Referring to methods for solving linear systems of equations.

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Open Source Linear Algebra Library

I am making a code in C that requires the equivalent of Matlab's '\' command for a linear system of the form AX=B where A is an NxN matrix and X, B are Nx1 vectors- i.e a code that performs X=A\B that ...
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Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$. We want to solve it with iterative method (assuming $A$ is good). First order iterative method is: $$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f), $$ The second degree ...
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Block-tridiagonal symmetric parallel solver (software recommendation)

What are the fastest numerical methods / (Fortran) packages to solve in parallel (MPI / OMP) a linear symmetric block - tridiagonal system (double complex) where the RHS is a block vector (complex)? ...
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62 views

Does algebraic multigrid reuse its coarse grids?

Admittedly, I'm new to the subject, so this is probably a really simple question. Let's assume I want to solve the (large sparse) linear system Ax = b multiple times with algebraic multigrid and b ...
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78 views

Parallel linear algebra without OpenMP

I have searched through the archives without success. Apparently, the question is simple: What linear algebra library can I use that is parallel (shared memory) but without OpenMP? As far as I've ...
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40 views

NONLINEAR ENERGY MINIMIZATION EXAMPLE

I am learning about FEM methods and nonlinear optimization. I would like to try my nonlinear trust region solver on some simple nonlinear problem. What would be good example to implement for ...
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153 views

Once and for all: Which FEM plattform should I use for a very large multiphysics simulation?

I'm fooling around with the decision on how to build a multiphysics simulation for too long now (also several questions in this forum). First, I thought it would be possible/necessary to write most of ...
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51 views

Pivoting in Crout algorithm

Consider the following code, found on wikipedia, that implements the Crout decomposition algorithm: ...
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83 views

Numerical method for solving a system with positive definite blocks

I have a system with below coefficient matrix $$ C = \begin{pmatrix} A & B^T \\ B & D \end{pmatrix},$$ where, $A$ and $D$ are square and positive definite. Furthermore, if $B$ be square, ...
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57 views

Solving linear system $Ax=b$ with Hessenberg matrix using lapack

I need to solve a linear system of the form $$Ax = b$$ where $A$ is upper Hessenberg matrix with the lower bandwidth equal to 1, $b$ is the RHS vector and $x$ is the solution vector. I have a C++ ...
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109 views

Robust smoothers for geometric multigrid

I'm searching for robust smoothers for geometric multigrids. By robust I mean: Effective for high order approximations (say spectral element, spectral Discontinuous Galerkin), Parallel (suitable ...
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“Cookbook” about iterative linear solvers and preconditioners

I'm using a lot of linear solvers and preconditioners, but mostly, they are magical black boxes to me. Since I'll also have to implement some of them in future, I would like to learn a bit more, ...
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Single Precision a x plus y (SAXPY) terminology

I've been reading books which refers to vector update operations of the form: y := y + ax, where y and x are vector variables and a is a scalar as SAXPY. I understand ax plus y part, but why "single ...
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One parameter family of linear systems

Suppose $A\in\mathbb{R}^{n\times n}$ is dense and $D\in\mathbb{R}^{n\times n}$ is a diagonal and nonnegative. Given a fixed vector $b\in\mathbb{R}^n$, define the following function $f:\mathbb{R}\...
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175 views

Solving a set of linear equations with block structure and weak coupling

I have a standard set of linear equations $Ax=b$ where the Hessian matrix $A$ has the special block structure as shown: $A= \begin{pmatrix} T & U\\ U^T & V \end{pmatrix}$, $x= \begin{...
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solving tridiagonal system with multiple right hand sides

I need to solve a tridiagonal system (positive definite, diagonally dominant) $Ax = b$ in a time stepping loop. $A \in \mathbb{R}^{N \times N}$ remains constant but $b$ changes during each time ...
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91 views

Linear Systems with Multiple Right Hand sides

I am interested in solving a sequence of linear systems of the form: $$A x_i = b_i$$ That is, all the systems use the same matrix $A$ but they have different right hand sides. The matrix $A$ is sparse ...
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64 views

Methods for solving rectangular, full-rank systems of equations — which is best?

