Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

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7
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1answer
80 views

Bad scaling versus collinearity

I was trying to solve a linear system: $$ \mathbf{A}\mathbf{x} = \mathbf{y} $$ but the conditioning number was quite bad (around $10^{17}$). I thought that the system was singular, but after scaling ...
10
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1answer
15k views

in matlab, what differences are between linsolve and mldivide?

in matlab, both linsolve and mldivide are used for solving a system of linear equations, in all of determined, overdetermined and underdetermined cases. Reading their documents, I was wondering what ...
10
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5answers
9k views

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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0answers
251 views

Memory allocation error with GSL ODE solver applied to system of 4 ODEs

I am trying to solve a (large) system of ODEs with GSL solvers. When I use driver method I get an error message of could not allocate space for gsl_interp_accel, ...
5
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3answers
275 views

Take advantage of the sparsity of b in AX=b

There is a lot of info about how to use the sparsity pattern of A in order to solve $Ax = b$. However I can't find much about using the sparsity pattern of b. Let me take a concrete example: Let us ...
4
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1answer
289 views

Sparse LU for block-sparse matrices

I frequently need to solve linear systems with sparse matrices of moderate dimension (say a few thousand). These matrices are composed entirely of small dense blocks (typically 5-10 in dimension), and ...
12
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2answers
6k views

solve $xA=b$ for $x$ using LAPACK and BLAS

I am porting an existing code from MATLAB to C++ and have a linear system to solve $xA=b$ (rather than the more typical form $Ax=b$) The matrix $A$ is dense, and of general form, but is no larger ...
12
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5answers
2k views

Repeatedly solving $\mathbf{A} \mathbf{x} = \mathbf{b}$ with same $\mathbf{A}$, different $\mathbf{b}$

I am using MATLAB to solve a problem that involves solving $\mathbf{A} \mathbf{x}=\mathbf{b}$ at every timestep, where $\mathbf{b}$ changes with time. Right now, I am accomplishing this using MATLAB'...
9
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1answer
1k views

preconditioner for a matrix-free method to solve Ax=b

I need to solve Ax=b, but I realize that even if it is sparse, storing the matrix coefficients of my problem will take too much memory. So now I'm considering using a matrix-free method, because the ...
12
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3answers
2k views

Sparse linear solver for many right-hand sides

I need to solve the same sparse linear system (300x300 to 1000x1000) with many right hand sides (300 to 1000). In addition to this first problem, I would also like to solve different systems, but with ...
4
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3answers
171 views

Numeric solution of simple but possibly singular linear system

I have a simple (and small) linear homogeneous system $Ax=0$, where the entries of the $N\times M$ matrix $A$ are small integers. I do not need fancy methods which efficiently solve almost singular ...
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2answers
343 views

Checking for error in conjugate gradient algorithm

What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
8
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1answer
568 views

Using algebraic multigrid for preconditioning convection-diffusion operators

I implemented a Navier Stokes based on FEM discretization and PETSc for solving the linear system of equations. To create an efficient solution procedure, I follow the paper "Efficient preconditioning ...
3
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1answer
198 views

How do you formulate the linear least-squares method for radiometric calibration?

In Debevec and Malik (mentioned similarly in Forsyth and Ponce's Computer Vision: A Modern Approach) they highlight a method of solving the camera response function using linear least-squares. We ...
14
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3answers
5k views

What are the symptoms of ill-conditioning when using direct methods?

Suppose we have a linear system and we know nothing about its conditioning and have no preliminary information about the solution. We blindly apply Gaussian elimination and obtain some solution $x$. ...
9
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3answers
1k views

Recommendations for a lightweight/no-install C or C++ based dense linear algebra solver

Most of my programming is one-off research codes in C for my own use. I have never distributed any code to other than close collaborators. I have developed an algorithm that I am publishing in a ...
12
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2answers
331 views

Efficient preconditioner for Augmented Lagrangian

I want to solve a non-linear problem with non-linear equality constrains and I'm using a augmented Lagrangian with a penalty regularization term that, as well known, spoils the condition number of my ...
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3answers
181 views

