# Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

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685 views

### Lost on Matrix Inversion

I try to implement some big matrix inversion. My system configuration is Hardware:- Memory: 62.8GiB, Processor: Intel Xeon(R)CPU E5-2670 v3 @2.30GHZ*48 To implement matrix inversion I am using ...
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### Use of GPU with respect to CPU

I have research work where I need to compute a matrix inversion. The matrix has a size $31300\times31300$. I am using a universal java matrix package to invert this matrix. But as the dimension of the ...
223 views

### How can a CG solver solve a non positive definite sparse matrix

I am using the CUSP CG solver and I ran it on couple of sparse matrices from the University of Florida sparse matrix collection. The solver was able to solve non positive definite sparse matrices. My ...
196 views

### Can the Power Method be used here?

Given a set of $n$ points on which a triangulation is performed, it is possible to construct coefficients $\lambda_{ij}>0$ such that each point $x_i$ is a convex combination of the points connected ...
283 views

### Solving linear system of the form $ABx=b$

I would like to solve a linear system of the form $ABx=b$ in parallel, where $A$ and $B$ are large, sparse matrices. Currently, I am forming the system matrix $AB$ explicitly, but I would like to ...
381 views

### How "sparse" should a sparse matrix be to see benefits?

I have a matrix, whose size scales as $2^N$ (assume even $N$). In each row of the matrix, only about $2^{N/2}$ of the entries are filled ($N$ can be somewhere between 10 and 40, depending on what's ...
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### Efficiently solve linear system with matrix quadratic form

Take the system $$A^TCAx=b$$ where $$A\in\mathbb{R}^{n\times m},\;C\in\mathbb{R}^{n\times n},\;x,b\in\mathbb{R}^{m}, \;m\leq n$$ and $$A^TA=I$$ and $$Cy=d$$ can be solved efficiently in general (...
94 views

### Partitioning SPD matrix with METIS to preserve block SPD-ness

I am using the METIS to partition a matrix and then using domain decomposition to solve the subdomains in parallel using the Restricted Additive Schwarz method. I am currently trying to solve some ...
668 views

### Finite Elements: using preconditioned conjugate gradients with incomplete cholesky decomposition

I have to write a little finite elements code in C. I was asked to implement the conjugate gradients method, which I have done. Now, I am looking to improve further the efficiency of my program by ...
149 views

### Iteratively solving a sparse, ill-conditioned system

I have a sparse (density = 0.2%), ill-conditioned system that I am trying to solve, with no luck. Background I have a sequence of sampled data, where two of every 8 samples have been zeroed due to a ...
140 views

### Reordering algorithm for minimization of ram usage of a skyline matrix

The stiffness matrix of $Ax=B$ system of linear equations, where $A$ is an $n\times n$ symmetric matrix stored in the form of symmetric skyline matrix, that is associated with a finite element model ...
356 views

### Automatic Differentiation - reverse accumulation of linear system solve

I am studying the reverse mode of automatic differentiation. The reverse mode of automatic differentiation allows the efficient computation of a the derivative of a single dependent variable $y$ with ...
815 views

### Conjugate Gradient for non symmetric matrix

I have a large sparse matrix which is symmetric for the location of non zero values, but the values are different. Could I still use the CG method? I don't have much knowledge of linear algebra, the ...
111 views

### Solve $A^{-1} b$ when one column is replaced

Given square matrix $A_0$, vector $b$, vector $A_0^{-1}b$ and matrices $A_1, A_2, \dots, A_k$, in which each $A_i$ is generated from $A_{i-1}$ by replacing one single column, I would like to find an ...
101 views

### Acceleration of matrix geometric series

Suppose we want to find $x$ such that: $$x=b+Ax$$ where $A$ is a large sparse square matrix with eigenvalues in the unit circle. There are two representations of the solution: 1) $$x=(I-A)^{-1}b,$$...
196 views

### Efficiency of parallel direct linear solver

I am currently working on solving a positive definite symmetric systems in parallel. The parallel direct solver I used is MUMPS. However, the performance and efficiency of the parallel direct solver ...
158 views

### Which iterative method and preconditioner from petsc should be used when solving linear algebra in parallel?

I am currently trying to parallelize the incompressible flow solver code. However, when I run the code I realise that the parallel code takes much longer time than sequential code to finish one ...
124 views

I am reading Chapter 5 of Saad's iterative methods book, and I don't understand section 5.2.1 about the two propositions of optimality results. In the statements of the propositions, what does it mean ...
408 views

### MA57 vs HSL_MA57: symmetric indefinite solvers

What are the differences between MA57 and HSL_MA57 solvers? I'm in an optimization class that will make use of symmetric indefinite factorizations, and I'm trying to learn about the distinction ...
60 views

### Why do higher order finite elements (Q2) do not perform well for large Peclet number flows, as compared to Q1 finite elments?

I a solving the 2d steady state convection-diffusion problem on the famous flow around a cylinder in rectangular domain benchmark. My numerical results show that with Q1 finite elements, the solver is ...
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### How to verify solution to pre-conditioned linear systems solver?

