# 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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### Implementing matrix term version of Gauss-seidel

I am trying to implement the below description from Ch. 11 of Heath's "Scientific Computing An Introductory Survey" of the Gauss-Seidel iterative method for solving a system of linear ...
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### How to find the optimal SSOR parameter

The symmetric successive overrelaxation method features the iteration matrix $$P=\left(\frac{D}{\omega}+L\right)\frac{\omega}{2-\omega}D^{-1}\left(\frac{D}{\omega}+U\right)$$ Either as a stationary ...
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### Solution to minimization problem when variables factor in 2 analytical problems

I am asking a follow up question to this question, but I could probably have written it as an answer instead. However, I don't know if what I am doing here makes sense or is too complicated for my ...
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### Iterative solution for a minimization problem involving matrix equations

I have a real valued function $F$ for which I am looking to find its global minimum. The function is well behaved and I can obtain its Jacobian. I could also compute the Hessian but the function ...
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I have matrices ($S_0$ thought $S_N$) and I have a recurrence relation that link successive matrices together. $$S_i + S_i(aS_{i-1}^{-1})S_i=C_i+aS_{i+1}$$ We can assume for this problem that $S_0=S_N=... 0 votes 2 answers 119 views ### Eigenvectors of Laplacian I am studying introduction to Multigrid methods. In all tutorials, authors write that eigenvectors of Laplacian (1D, finite difference) are given as$w_k(x_i) = \sin(k \pi x_i),$where$x_i$is a ... 1 vote 1 answer 82 views ### Memory issues with iterative solvers Was trying to implement a poisson 2d solver using Conjugate Gradient Method, so from 10x10 grid the matrix becomes 100x100 (since we have 100 nodes to find the values at), 100x100 grid goes to ... 3 votes 1 answer 1k views ### Solving Kepler's Equation with Newton-Raphson Method Note (2022/03/07): This question is solved. Unfortunately, I'm not able to accept the correct answer by Lutz Lehmann, because I screwed up my registration and the account which posted this question is ... 2 votes 2 answers 90 views ### Possible to use Iterative FD methods to solve a transformed non square domain [matlab]? For the 2-D Poisson equation $$-(u_{xx}+u_{yy}) = f \ \ \text{where} f = 1$$ For boundary conditions $$\frac{\partial u}{\partial n} = 0 \ \text{on AB and AD}$$ $$u = 0 \ \ \ \text{on BC and CD no-... 1 vote 0 answers 73 views ### Slow convergence of Stokes solver used with the Immersed Boundary method I am using Immersed Boundary Method to simulate elastic particles in 3D Stokes flow. Specifically, one has \nabla ^2 \mathbf{u}-\nabla p + \mathbf{f}(t) = 0, \nabla \cdot \mathbf{u} \; , where \... 4 votes 0 answers 51 views ### Pass forward intermediate results during iterative optimization To investigate a counter-current flow heat exchanger while considering temperature dependent physical properties (such as specific heat c_\textrm{p,i}, heat conductivity \lambda_\textrm{i}, ... 3 votes 1 answer 470 views ### Incomplete Cholesky preconditioner for CG efficiency I am currently solving the harmonic equation using a P1 FEM discretisation. The resulting matrix A is SPD and fairly sparse so I use a preconditioned conjugate gradients (CG) solver to find a ... 3 votes 1 answer 526 views ### When do not use preconditioners for sparse linear system of equations? I'm implementing a solver of Finite Element Method, and to solve the linear system of equations I'm using gmres from MKL of Intel. Exists the option with and without a preconditioning. In what case it ... 0 votes 0 answers 84 views ### Help with debugging block GMRES I have written block version of GMRES by referring  and MATLAB implementation of gmres. I need to write it for complex matrices. My block implementation when run on single RHS is giving correct ... 6 votes 2 answers 151 views ### Integrating exponential of second degree polynomials I'm looking to compute the value of the following integral, for small values of |a|.$$u_n(a,b)=\frac{1}{2}\int_{-1}^1 x^ne^{ax^2+bx}\mathrm{d}x$$In this equation, a,b \in \mathbb{R} and n \in \... 3 votes 1 answer 326 views ### Doubt regarding GMRES(m) and preconditioned GMRES I have the two following algorithms for GMRES(m) and left preconditioned GMRES. GMRES(m) Left preconditioning I would like to know if anyone could explain why steps 10 through 12 are not used in the ... 