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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Approximately "solving" a linear system of equations without a feasible solution

A linear system of equations has the form $Ax = b$, where a matrix $A$ and a vector $b$ are given, and I wish to find a solution vector $x$. Suppose that the system $Ax = b$ has no feasible solution. ...
sara's user avatar
  • 311
9 votes
2 answers
943 views

Which Sparse Matrix Solver Libraries can I run on Android?

The title says most of it. I'm looking for a lightweight and easy-to-use library that I can use for Android (NDK) projects. For dense stuff I like using Eigen but I haven't found many comprehensive (...
rsp1984's user avatar
  • 435
7 votes
4 answers
2k views

precision vs matrix condition number

I have an application in which I am computing a quantity which is approximated by an average over $M$ points. In theory, the average converges to the correct quantity when $M$ is infinite. In practice,...
yannick's user avatar
  • 375
2 votes
2 answers
1k views

Lanczos solver implementations in MATLAB/C++ give different results

I have transferred my MATLAB Lanczos solver for symmetric eigenvalue solvers to C++ with the help of Intel MKL and MTL4 libraries. I have some wrapper templates for MKL routines. However during the ...
Umut Tabak's user avatar
9 votes
1 answer
2k views

Schrodinger equation with periodic boundary conditions

I have a couple of questions regarding the following: I am trying to solve the Schrodinger equation in 1D using the crank nicolson discretization followed by inverting the resulting tridiagonal ...
WiFO215's user avatar
  • 143
4 votes
1 answer
348 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,...
Paul's user avatar
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3 votes
3 answers
1k views

What is the best way to solve Ax = b (with A large, spd, sparse, banded and poorly conditioned)?

I'm trying to solve $Ax = b$ given a vector $b$ and a large, symmetric positive definite, sparse, banded matrix $A$ that has a very poor condition number. I know several iterative methods that ...
user avatar
8 votes
3 answers
3k views

Solving a non-symmetric non-diagonally dominant sparse system the best way

I faintly recall from my early "numerics" lectures that iterative linear solvers for $Ax=b$ often require that when $A$ is decomposed as $$A=D + M$$ where D is a diagonal matrix and $M$ has zero ...
Lagerbaer's user avatar
  • 487
18 votes
6 answers
2k views

How to reorder variables to produce a banded matrix of minimum bandwidth?

I'm trying to solve a 2D Poisson equation by finite differences. In the process, I obtain a sparse matrix with only $5$ variables in each equation. For example, if the variables were $U$, then the ...
Paul's user avatar
  • 12k
5 votes
2 answers
594 views

How to parallelize a banded direct solver?

I have a linear system whose matrix that is diagonally dominant, non-symmetric, but banded. Since the band-radius is 2 (producing only 5 variables per equation), a banded direct solver (gaussian ...
Paul's user avatar
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11 votes
1 answer
8k views

Are there any heuristics for optimizing the successive over-relaxation (SOR) method?

As I understand it, successive over relaxation works by choosing a parameter $0\leq\omega\leq2$ and using a linear combination of a (quasi) Gauss-Seidel iteration and the value at the previous ...
Paul's user avatar
  • 12k
17 votes
1 answer
3k views

Can a Krylov subspace method be used as a smoother for multigrid?

As far as I am aware, multigrid solvers use iterative smoothers such as Jacobi, Gauss-Seidel, and SOR to dampen the error at various frequencies. Could a Krylov subspace method (like conjugate ...
Paul's user avatar
  • 12k
10 votes
3 answers
3k views

How to construct a prolongation and restriction operator for an algebraic multigrid solver?

I am trying to solve a linear system of equations that is sparse, but lacks any kind of banded structure. I have heard that there is a way to extend the principles of a multigrid solver for implicit ...
Paul's user avatar
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11 votes
1 answer
2k views

How can one parallelize a multigrid method for solving a linear system of equations?

As I understand it, the multigrid method solves a linear system by solving a coarser version of the same problem (there by eliminating low frequency error) then projecting back to the fine grid to ...
Paul's user avatar
  • 12k
3 votes
0 answers
126 views

Up-/downdating methods for a series of normal equations

In an application I have to solve a series of positive definite linear systems of the form $A^TA x = A^Tb$ (i.e. normal equations). The next system is obtained from the previous one by adding and/or ...
Dirk's user avatar
  • 1,738
6 votes
2 answers
451 views

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 ...
Paul's user avatar
  • 12k
9 votes
3 answers
1k views

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 \...
shuhalo's user avatar
  • 3,620
7 votes
3 answers
1k views

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) ...
Inquest's user avatar
  • 3,374
10 votes
1 answer
4k views

What is a good stop criterion when using an iterative method to find eigenvalues?

I read this answer, and realized I have been using the difference between sucessive iterates to define a stop criterion for an iterative method of finding eigenvalues/vectors. What are good stop ...
Dan's user avatar
  • 3,355
17 votes
2 answers
5k views

Stopping criteria for iterative linear solvers applied to nearly singular systems

Consider $Ax=b$ with $A$ nearly singular which means there is an eigenvalue $\lambda_0$ of $A$ that is very small. The usual stop criterion of an iterative method is based on the residual $r_n:=b-Ax_n$...
Hui Zhang's user avatar
  • 1,319
4 votes
2 answers
212 views

Looking for a mathematical proof of stability in floating point arithmetic of CG - any reference?