I am solving Ax=b. A has a very large condition number (> O(10^10)) I am using the conjugate gradients method with point jacobi pre-conditioning. I obtained a solution 'x' that "looks" reasonable. ...
67 views

### Implicit solution to Sylvester equation

Suppose a matrix $M\in\mathbb{R}^{n\times n}$ is defined as the solution to a Sylvester equation $$AM+MB=C,$$ for some fixed (known) matrices $A,B,C$. In the regime where $n$ is large, we may with ...
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### Solving for $C$ in $Q = YCZ$ using least squares in Matlab

I am trying to solve for the matrix $C$ in $Q = YCZ$ in matlab. I have preliminary results but they don't seem realistic. Here, $Q$ is $n \times m-1$, $Y$ is $n \times p$, $C$ is $p \times m$ and $Z$ ...
219 views

### Method to solve linear, first order ODE of generalized matrix matrix form

The equation and its meaning: Consider two sets $(A)_{l=0,...,m_a},$,$(B)_{l=0,...,m_b}$ of hermitian matrices and a set of positive semidefinite matrices $(C)_{l=0,...,m_c}$. Each matrix has the ...
62 views

### Parallel dense solve with submatrices from mesh refinement with Petsc

For a Bounday Element Method problem I require the solution of a system of linear equations with multiple right-hand sides. Though this is a dense system, I still want to do it via Petsc in parallel. ...
43 views

### Iterative solver, which balances the equations with the biggest errors

I am trying to understand the mathematical side of an algorithm, which approximates the solution of coulomb's law. Their approach is to balance the two elements with the biggest positive/negative ...
218 views

### Iterative linear solvers compatible with automatic differentiation?

I'm using automatic differentiation on a function that contains a sparse nonsymmetric linear system to be solved. I was using BiCGStab to solve this part of the function, but noticed the derivatives ...
354 views

### Practical reference on sparse linear solvers for PDEs (Navier-Stokes, Poisson) and on learning PETSc

My background is mainly engineering and applied research and I have been a developer or some CFD software, but mostly at high level without worrying about linear solvers and the like. This has been ...
910 views

### Linear solvers: How to deal with a singular system? (Poisson equation with Neumann boundary conditions)

The question is in the context of iterative numerical solution of large PDE systems with Finite Differences or Finite Elements: Stating the Poisson equation with Neumann boundary conditions will lead ...
482 views

### Algebraic multigrid in PETSc

Consider a potential/poisson equation on a very large, complicated geometry. Currently, an self-written FEM and linear solvers from NumPy are used. Performance is, of course, not good enough for ...
61 views

### Scaling for a nonsymmetric eigenvalue problem

I have an eigenvalue problem emerging from the internal vibro-acoustic coupling. The eigenvalue problem is nonsymmetric but it was proven in literature that it results in real eigenvalues and ...
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### Fast c++ library to solve very big sparse systems

I am working on a project with electrical circuits, where I am trying to compute the voltages at all the nodes of an electrical circuit. I know that the electrical circuit is a perfect grid, so each ...
220 views

### Fast counting of all submatrices of a binary matrix with a full column rank

I have a binary full-rank matrix of size, say, $25 \times 50$. I need to count how many subsets of its columns form matrices with a full column rank, i.e. all the columns in the subset are linearly ...
1k views

### Which is the best subroutine available for solving sparse linear system of equations [closed]

I am trying to solve the system of linear equations: $AX=B$. For this currently I am using Intel MKL Pardiso solver. It works well when the order of $A$ is around $13500\times13500$ and below. Above ...
925 views

### Tridiagonal Solver in Python

I have a code I'm working on that involves solving a 1D Schrodinger equation using a Crank-Nicolson time step. The code is written in NumPy/SciPy, and I was doing a bit of profiling and discovered ...
255 views

### How to reshape matrix into row-major order for MKL DSS?

I would like to use MKL to solve a sparse linear system. I chose the DSS (Direct Sparse Solver) interface, which implements the following steps: ...
2k views

### How to get multiple solutions for a optimization problem using any kind of software

I have a optimization problem in which the optimal objective value occurs at multiple point in the feasible space. If I run my problem in LINGO software then it gives me the optimal objective value at ...
107 views

### Writing a non-square linear system in standard form $A\cdot{x}=b$

I have spend the last few days working my way through an interesting paper and I'm building a numerical model so I can apply the method. However, I am getting stuck at an "it can be shown" step. I am ...
3k views

### Iteratively solving 3D Poisson equation in MATLAB

I have written a function that sets up a sparse matrix A and RHS b for the 3D Poisson equation in a relatively efficient way. The set-up is nothing fancy: I have extended the 2D 5-point stencil to an ...
111 views

### Iterative single variable solutions in large linear systems

I have a system where $A$ is a large $n\times n$ marix with fast MVMs. It may have many nonzero entries (albeit in a structured way so as to allow fast MVMs), and is not necessarily diagonally ...
320 views

### Test matrices for large sparse overdetermined system of linear equations

I'm working on some c++ code to solve (conjugate gradient, least squares conjugate gradient, LSQR,..) large sparse overdetermined systems of linear equations. There is a twist to my matrices and the ...
104 views

### Optimization based integration for MPM

I'm considering implementing (just for simplicity) the unconstrained implicit optimization based integration for Material Point Method as described in Chenfanfu Jiang's thesis on MPM (the minimization ...
I have a linear system of the form $\mathbf{A}\mathbf{x} = \mathbf{f}$. If the length of the vector $\mathbf{x}$ is $N$, meaning that there are $N$ unknowns, then the matrix $\mathbf{A}$ has seven ...