Suppose I have a large, sparse, $m \times n$ matrix $A$, with $m \gt n$ and $\text{rank}(A) = n$. I wish to solve $Ax=b$. Suppose I know that $A$ has the following characteristics: $A$ is somewhat ...
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69 views

role of initial guess for iterative linear solver

Suppose we use a preconditioned iterative solver for a linear system. If the initial state for the solver can be chosen very close to the exact solution - does this reduce requirements for the ...
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118 views

Solving linear systems with ill-conditioned matrices

As per suggestions of the people from MathOverflow, I'm reposting my question here: I'm currently trying to solve a linear system $Ax = B$, where the matrix $A$ is ill conditioned (i.e. nearly ...
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105 views

How to assemble Global matrix (for coupled) problem?

I'm trying to assemble global matrices for the following system. $$ \begin{bmatrix} K& Q\\ Q^T&S \end{bmatrix} \begin{bmatrix} u_h\\ p_h \end{bmatrix} = \begin{bmatrix} f_u\\ f_p \end{...
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Solve for $D$ in $R^{T}DSDR = Id$

Given that $R$ is a rectangular matrix, $D$ is a diagonal, square matrix and $S$ being a square matrix along with the fact that both $D$ and $S$ are invertible. $S$ in this specific case can be ...
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87 views

Resources for solving mixed left and right matrix equations

I'm looking to solve a matrix equation and not sure where to start looking for resources. The equation is $$AX + XB = C\,,$$ where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, $C\in\...
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Mathematical Complexity of Sparse Solvers

For a system $\mathbf{x=Da}$, there exist a lot of algorithms to estimate sparse vector $\mathbf{a}$. I wish to know the big-O mathematical complexity of 1) orthogonal matching pursuit (OMP) both ...
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Solving system of equations with zeros on diagonal [closed]

I am using finite elements, Newton Raphson technique and MUMPS direct solver, to solve for $P$ in this equation: $$\frac{\delta K}{\delta x} \frac{\delta P}{\delta x}= \frac{\delta K B}{\delta x} + A$...
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Implementation of a direct solver in Fortran 90?

My question may be elementary, but it is quite essential as I am getting confused. Here I am supposed to solve the following equation: $Ax=B$ From my understanding I have options of using either ...
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158 views

Skyline solver for AX=B where A is symmetric skyline matrix

I am looking for a simple subroutine in Fortran 90 (GNU Compiler) to solve linear equation of the type $AX=B$, where $A$ is an $n\times n$ symmetric matrix stored in the form of symmetric skyline ...
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255 views

What category is this problem?

My first question, please excuse me if its too basic. I have a matrix of evenly spaced geographical points; say 10 x 10, which I will call seed points. Each seed ...
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91 views

Finding the matrix inverse given a solver for the matrix equation $Ax=b$

So I'm given a solver that can solve for $x$ in the matrix equation $\underset{=}{A} \underline{x} = \underline{b}$ where $b$ can be anything we specify. (NB: A is an NxN matrix). I now want to find ...
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Is the “practical” complexity of linsolve direct solver O(n^2) ?

I recently timed the linsolve direct solver and I was kind of shocked to see that the solver seemed to be scaling quadratically even upto a 1000 dimensions. Specifically I ran the following code and ...
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120 views

How efficient (compared to “normal” methods) is using a sparse finite difference matrix to solve differential equations?

Any differential operator can be written as a finite difference matrix acting on a vector of values. I recently used it to solve $\Delta A = j$ by writing the Laplacian as a finite difference ...
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Which preconditioning for large linear elasticity problem?

The problem I want to solve is the displacement formulation of the linear elasticity : $$ \nabla \cdot \sigma = 0 \quad \text{in} \quad \Omega \\ \sigma = \lambda ( \nabla \cdot u ) I + \mu (\nabla \...
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252 views

Solving a system of linear equations with only an approximate solution

I have a system of linear equations that is derived partially from experimental data. Theoretically, the system should have a single, exact solution; however, experimental error causes it to not have ...
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Solving new linear system that comes from an $p$ enrichment

Let's say I am solving a simple Poisson problem using a Mixed (DG) finite element method. If we use orthogonal polynomials as basis functions we can write the finite-dimensional linear system as $$ ...
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Does Conjugate Residual really have convergence properties similar to that of Conjugate Gradient?