1 vote 1 answer 32 views ### Find best matching ranges below limit in defined set of numbers I am trying to calculate the best set of cuts for some wood cutting to reduce the waste. So: Given a set of numbers, the goal is to find the best matches below a limit (size of wood beam). Example of ... 4 votes 1 answer 134 views ### Solving for a single element of a solution of a linear system I wish to solve a linear system A x =b in which A is dense but not too large, say no larger than 10\times10. However, I am not interested in the full solution vector x = [x_0, x_1, \dots], ... 4 votes 0 answers 182 views ### Stable iterative solver for complex symmetric linear systems I am interested in the iterative solution (preferably Krylov-type solvers) of a problem \boldsymbol{A}x=b, with x,b\in\mathbb{C}^{n\times1} and \boldsymbol{A}\in\mathbb{C}^{n\times n}. \... 2 votes 1 answer 58 views ### Simulate circular mold spread using cellular automata - square emerges instead I am trying to simulate the spread of mold in a petri dish using a cellular automata based approach. Thanks to the answer in my other question Stochastic cellular automata - algorithm limited by 1 ... 4 votes 1 answer 104 views ### Stochastic cellular automata - algorithm limited by 1 cell per timestep Context Let's say I am trying to model the spread of mold in a petri dish, using a stochastic cellular automata approach. The petri dish can be thought of as a grid of 1mm x 1mm squares, each called ... 0 votes 1 answer 175 views ### Best search algorithm for optimal weight factor in SOR method I had written an algorithm that searches for the optimal weight parameter to be implemented in the successive-over relaxation (SOR) method which worked cleanly by vectorizing the interval and for ... 3 votes 0 answers 189 views ### How to obtain smallest eigenvalues with Arnoldi iteration I understand that the Arnoldi iteration produces a basis which tends to include in its span the eigenvectors corresponding to eigenvalues of large magnitude (hence the analogy between the last vector ... -1 votes 1 answer 57 views ### relres in gmres MATLAB I think the relres in MATLABis the form that relres = norm(M(b-Ax))/norm(M\b),when it smaller than tol then stop the iteration. I want to know how to change relres to norm((b-Ax))/norm(b). Or use ... 7 votes 2 answers 229 views ### 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 ... 0 votes 2 answers 293 views ### Why minimizing with respect to A-norm? Assume solving the linear system A \textbf x = \textbf b, with an A so large that nothing but iterative methods may be employed. Assuming A induces a norm, I realized that it is often desired to ... 1 vote 0 answers 36 views ### Approximation in the derivation of the Arc Length method I am studying the proof of the Arc Length method in section 2.2 of this thesis. In equation (2.2) the author introduces the supplementary conditions$$ (\Delta {\bf u} + \delta {\bf u})^T \cdot (\... 3 votes 0 answers 155 views ### Python routine to calculate shape resonances of H2 I am currently doing a project in which my aim is to write a program that can be used to calculate single and multi-channel shape resonances. So I'm looking at bound states and quasi-bound states. ... -1 votes 1 answer 44 views ### Automatic selection of the SLE solver and preconditioner during simulation To simulate the physical process necessary to solve the arising systems of linear algebraic equations. The SLE matrix has a highly sparse form. There are a couple dozen non-zero elements in the string,... 1 vote 0 answers 151 views ### Upper bound on condition number in linear preconditioning I'm studying iterative methods for solving linear system, and I find the following setting in Wikipedia: Consider a matrix splitting$A = M-N$, where$A,M,N$are all symmetric and positive definite ... 2 votes 1 answer 376 views ### Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular I am reading the Chapra and Canale book on numerical methods, and was working through the chapters on solving linear systems. Now the book goes through direct methods including Gaussian Elimination, ... 1 vote 0 answers 35 views ### Problem with recursive implementation of Subspace Iteration method in Numpy I am having trouble with implementing the method of subspace iteration to find the eigenvalues and vectors of a random, symmetric matrix, A that is mxm with m = 10. ... 1 vote 0 answers 286 views ### Minimax optimization with an oracle I have an optimization problem of the following form: $$\min_y\left[\max_x f(x,y)\right].$$ It is fairly straightforward to minimize$f(x,y)$over$y$with$x$fixed, and similarly to maximize$f(x,...
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 ...