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 ...

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80 views

Test set for linear solvers

Lets assume I have a iterative linear system solver, e. g. this one. Whats the typical approach on verifying and testing this kind of solvers? Is there a standard test set of linear systems one ...
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2answers
106 views

What are some reasons that Conjugate Gradient iteration does not converge?

I would greatly appreciate it if you could share some reasons the Conjugate Gradient iteration for Ax = b does not converge? My matrix A is symmetric positive definite. Thank you so much! Edit with ...
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1answer
124 views

How to test convergence of an algorithm for constrained optimization

I am applying an iterative method (projected newton) to an optimization problem. Theoretically, the method should converge linearly. I would greatly appreciate it if you could share how should I test ...
4
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1answer
54 views

Iteratively refine bounds on exp for Metropolis criterion

In Monte Carlo simulations, using the Metropolis criterion, one often has to compare a random number $a$, $0 \leq a < 1$, to the Boltzmann distribution $exp(-\beta\Delta E)$, where $\Delta E$ is ...
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2answers
62 views

Performance metrics to compare initial-boundary value problem solutions

I am comparing the performance several finite difference methods of solving an initial-boundary value problem. There are several dimensions to this comparison: Number of cells Number of timesteps ...
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3answers
248 views

Eigenvalues of a sparse banded nonsymmetric matrix from an elliptic operator

I have a sparse matrix coming from the discretization of a 3D elliptic PDE. The matrix is banded with seven non-zeros diagonals. The sparsity pattern of the matrix looks like this (the actual matrix ...
4
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1answer
133 views

Linear equation system: Direct solver works, iterative solver does not

I have to solve for x in b = A*x, where a is sparse. This works fine with Matlab's mldivide: x = A \ b. Since I will have to use an iterative algorithm for very large A, I'm currently testing Matlab's ...
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1answer
95 views

Finding null vectors of a parameter-dependent matrix

I have dense complex matrices $M(z)$ in which each element $M_{ij} = M_{ij}(z)$ depends on a complex parameter $z$. I need to find $z$ such that the matrix $M$ gets singular, i.e. I am looking for ...
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1answer
66 views

Finding Interior eigenvalues using Davidson algorithm

Is it possible to find interior eigenvalues closer to some lambda using Davidson method. I was searching online but found that most people use Jacobi-Davidson method for that. Thanks
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2answers
125 views

What sparse linear programming solver it is better to use?

I have the following LP problem: $$ \min \limits_{\varepsilon, x_{1}, \ldots, x_{n}}f(\varepsilon, x_{1}, \ldots, x_{n}) = \varepsilon \;\;\;\;\; s.t. \;\;C x \geq 0, \;\; x_{i}^{0} - \varepsilon ...
2
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0answers
62 views

Modal analysis of structure with aerodynamic damping

I'm using modal decomposition to predict the steady state response of a beam structure to harmonic loading. The structure itself is very lightly damped, but we know from experiments that the ...
7
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3answers
284 views

Sort of problems where SOR is faster than Gauss-Seidel?

Is there any simple rule of thumb to say if it is worth to do SOR instead of Gauss-Seidel? ( and possible way how to estimate realxation parameter $\omega$) I mean just by looking on the matrix, or ...
2
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2answers
257 views

What is the difference between “Newton-type” and “Newton-like” iteration?

Is there any clear classification between different iterative methods? What is the difference between Newton-type and ...
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0answers
72 views

full rank update to cholesky decomposition for multivariate normal distribution

This question is a specialization of full rank update to cholesky decomposition, to which I hope to get a more positive answer. When calculating the minus log of the multivariate normal distribution, ...
2
votes
1answer
114 views

full rank update to cholesky decomposition

Let $A$ be a real, symmetric, positive definite matrix. It has at least 500 rows, possibly much more. I compute its Cholesky decomposition, which allows me to calculate $det(A)$ $A^{-1}X$ for some ...
4
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0answers
105 views

Conjugate residual/gradient convergence checking in practice

Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
5
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1answer
212 views

Inverse iteration to find the null singular vector of a rank-deficient matrix

I have an $n \times n$ unsymmetric matrix $A$ that results from the discretization of an ill-posed Poisson problem, and thus is rank-deficient with null space of dimension one. I want to compute just ...
4
votes
1answer
121 views

Factorization for reweighted least squares

I am solving a problem using an iteratively-reweighted least squares method: http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares Essentially this requires solving a number of ...
6
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1answer
63 views

