Questions tagged [linear-solver]
Referring to methods for solving linear systems of equations.
287
questions
2
votes
1answer
47 views
Partially Banded Matrix
I have a somewhat peculiar Jx=R system that I need to solve. The matrix J is 2N -by- 2N. The first N rows have all entries filled. The next N rows are banded in two places, i.e. for the (N+k)th row, ...
0
votes
1answer
49 views
Method to calculate solution of a linear equation system?
I am searching a solution method for the following equation system of equation systems:
Let $A, B \in \mathbb{R}^{n \times n}$ be s.p.d. Matrices and $O$ be the zero matrix of the same size. Further ...
5
votes
0answers
28 views
Symmetric sparse direct solvers in scipy
scipy.linalg.solve, in its newer versions, has a parameter assume_a that can be used to specify that the matrix $A$ is symmetric ...
1
vote
0answers
38 views
Hybrid Ellpack-Itpack (ELL) + COO Sparse Matrix Representation decomposition threshold
Hybrid ELL-COO sparse matrix representation can be done as in the picture, I was looking intensively, however I couldn't find out what is the threshold of decomposing the original matrix into ELL part ...
1
vote
0answers
48 views
implementation for coppersmith matrix multiplication
Is there any online implementation for the coppersmith matrix multiplication I have searched alot but can not find any? and if there is not any why is that Isn't this algotithm much faster than ...
0
votes
0answers
70 views
What algorithm do BLAS and ATLAS use for matrix multiplication?
I have searched and what I understood was that they use the naive one with several memory and cache optimizations.
However, I wanted to know whether they are using the Strassen or the Coppersmith-...
1
vote
1answer
86 views
Lapack symmetric update $B^{-1}AB^{-T}$
Does Lapack have a routine that, given symmetric $A=A^T$ and $B$, computes the symmetric matrix $B^{-1}AB^{-T}$ (while preserving symmetry exactly)?
It would be enough to have this routine for ...
0
votes
1answer
45 views
The proper way to assess the error of Jacobi iteration (for 2D Poisson equation)?
Motivation: I'm using 2D regular grid (it's actually a quadtree but I can still treat it as a finite difference thing if I weight-average the solution over smaller scale cells for the purpose of ...
2
votes
2answers
139 views
Why iterative method: AMG preconditioned PCG is slower than Matlab direct method 'A\b'?
Recently, I have met a question that
a saying goes that for large linear system: iterative methods are required because of memory problem of direct methods.
But when I implement some experiments ...
3
votes
1answer
67 views
Why are the round-off errors when solving the linear system $Ax = b$ of order $\varepsilon_\text{mach} x_j$?
I was reading a paper on arXiv where, in Section 2.4, the authors are discussing the error that arises in the solution of a linear system
$$Ax = b,$$
or, to match up better with the paper,
$$\Phi \...
1
vote
1answer
84 views
Sparse matrix inversion
I have the impedance matrix $Y$, formulated from an electrical network by augmented nodal analysis. The matrix $Y$ is shown as an image to illustrate its feature visually, where all the white blocks ...
0
votes
1answer
93 views
How to set an initial guess for the iterative solver in Comsol?
How to set the initial guess for the iterative solver GMRES or FGMRES for linear problems (Helmholtz equation of RF module) in Comsol?
4
votes
0answers
66 views
Complex differentiation of linear solvers
I have a linear system $$Ax=b$$ which I'm solving approximately, and I need to take the frechet derivative of x with respect to z. Were I solving the problem exactly (either analytically or to machine ...
0
votes
0answers
73 views
Algebraic multigrid for coupled equations
As far as I understand is algebraic multigrid(AMG) a method that was intentionally developed to solve linear systems where every grid point or node has a single DOF. When AMG should now be used for ...
7
votes
0answers
75 views
How to construct an effective preconditioner for this particular problem
A quick introduction to my problem
I am currently developing a method for simulation of water waves in three dimensions based on potential flow theory. The computational bottleneck of the method is ...
1
vote
0answers
41 views
Stability of SVD, Eigendecompositions, and pseudoinverse procedures in modern LAPACK routines
I have proposed an optimisation algorithm which I claim has improved upon the previous algorithm in a number of ways.
One of these claims is that my proposed solution requires no explicit SVD and ...
2
votes
1answer
172 views
A fast way to check if a Matrix is ill-conditioned, and turning it into well-conditioned
I'm running a simulation, and some linear solvers are returning a message of ill-conditioned matrix.
Hence, I'm looking for a fast, easy to implement, method to detect if a matrix is ill-conditioned, ...
6
votes
2answers
113 views
Efficient approach for solving matrix plus diagonal matrix system that varies in time
When solving a system of ODEs, as part of a preconditioner, I get the system
$(A + D(t))x = b(t)$
where $A$ is a sparse matrix and $D(t)$ is diagonal. I'm currently solving this by taking the LU-...
2
votes
1answer
74 views
Assessing numerical error in solving a least squares problem
I have a linear system of the type
$$Ax = b$$
I want to minimise $|b - Ax|^2$.
I know there are different approaches to directly solve the system (Normal equation + Cholesky, QR decomposition, SVD ...
2
votes
0answers
78 views
Richardson's Iteration, Gradient Method and Spectral Radius
Richardson's iteration introduce a scalar $\alpha$ to the update formula:
$$ \textbf{x}^{(k+1)} = \textbf{x}^{(k)} + \alpha \textbf{r}^{(k)} $$
And compute $\alpha$ by minimizing the spectral radius:...