Equivalence of linear systems, solving one instead of the other

This question is related to recently posted one, but I guess it deserves a separate attention. Suppose a symmetric matrix $L\in\mathbb{R}^{n\times n}$ is given, and a rectangular matrix $A\in\mathbb{...
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3answers
958 views

Convergence of the gradient descent and linear vs non-linear fixed point iteration

Suppose a system $$Ax=b$$ is given, with $A\in\mathbb{R}^{n\times n}$ being a symmetric positive-definite matrix, and some non-zero $b\in\mathbb{R}^n$. The gradient method with optimum step length can ...
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3answers
14k views

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 ...
3
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2answers
604 views

Solving Poisson equation with free boundaries and adaptively refined mesh

Assume we want to solve the Poisson equation $$ \Delta u = f $$ with free (Neumann) boundary conditions. So, the right hand side function $f$ must fulfill the compatibility condition to integrate to ...
2
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3answers
405 views

derivative of linsolve

Consider a vector $\mathbf{g} \in \mathbb{R}^{m}$ and a matrix $\mathbf{A} \equiv \mathbf{A(g)} \in \mathcal{M}_{p\times q} [\mathbb{R}]$, a function of $\mathbf{g}$. Furthermore, let $\mathbf{S} \...
6
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2answers
4k views

When to stop Gauss-Seidel-iterations?

I want to have an estimation, that my solution has an error, let's say less than 1e-8. Usually, I stop the Gauss-Seidel algorithm, when the residual is "small enough" and this is already the problem. ...
11
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1answer
688 views

Solving huge dense linear system?

Is there any hope in solving the following linear system efficiently with an iterative method? $A \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^n, b \in \mathbb{R}^n \text{, with } n > 10^6$ $Ax=...
6
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3answers
272 views

efficiently solving a low rank linear parametric systems?

I have a large number of systems of the form: $Ax=b_i$ To solve for a large numbers of such $b_i\;1\leq i \leq k$ but where $A$ is fixed (A is a rank $p$ general --i.e. non sparse, non PSD-- ...
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3answers
4k views

Single versus double floating-point precision

Single precision floating point numbers take up half the memory and on modern machines (even on GPUs it seems) operations can be done with them at almost twice the speed compared to double precision. ...
7
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3answers
245 views

Wanting to learn about matrix solvers

Edit: I was advised to replace the question with a more specific one. Coming from a very theoretical background, I'm pretty ignorant about what practical matrix solvers exist. (I have been, and will ...
7
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3answers
701 views

Solving shifted linear systems with LU factorization

I am interested in solving a sequence of shifted linear systems $(A+\sigma I)x = b$ for various values of $\sigma$. The matrix $A$ is sparse and not too large, so I have its LU factorization available....
8
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2answers
1k views

Taxonomy of ILU preconditioners

I learned that for BiCGStab solver for sparse linear systems it's pretty much always necessary to use a preconditioner. I realized by now that choosing a good one is problem dependent. Surfing the ...
22
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3answers
797 views

Solving $(G^TA^{-1}G)x = b$ without inverting $A$

I have matrices $A$ and $G$. $A$ is sparse and is $n\times n$ with $n$ very large (can be on the order of several million.) $G$ is an $n\times m$ tall matrix with $m$ rather small ($1 \lt m \lt 1000$) ...
2
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0answers
61 views

Complexity of direct solvers? [duplicate]

Possible Duplicate: How to reorder variables to produce a banded matrix of minimum bandwidth? What is the time and space complexity of direct sparse solvers (e.g., UMFPACK, SUPERLU, PARDISO, etc.)...
2
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0answers
217 views

Is it possible to run a Solver Foundation solver against a model containing linear and non-linear elements?

This is a follow up question to one I made previously about non-linear equations and ranged real numbers in Solver Foundation. I acknowledge that where possible, rewriting a problem that is non-...
4
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2answers
189 views

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 ...
11
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1answer
714 views

Projecting out the null-space of $A$ from $b$ in $Ax=b$

Given the system $$Ax=b,$$ where $A\in\mathbb{R}^{n\times n}$, I read that, in case Jacobi iteration is used as a solver, the method will not converge if $b$ has a non-zero component in the null-space ...
6
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3answers
4k views

How to find QR decomposition of a rectangular matrix in overdetermined linear system solution?