I have coded up a toy implementation of Conjugate Residual and have been testing it. Both wikipedia and the Saad claim that Conjugate Residual and Conjugate Gradient have similar convergence behavior....
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FLOPS of a linear system

I have two questions that I want to ask. Consider the following system: $$BM = A$$ i) $B$ is a $n$ by $n$ tridiagonal matrix and $A$ is a diagonal matrix. What is the leading order computation cost ...
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115 views

Does this partial eigen-expansion have a name?

This question is a follow-up to this one. Let $A\in \mathbb{R}^{n\times n}$ be large, sparse, symmetric and positive definite. Suppose for I already know $m<n$ eigenpairs of $A$, corresponding to ...
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39 views

Suggestions for an out-of-core sparse solver

I have a sparse $2\times10^5$ by $2\times10^5$ matrix with $3.2\times10^9$ non-zero elements. I want a sparse solver with out-of-core functionality. I have attempted to use Intel's ...
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85 views

Solve FEM matrix from coupled system

I'm developing an FEM solver for a coupled system. I have diffusion and potential equations which result in positive definite matrices for each equation, but the coupling makes the overall system ...
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78 views

What are some ideas to preprocess / precondition the following linear system?

Let $A\in \mathbb{R}^{n\times n}$ symmetric and positive semidefinite, and $\omega\in \mathbb{R}\setminus\{0\}$. I am interested in solving the following linear system for a range of values of $\...
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What strategies / decompositions would be useful to solve the following linear system repeatedly if I only care about time to solution?

Let $A\in \mathbb{R}^{n\times n}$ be symmetric positive semidefinite, and $B\in \mathbb{R}^{n\times n}$ be symmetric positive definite. Suppose $B$ is block diagonal so it is easy to invert. (We ...
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119 views

What factors are relevant when deciding between GMRES and schur complement solves?

Suppose I have a linear system $$ \left\lbrack \begin{array}{cc} M_1& S\\ S^{\mathrm{T}}& M_2 \end{array} \right\rbrack \left\lbrack \begin{array}{c} X\\ Y\end{array} \right\rbrack= \...
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199 views

What is a good way to solve the following linear system? (repeatedly)

Let $n,m\in \mathbb{N}$ be such that $m\ge n$. Let $M_1\in \mathbb{R}^{n\times n}$, $\{M_{2},M_{3} \} \subset \mathbb{R}^{m\times m}$ be symmetric positive definite and computationally cheap to ...
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Large scale triangular least squares

I have to solve the following least squares problem: \begin{equation} \| \left[ \begin{smallmatrix} \mathbf{L} \\ \mathbf{I} \end{smallmatrix} \right]\mathbf{x} - \mathbf{b} \|_2^2 \end{equation} ...
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106 views

Best solver/preconditioner for least-squares finite element method

I have seen a lot of literature, lecture videos, etc. on solvers/preconditioners for non-symmetric and/or indefinite systems. However, now I want to solve the mixed poisson/Darcy equation using the ...
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146 views

Appropriate iterative linear solver for an eigenvalue problem

I'm trying to solve a generalized eigenvalue problem $$Ax = \lambda Bx, \quad A = A^\top > 0,\; B = B^\top > 0$$ with $\lambda \approx \sigma$ using Rayleigh Quotient Iteration (RQI) (RQI is ...
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106 views

Under what circumstances can two (nearly) identical sparse matrices give different solutions to Mx = b?

Suppose I have two sparse matrices, $A$ and $B$, of size $N \times N$. They each have the same sparcity pattern ("footprint"). They each also have values which in theory should be identical, but aren'...
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107 views

Choosing preconditioner for unsymmetric pressure-velocity coupled system

I'm working with pressure-velocity coupled systems. It means that instead of solving 4 different linear systems in segregated approach (1 for pressure and 3 for Ux, Uy, Uz), we can solve only one ...
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102 views

solve linear system of equation of a large sparse symetric positive definite matrix

I want to invert large matrices ($10^4 \times 10^4$ to $10^6 \times 10^6$) but sparse (less than $100$ non-zero entries per line) on clusters with $16$ to $48$ processors per node. I'm looking for an ...
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Evaluating a quadratic form with an inverse of a sparse PD matrix, comparison between using the inverse vs using a Cholseky decomposition

I have the following quadratic form I need to evaluate: $x^T A^{-1} y$, where $A$ is a sparse positive definite matrix, $x, y$ are sparse vectors. Now assume that I am given for free both $A^{-1}$ ...