2
votes
1answer
61 views
Right-preconditioning and fixed point linear iterations
Given a linear system $A\textbf{x}=\textbf{b}$, we can express it into the easier-to-solve right-preconditioned form:
$$ AM^{-1}\textbf{y}=\textbf{b}, \quad \textbf{y}= M\textbf{x} $$
On the other ...
9
votes
3answers
346 views
fastest linear system solve for small square matrices (10x10)
I am very interested in optimizing the hell out of linear system solving for small matrices (10x10), sometimes called tiny matrices. Is there a ready solution for this? The matrix can be assumed ...
3
votes
0answers
63 views
Numerical analysis, pivoting and incomplete LU decomposition
When doing LU decomposition, the algorithm will break down if any of the diagonal element $x_{ii}$ is zero. Therefore, we can use pivoting on the matrix such that $x_{ii}$ is no longer zero. That is ...
6
votes
1answer
108 views
numerical solution of an under-determined linear equation in high dimensions
I need to solve a linear regression problem $$Ax=y$$ which is hugely underdetermined. I have around $10^6$ features but only $10^3$ equations. So $A$ is a $1,000\times 1,000,000$ matrix and $y$ a ...
2
votes
2answers
129 views
Asymptotic Complexity of Gaussian Elimination using Complete Pivoting
I would like to know the algorithm asymptotic complexity with Complete Pivoting. With partial pivoting, it is known to be $O(n^3)$.
Is it the same for complete pivoting?
1
vote
1answer
89 views
Conjugate gradient - ill-conditioning and numerical tolerance
I would like to solve system $Ax=b$, where $A$ is SPD, but very ill-conditioned ($\text{cond}(A)>10^{11}$). I am interested in using UNpreconditioned version of the conjugate gradient method.
Is ...
3
votes
1answer
997 views
Which C++ linear algebra library is probably the fastest on solving huge sparse [square matrix] linear system?
I am developing a 2D CFD solver for fluid-particle interaction. To solve Navier-Stokes equations on a grid of size $10000\times 10000$ cells (or >1 million cells), a large linear system $Ax=b$ with $A$...
2
votes
1answer
235 views
Memory and time requirements of the scipy sparse spsolve
I have a system of fairly large set of linear equations (approximately 30K equations). I am using scipy.sparse.spsolve to solve these equations. Initially, I tried ...
0
votes
1answer
36 views
Can LINCS algorithm be used for colliding molecules?
Supposing that one molecule is static and one is dynamic,
can the dynamic one be solved with LINCS for its shape (angle, bond length) constraints and also keep collisions with static molecule off, ...
2
votes
1answer
92 views
Is steady linear elasticity inherently ill-conditioned?
Compared to the transient PDE for linear elasticity, the steady equations appear to less well-conditioned. Are they inherently ill-conditioned without the transient term?
The condition number for ...
2
votes
1answer
108 views
Condition number of matrix and effects of round off errors
In my numerical linear algebra class, I learned that for some matrices, it could have an element that is a very small number that is approximately 0 (and many orders of magnitude different from all ...
3
votes
1answer
142 views
Derivatives of Approximate Matrix inverses
I am cross posting this question to the mathermatics stack exchange. please find it either at this link, https://math.stackexchange.com/q/2952989/430980, or below:
I have a question concerning the ...
8
votes
0answers
181 views
Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system
Problem
Solving a non-linear system of equations.
The number of variables is the same as the number of equations.
When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
2
votes
2answers
125 views
Memory/speed tradeoff for many small matrix inverses
Problem
In the case of a finite element code, I have many small (order of 30x30) matrix inverses (or LU factorizations), one per finite element. These matrix inverses never change and must be applied ...
0
votes
1answer
60 views
How to access solution to linear system in PETSc?
I have just started with PETSc hence it might seem like a very stupid question but I couldn't find any answer in manual.
After Calling KSPSolve, where can I access the soluition for my linear system?
...
3
votes
1answer
136 views
How many operations are needed for LAPACK's zgesv to solve a linear system?
I have a linear system of complex numbers. I am using LAPACK' zgesv (actually I am using intel MKL LAPACKE, but I am assuming the algorithm is the same). No assumption can be made about the system.
I ...
2
votes
2answers
290 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 ...
0
votes
1answer
86 views
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 ...
3
votes
1answer
125 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 ...
2
votes
1answer
154 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 ...
6
votes
1answer
158 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 ...
4
votes
2answers
197 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 ...
5
votes
0answers
201 views
conjugate gradient for Newton's method with non positive definite Hessian matrix
I want to minimize a non-linear function $f(x)$ using Newton's method. At each optimization step, I compute a descent direction $d$ to update $x$ using a second-order approximation of $f(x)$:
$$ \...
1
vote
1answer
199 views
How to use CSDP to express a semidefinite program?
I am trying to use CSDP and am struggling with it. Consider, for example, the following semidefinite program
$$\begin{array}{ll} \text{minimize} & 0\\ \text{subject to} & Q - A' Q A - \...
2
votes
0answers
110 views
Efficiently solve linear system with matrix quadratic form
Take the system $$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 (...
1
vote
1answer
78 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 ...
2
votes
1answer
129 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 ...
3
votes
0answers
110 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 ...
1
vote
1answer
104 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 ...
2
votes
2answers
203 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 ...
