# 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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### How to choose a method for solving linear equations

To my knowledge, there are 4 ways to solving a system of linear equations (correct me if there are more): If the system matrix is a full-rank square matrix, you can use Cramer’s Rule; Compute the ...
11k 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 ...
2k views

### What guidelines should I use when searching for good preconditioning methods for a specific problem?

For the solution of large linear systems $Ax=b$ using iterative methods, it is often of interest to introduce preconditioning, e.g. solve instead $M^{-1}(Ax=b)$, where $M$ is here used for left-...
2k 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 ...
2k views

### Intuitive motivation for BFGS update

I am teaching a numerical analysis survey class and am seeking motivation for the BFGS method for students with limited background/intuition in optimization! While I don't have time to prove ...
568 views

### What is the current state of polynomial preconditioners?

I wonder what has happened to polynomial preconditioners. I am interested in them, because they appear to be comparatively elegant from a mathematical perspective, but as far as I have read in surveys ...
9k views

### Understanding the “rate of convergence” for iterative methods

According to Wikipedia the rate of convergence is expressed as a specific ratio of vector norms. I'm trying to understand the difference between "linear" and "quadratic" rates, at different points of ...
262 views

### Non-monotonic convergence in fixed-point problem

Background I am solving a variant of the Ornstein-Zernike equation from liquid theory. Abstractly, the problem can be represented as solving the fixed point problem $A c(r)=c(r)$, where $A$ is an ...
539 views

### Using fixed point iteration to decouple a system of pde's

Suppose I had a boundary value problem: $$\frac{d^2u}{dx^2} + \frac{dv}{dx}=f \text{ in } \Omega$$ $$\frac{du}{dx} +\frac{d^2v}{dx^2} =g \text{ in } \Omega$$ $$u=h \text{ in } \partial\Omega$$ My ...
307 views

### What is the underlying structure of scientific code performance?

Consider two computers with different hardware and software configurations. When running the exact same serial Navier-Stokes code on each platform it takes x and y time to execute one iteration for ...
708 views

### Smallest eigenvalue without inverse

Suppose $A\in\mathbb{R}^{n\times n}$ is a symmetric, positive definite matrix. $A$ is big enough that it's expensive to solve $Ax=b$ directly. Is there an iterative algorithm for finding the ...
687 views

### Projecting out the null-space of $A$ from $b$ in $Ax=b$

Given the system $$Ax=b,$$ where $A\in\mathbb{R}^{n\times n}$, I read that, in case Jacobi iteration is used as a solver, the method will not converge if $b$ has a non-zero component in the null-space ...
430 views

### How to establish that an iterative method for large linear systems is convergent in practice?

In computational science we often encounter large linear systems which we are required to solve by some (efficient) means, e.g. by either direct or iterative methods. If we focus on the latter, how ...
1k views

### Basin of attraction for Newton's method

Newton's method for solving nonlinear equations is known to converge quadratically when the starting guess is "sufficiently close" to the solution. What is "sufficiently close"? Is there literature ...
6k 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 ...
914 views

### Why does iteratively solving the Hartree-Fock equations result in convergence?

In the Hartree-Fock self-consistent field method of solving the time-independent electronic Schroedinger equation, we seek to minimize the ground state energy, $E_{0}$, of a system of electrons in an ...
1k views

### Which iterative linear solvers converge for positive semidefinite matrices?

I want to know which of the classic linear solvers (e.g Gauss-Seidel, Jacobi, SOR) are guaranteed to converge for the problem $Ax=b$ where $A$ is positive semi definite and of course $b \in im(A)$ (...
148 views

### What iterative method can effectively solve a linear system with this kind of spectrum

I have a linear system with matrix which eigenvalues are uniformly distributed on the unit circle like this: Is it possible to solve this kind of system effectively by iterative method, maybe with ...
4k views

### How to remove Rigid Body Motions in Linear Elasticity?

I want to solve $K u = b$ where $K$ is my stiffness matrix. However some constraints may be missing an therefore some rigid body motion may be still present in the system (due to eigenvalue zero). ...
1k views

### What is a robust, iterative solver for large 3-d linear-elastic problems?

I'm diving into the fascinating world of finite element analysis and would like to solve a large thermo-mechanical problem (only thermal $\rightarrow$ mechanical, no feedback). For the mechanical ...
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### Numerical iterative method, estimating error

Given iterative method: $x_{n+1}=0.7\sin x_n +5 = \phi(x_n)$ for finding solution for $x=0.7\sin x +5$, I want to estimate $|e_6|=|x_6-r|$ as good as possible, with $x_0=5$, where $r$ is exact ...
282 views

### Iterative linear solver for “ugly” saddle point system

I am a graduate student majoring scientific computing. The numeric model I made caused a very ugly-looking saddle-point linear system. It is not symmetric at all and I will attach the sparsity pattern ...
596 views

### On Vanilla Preconditioners for solving dense $Ax=b$ iteratively

I am looking for preconditioners which don't assume anything about the matrix or its origins. I basically want to be able to type in the following in MATLAB and have quick solving time: ...
3k views

### Simulated Annealing proof of convergence

I implemented downhill simplex simulated annealing algorithm. Algorithm is very hard to tune, w.r.t. parameters including cooling schedule, starting temperature... My first question is about ...