# Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

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### How to implement the gmres method using Householder transformation instead of the Gram-Schmidt?

For Generalized Minimal Residual method GMRES, we usually use the Modified Gram-Schmidt MGS to generate an orthonormal basis of ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
235 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, ...
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### 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-...
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### 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 ...
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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:...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 $A$...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...