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### How to choose a method for solving linear equations

To my knowledge, there are 4 ways to solving a system of linear equations (correct me if there are more): If the system matrix is a full-rank square matrix, you can use Cramer’s Rule; Compute the ...
13k views

### Why is my iterative linear solver not converging?

What can go wrong when using preconditoned Krylov methods from KSP (PETSc's linear solver package) to solve a sparse linear system such as those obtained by discretizing and linearizing partial ...
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### What guidelines should I use when searching for good preconditioning methods for a specific problem?

For the solution of large linear systems $Ax=b$ using iterative methods, it is often of interest to introduce preconditioning, e.g. solve instead $M^{-1}(Ax=b)$, where $M$ is here used for left-...
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### Best choice of solver for a large sparse symmetric (but not positive definite) system

I am presently working on solving very large symmetric (but not positive definite) systems, generated by some certain algorithms. These matrices have a nice block sparsity which can be used for ...
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### Krylov Subspace Methods for Dense Systems

I am currently researching on the viability of using KS methods for solving large dense systems. What I wish to prove (or disprove) is that methods like CG, BiCG and QMR are as good (if not better) ...
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### What is a robust, iterative solver for large 3-d linear-elastic problems?

I'm diving into the fascinating world of finite element analysis and would like to solve a large thermo-mechanical problem (only thermal $\rightarrow$ mechanical, no feedback). For the mechanical ...
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### Iterative methods for indefinite systems without block structure

Indefinite systems of matrices appear for example in the discretization of saddle point problems by mixed finite elements. The system matrix can then be put in the form \begin{pmatrix} A & B^t \...
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### What guidelines should I use when choosing a scalable linear solver?

There are many different linear solvers, some which work best for diagonally dominant matrices, some for symmetric, some for positive definite ones, some for banded matrices, etc... There are direct ...
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### Intersection of hyperplanes

A very basic question but i couldn't find another post about it: Given $p$ non parallel hyper-plane in $\mathbb{R}^p$: \$\left(\begin{array}{cccc} c_{11} & a_{11} & .... & a_{1p} \\ ... &...
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After reading the first answer here about how the best way to find the most performant sparse solver is to try almost everything, I began to wonder if there was any past work on libraries or research ...
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### 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 ...
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### What method to solve a sparse complex symmetric (non-Hermitian) system?

I have a sparse system (about 78% of zero entries) that is complex and symmetric (but not Hermitian). The following figure shows the structure of the problem. The off-diagonal blocks are incidence ...