While trying to find cell-centered gradients in finite volume method computation of incompressible fluid flow I get over-determined linear system. This is a well known "cell based least-square" ...
11
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4answers
5k views

What are the best Python packages/interfaces to sparse direct solvers?

Please list the Python package (petsc4py, etc...) and the sparse direct solvers it supports. One (community-wiki) answer per package, please.
6
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1answer
1k views

How does a Sparse Direct Solver know about dimensionality of a problem being solved?

It is claimed that the time and memory complexities of sparse direct solver are $O(N^2)$ and $O(N^{4/3})$ for 3D problems and $O(N^{1.5})$ and $O(N \log N)$ for 2D, respectively. But how does a ...
1
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3answers
245 views

Unique coordinates (solutions) in a single Gauss-Seidel iteration

I managed to reduce certain computational problem to the Gauss-Seidel solution of the following linear system: $$Ax=Ly,$$ where $A, L\in\mathbb{R}^{n\times n}$ are weighted Laplacian matrices (...
5
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2answers
2k views

How does matrix scaling influence linear solvers?

For instance, in MUMPS there is option to scale matrix s.t. all rows/columns have the same norm. This claims to decrease condition number and improve numerical properties of the matrix: ftp://cuter.rl....
3
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2answers
424 views

What numerical methods are recommendable for simulating two phase immiscible fluid flow through a pipe with high capillary pressure?

I'm simulating two phase immiscible drainage (air displacing water) in a rectangular domain of size .6mm x 2.4mm (2 dimensions) using Ansys FLUENT software. I am using an implicit Volume of Fluid ...
3
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1answer
70 views

2D Jacobi line maintenance?

Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
8
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2answers
129 views

Initial guesses for perturbed linear systems

Suppose you solve a linear system $Au = f$ by an iterative method, e.g. conjugate gradients or Richardson iteration. Then you try to solve a linear system that is slightly perturbed in the matrix and ...
9
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2answers
1k views

Safe application of iterative methods on diagonally dominant matrices

Suppose the following linear system is given $$Lx=c,\tag1$$ where $L$ is the weighted Laplacian known to be positive $semi-$definite with a one dimensional null space spanned by $1_n=(1,\dots,1)\in\...
10
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2answers
1k views

Which iterative linear solvers converge for positive semidefinite matrices?

I want to know which of the classic linear solvers (e.g Gauss-Seidel, Jacobi, SOR) are guaranteed to converge for the problem $Ax=b$ where $A$ is positive semi definite and of course $b \in im(A)$ (...
5
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2answers
530 views

Recommendation for a good article/book for frontal methods?

Can someone provide an article or book that explains the principle used in frontal solvers? Some examples also may help understand the frontal methods better.Thanks in advance!
10
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2answers
1k views

Selection of linear solver for GPGPU computation (OpenCL)

I have already developed a working solution of the Finite Element Method to solve heat transfer problems using GPU and OpenCL using the Conjugate Gradient method. The main disadvantage of this method ...
15
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1answer
715 views

Are there any open source inverse-based multilevel ILU implementations?

I am very impressed with the serial performance of multilevel inverse-based ILU preconditioners, particularly for heterogeneous Helmholtz, but I am surprised to not be able to find any open source ...
3
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1answer
285 views

How to solve a problem with structure similar to a finite difference discretization of the 2D Poisson equation, but with non-symetric coefficients?

Recently, I've been asking about methods to solve a finite difference discretization of the 2D Poisson equation (see here and here) of the form: $$U_{i-1,j} + U_{i+1,j} -4U_{i,j} + U_{i,j-1} + U_{i,...
21
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2answers
9k views

Libraries for solving sparse linear systems

There are a number of different libraries out there that solve a sparse linear system of equations, however I'm finding it difficult to figure out what the differences are. As far as I can tell there ...
51
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4answers
7k views

What guidelines should I follow when choosing a sparse linear system solver?

Sparse linear systems turn up with increasing frequency in applications. One has a lot of routines to choose from for solving these systems. At the highest level, there is a watershed between direct (...

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