adjacency matrix $A$ in CSR format represented by the arrays ia (row indexes) and ja (...