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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7
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1answer
610 views

Decreasing spectral radius

I have a $4\times 4$ matrix and I want to use Jacobi iteration on it. Right now the spectral radius is higher than $1$. I know that the method is guaranteed to converge if the matrix is diagonally ...
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2answers
277 views

Krylov subspace iterative methods in floating point arithmetic

Is there any work that considers Krylov subspace iterative methods in floating point arithmetic? I'm especially interested in how rounding errors influence the convergence and the accuracy of the ...
9
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2answers
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). ...
4
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2answers
261 views

Does right-hand side influence convergence rate of a Krlylov supspace method?

Consider general system $Ax=b$. Does convergence of the Krylov subspace methods depend on actual vector $b$ assuming initial guess is zero? I mean such factors as locality of the source (with the ...
15
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1answer
626 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 ...
6
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3answers
1k views

Eigenvectors with the Power Iteration

To compute the eigenvector corresponding to dominant eigenvalue of a symmetric matrix $A\in\mathbb{R}^{n\times n}$, one used Power Iteration, i.e., given some random initialization, $u_1\in\mathbb{R}^...
5
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4answers
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 ...
2
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3answers
1k views

Convergence of the gradient descent and linear vs non-linear fixed point iteration

Suppose a system $$Ax=b$$ is given, with $A\in\mathbb{R}^{n\times n}$ being a symmetric positive-definite matrix, and some non-zero $b\in\mathbb{R}^n$. The gradient method with optimum step length can ...
33
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3answers
14k views

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 ...
8
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5answers
365 views

Iterative solution to a nonlinear equation

I appologize in advance if this question is silly. I need to compute the root of \begin{equation} u -f(u) =0 \end{equation} Where $u$ is a real vector and $f(u)$ is a real-vector valued function. ...
9
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3answers
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 ...
8
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1answer
351 views

Is the sparsity pattern of a linear system important for iterative (KSP) solvers?

Pretty much the question. Given a general sparse, non-symmetric (both numerically and structurally) matrix, how important is the sparsity pattern (i.e. row/column permutation of matrix/vector) for ...
9
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1answer
370 views

Solving a system with a small rank diagonal update

Suppose I have the original large, sparse linear system: $A\textbf{x}_0=\textbf{b}_0$. Now, I do not have $A^{-1}$ as A is too large to factor or any sort of decomposition of $A$, but assume that I ...
9
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4answers
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 ...
11
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1answer
745 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 ...
3
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2answers
188 views

How to establish that an iterative method can be applied to large matrices whose size may reach 10^3?

I have an iterative method for computing the Moore-Penrose generalized inverse of matrices, that is $$X_{k+1} = ((I-\beta X_{k}A)^t) + X_{k}$$ with initial approximation: $$X_{0} = \beta AA^t$$ ...
1
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1answer
723 views

Gradient descent to stationary, or accumulation point

I recently came across the notion of an accumulation point as a result of a certain gradient descent variation. The following definition was found: An accumulation point $P$ is such that there are an ...
1
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3answers
253 views

Unique coordinates (solutions) in a single Gauss-Seidel iteration

I managed to reduce certain computational problem to the Gauss-Seidel solution of the following linear system: $$Ax=Ly,$$ where $A, L\in\mathbb{R}^{n\times n}$ are weighted Laplacian matrices (...
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1answer
1k views

A Comparison between GMRES, QMR and LU for Dense Matrices

As I see it, there are 3 ways to solve unstructured dense system of equations: GMRES, QMR and LU. Has anyone done a comparison for these three? As far as I know, LU is the preferred choice and it is ...
9
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2answers
1k views

Safe application of iterative methods on diagonally dominant matrices

Suppose the following linear system is given $$Lx=c,\tag1$$ where $L$ is the weighted Laplacian known to be positive $semi-$definite with a one dimensional null space spanned by $1_n=(1,\dots,1)\in\...
5
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3answers
621 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: ...
4
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4answers
230 views

Determining the algorithmic complexity

A few of the iterative matrix algorithms (CG,GMRES etc.) I have authored are acting rather funny. They converge to the right answers but take abnormally long time to run. I am in the process of ...
10
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2answers
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 $b \in im(A)$ (...
7
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1answer
604 views

Jacobi iteration to reduce the quadratic function

Given certain function $f(X)$ which is quadratic in $X\in\mathbb{R}^{n\times d}$, $$\frac{1}{2}tr(X^TAX) - tr(Y^TBX)$$ for positive definite weighted Laplacian matrices $A, B\in\mathbb{R}^{n\times n}...
6
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1answer
281 views

Question about the smoothing operators in multigrid methods for nonlinear PDEs

Suppose we are dealing with a nonlinear problem, say $$ A u := L u + G(u) = f $$ the nonlinearity of the operator $A$ is the polynomial type, ie, $L$ is a linear operator, and $G(u) = u^k$, or ...
11
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1answer
456 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 ...
10
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3answers
990 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 ...
20
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3answers
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-...
10
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1answer
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 ...
15
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1answer
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 ...
9
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3answers
893 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 \...
6
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3answers
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) ...
8
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1answer
3k 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 ...
27
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2answers
12k 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 ...

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