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 ...
5
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3answers
120 views

Newton's method for a given polynomial

Let $f(x)=\frac{1}{5}x^5+\frac{1}{3}x^3+x-1$ Show that $f$ has only one zero $r$ in interval $(0,1)$ To find approximation of $r$ we apply Newton's method ...
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0answers
63 views

Increase convergence of non-linear equations resulting from ODEs

I am trying to solve a set of couple ODEs: $V_l(r) - r W_l(r) - f1(r) W_l' = 0\tag 1$ $r^2 h''_l(r) + f2 r h_l'(r) + f3 h_l(r) - f4 U_l(r) = 0 \tag 2$ $\kappa (U_l + h_l) + V_{l+1} + W_{l+1} = ...
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1answer
382 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 ...
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2answers
222 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 ...
4
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0answers
129 views

Understanding the meaning of Computational Order of Convergence

I am a postgraduate student with interest in numerical methods for solving nonlinear systems of equations. I have read some papers that discussed about 'computational order of convergence' for some ...
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2answers
300 views

Convergence/stagnation of BiCGStab(l)

I am solving 3D time-harmonic Maxwell FDFD problems (which result in huge sparse linear systems) using BiCGStab(l). I have tried out a bunch of different methods and for my specific use case, it seems ...
7
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2answers
139 views

Hartree Fock iteration problem

I am writing a program to compute the ground state energy for any closed shell atom using Hartree Fock Roothaan method, with GTO basis. The code works for the simplest case, the helium, but it fails ...
5
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1answer
70 views

Can we compare the speed of convergence of two different iteration methods of same order looking at their error estimates?

I have a two iterative method for approximating the inverse of given square matrix $A$ whose error terms are given as follows Error estimate of method $1$: $\lVert A^{-1} - X_{k}\rVert \leq ...
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0answers
44 views

Question about Logarithmic convergence

I examine the following recursion $X_{n+1}=\frac{t_n}{t_{n+1}}X_n+\frac{Y_n}{t_{n+1}}$ where $X_n,Y_n$ are positive finite random variables and $t_n$ the time. I have shown that $\lim_{n \to \infty} ...
13
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1answer
239 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 ...
2
votes
1answer
185 views

Is it possible to ensure global convergence of a fixed point iteration?

Suppose I have a fixed point iteration of the form $$x_{n+1}=f(x_n).$$ Suppose further that after some initial testing, I found that it does not converge to an a priori known fixed point $x^*$. I ...
2
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1answer
247 views

Convergence of GMRES

From what I understand the GMRES method is (using Arnoldi Iterations/Modified Gram-Schmidt): The first vector of the Krylov subspace span of A is the normalized vector $\frac{\vec b - A\vec x_0} {|| ...
3
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1answer
130 views

Jacobi Iteration diverges?

I'm working through a problem in a textbook as follows: "Consider the $d \times d$ Toeplitz matrix $$ A = \left[ \begin{array}{ccccc} 2 & 1 & 0 & \cdots & 0 \\ -1 &2 ...
2
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0answers
57 views

What are Implications of Commutative Diagrams?

This question may be too broad. But I really want to know some concrete explanations. I often find various commutations appear here and there, which concerns the application order of two operators. ...
8
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2answers
185 views

Looking for an algorithm that allocates climbing hold colors to wall sectors

I posted this question earlier on stackoverflow, where it was closed as off-topic. I hope it survives here. I our climbing gym, the routes need to be re-set from time to time. The following rules ...
5
votes
2answers
92 views

Levenberg optimizer halts quickly when given more variables, or fewer constraints

I'm using the g2o C++ optimization library to refine a GPS trajectory using accelerometer data. The program uses a Levenberg-Marquardt optimizer over data points representing the position and ...
5
votes
4answers
541 views

Fastest polynomial root finder for a given accuracy

I am looking for a very fast polynomial root finder, hopefully with a matlab code. I don't need very accurate results, 2-3 decimal places would be fine. Also the method should be able to optionally ...
10
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3answers
1k 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 ...
0
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2answers
163 views

Checking for error in conjugate gradient algorithm

What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
7
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1answer
142 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 ...
4
votes
2answers
130 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 ...
6
votes
2answers
902 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). ...
3
votes
2answers
142 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 ...
11
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1answer
229 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
565 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, ...
5
votes
4answers
687 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
231 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 ...
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2answers
2k 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
votes
5answers
320 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. ...
7
votes
2answers
259 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
205 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 ...