I am looking for a reference - paper, book, discussion, anything that has a mathematical proof for stability of the conjugate gradient method in floating point arithmetic. Something similar for ...
ECS's user avatar
  • 41
30 votes
2 answers
15k 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 ...
Jed Brown's user avatar
  • 25.6k
23 votes
3 answers
3k views

Can diagonal plus fixed symmetric linear systems be solved in quadratic time after precomputation?

Is there an $O(n^3+n^2 k)$ method to solve $k$ linear systems of the form $(D_i + A) x_i = b_i$ where $A$ is a fixed SPD matrix and $D_i$ are positive diagonal matrices? For example, if each $D_i$ is ...
Geoffrey Irving's user avatar
4 votes
1 answer
115 views

Applicability of combinatorial and support preconditioner

There are several correspondences between matrices and graphs, e.g., each matrix is the adjacancy matrix of a weighted graph. The terms support preconditioner or combinatorial preconditioner refer to ...
shuhalo's user avatar
  • 3,620
9 votes
1 answer
343 views

Nested dissection on regular grid

When solving sparse linear systems using direct factorization methods, the ordering strategy used significantly impacts the fill-in factor of non-zero elements in the factors. One such ordering ...
Victor Liu's user avatar
  • 4,480
7 votes
2 answers
895 views

Is it possible to dynamically resize a sparse matrix in the Petsc library?

This may be a Petsc newbie question, but... I'm using Petsc to solve a large sparse linear system. The initial creation of the matrix is fairly slow, which I'm given to understand is largely due to ...
batty's user avatar
  • 213
16 votes
2 answers
2k views

Estimation of condition numbers for very large matrices

Which approaches are used in practice for estimating the condition number of large sparse matrices?
Allan P. Engsig-Karup's user avatar
170 votes
8 answers
140k views

Recommendations for a usable, fast C++ matrix library?

Does anyone have recommendations on a usable, fast C++ matrix library? What I mean by usable is the following: Matrix objects have an intuitive interface (ex.: I can use rows and columns while ...
Geoff Oxberry's user avatar
8 votes
3 answers
866 views

What should be the criteria for accepting/rejecting singular values?

I am solving a system using singular value decomposition. The singular values (before scaling) are: ...
drjrm3's user avatar
  • 2,139
8 votes
2 answers
242 views

Is there one general approach to build a projection methods for different problems?

My question is probably going to be too general to answer it with a couple words. Could you please suggest a good reading in that case. Projection methods are used to reduce size of the solution space ...
danny_23's user avatar
  • 501
13 votes
3 answers
417 views

In what application cases are additive preconditioning schemes superior to multiplicative ones?

In both domain decomposition (DD) and multigrid (MG) methods, one may compose the application of the block updates or coarse corrections as either additive or multiplicative. For pointwise solvers, ...
Peter Brune's user avatar
  • 1,675
3 votes
2 answers
962 views

Linear regression via SVD not producing best fit with escalating polynomial degree

I am using a basic singular value decomposition (via LAPACK) routine in FORTRAN to solve an overdetermined system in the form of $A\cdot X = B$ where $\mathrm{size}(A) = [m,n]$ with $m > n$. My ...
drjrm3's user avatar
  • 2,139
33 votes
5 answers
16k views

Performance differences between ATLAS and MKL?

ATLAS is a free BLAS/LAPACK replacement that tunes itself to the machine when compiled. MKL is the commercial library shipped by Intel. Are these two libraries comparable when it comes to performance, ...
Stefano Borini's user avatar
8 votes
1 answer
1k views

How can I compute the Schur complement in PETSc?

How can I compute the Schur complement: $$ S = K_{bb} - K_{ba} K_{aa}^{-1} K_{ab} $$ where $$ K=\begin{pmatrix} K_{aa} & K_{ab} \\ K_{ba} & K_{bb} \end{pmatrix} $$ (in some ordering) is a ...
Peter Brune's user avatar
  • 1,675
14 votes
3 answers
2k views

Why is my parallel solver slower than my sequential solver?

I was playing around with PETSc and noticed that when I run my program with more than one process via MPI it seems to run even slower! How can I check to see what is going on?
Sean Farley's user avatar
  • 1,370
12 votes
3 answers
1k views

Efficient tridiagonal matrix algorithm implementation

I am solving a physical problem using implicit numerical scheme. This leads me to solving a linear equation with tridiagonal matrix. I've coded this algorithm from Wikipedia. I wonder if there is an ...
gmk's user avatar
  • 455
25 votes
2 answers
11k 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 ...
Andrew Spott's user avatar
  • 1,155
58 votes
4 answers
9k 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 (...
J. M.'s user avatar
  • 3,155
16 votes
1 answer
5k views

How can I estimate the condition number of a large sparse matrix using PETSc?

I have a PETSc Mat and would like to estimate its condition number.
Jed Brown's user avatar
  • 25.6k

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