# Questions tagged [iterative-method]

A method which produces a sequence of numerical approximations which converges (provided technical conditions are satisfied) to the solution of a problem, generally through repeated applications of some procedure. Examples include Newton's method for root finding, and Jacobi iteration for matrix-vector solves.

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### solving linear system whose symmetrized matrix is positive definite

Are there iterative methods for the solution of nonsymmetric linear systems $Ax=b$ that can take (theoretical or practical) advantage from knowing that $A+A^T$ is positive definite? These matrices are ...
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### How can I color my Mandelbrot set like this?

I have a background image of a fractal on my phone that I would like in a higher resolution with super sampling, and decided to write my own program for it. I've got down rendering a Mandelbrot set, ...
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### When will the Orthomin/CG iteration fails

I know that the the Conjugate Gradient iteration fails when $0\in \mathcal {W}(A^{H})$, which means there's a complex vector $x+iy$ such that $(x+iy)^{T}A^{H}(x+iy)=0$. I wonder how to derive a real ...
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### Is there any function to calculate condition number of sparse matrix in Eigen libraray?

The function JacobiSVD and BDCSVD can calcuate condtion number of a dense matrix via singular values. However I need to know condition number of a sparese matrix due to slow computation speed using ...
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### On the reordering of sparse matrices

I have been reading on different techniques used to reorder sparse matrices to achieve better performance, the most popular being the Cuthill-McKee or Reverse Cuthill-McKee algorithm. Most of those ...
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### Does right-hand side influence convergence rate of a Krlylov subspace method?

Consider general system $Ax=b$. Does convergence of the Krylov subspace methods depend on actual vector $b$ assuming the initial guess is zero? I mean such factors as the locality of the source (with ...
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### Effecient method for iterating over sparse dataset

Apologies if this isn't the appropriate forum for this question. I have a set of elements that I need to iterate over as part of a modeling workflow. The elements exists over a set of dimensions (i, ...
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### 1-D Conduction Steady state heat transfer using FD

I have tried to solve 1-D Conduction Steady state heat transfer problems in Matlab (see below). Here is the 1-D model: T''[x] == 0, T[0] == 100, T'[100] == 0 ...
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### Why does smoothed aggregation multigrid method used as preconditioner in conjugate gradient slows down the solution time?

I'm solving a system of linear equations obtained from the FEM discretization of a simple linear elasticity problem on a cube with zero displacements at one plane and a load on the opposite one. The ...
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### How to implement the Hessenberg QR Algorithm?

For context, I'm creating a linear algebra library from scratch for learning purposes in C. Right now I'm working on calculating eigenvalues but my implementation of the QR Algorithm is diverging. ...
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### Solving a nonlinear problem with a very small components with finite element method

In solving nonlinear hyperelastic solid mechanics problems, to converge to the correct solution we need to do step-by-step loading which makes the deformation at each step very small (for my ...
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### What online optimisation algorithm can be used for a noisy cost function?

I am trying to optimise a function, but the function can be noisy and give varying results for the same parameters. Furthermore, it needs to be online, as the data from each new iteration happens ...
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### Best way to check if SOR solution has converged for 2d matrix

I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10. I can obtain the fully converged solution for ...
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### How to solve for f(A)x=b without GMRES?

How to solve for $f(A)x=b$? For GMRES, an answer is given in this book chapter: http://link.springer.com/chapter/10.1007%2F978-3-642-58333-9_2. Ungated version: https://www.researchgate.net/profile/...
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### Residual of Poisson equation with periodic boundaries

I am trying to write a multigrid solver for Poisson's equation, $-\Delta u=f$, on the unit square, $\Omega=(0,1)^2$ with periodic boundaries. My primary source has been Multigrid by Trottenberg, ...