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14 votes
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Quality of eigenvalue approximation in Lanczos method

The convergence behavior you are seeing is actually expected. One of things that makes the Lanczos method so interesting is that it does a good job of simultaneously converging eigenvalues at both ...
Bill Greene's user avatar
  • 6,229
14 votes
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How does the QR algorithm applied to a real matrix returns complex eigenvalues?

In a nutshell, the QR algorithm applied to a matrix $A$ is an iterative procedure that converges to the real Schur decomposition: a unitary matrix $Q$ and a matrix $R$ in block upper triangular form (...
Christian Clason's user avatar
13 votes
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When do not use preconditioners for sparse linear system of equations?

In my experience, you always need (or better use) some form of preconditioning. The type and complexity of the precondition would vary depending on the task though. From Y. Saad, Iterative Methods for ...
Anton Menshov's user avatar
  • 8,702
12 votes
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Without positive definiteness, does an iterative solver work?

No, positive definiteness (and symmetry) are only precondition to using the Conjugate Gradient method. But there are plenty of other iterative methods such as MinRes and GMRES that can be used for ...
Wolfgang Bangerth's user avatar
12 votes

Iteration counts of AMG solver changes in parallel

This is something that can happen in almost any numerical algorithm running in parallel. It's important to know that floating-point addition is not associative due to round-off errors. Thus you can't ...
Brian Borchers's user avatar
11 votes
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Markov (Chain) image generators?

I've implemented this recently, basically it counts how many times each specific colour borders another colour to make up a frequency table. To generate an image, a random colour and position are ...
Jonno_FTW's user avatar
  • 226
10 votes
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How can a CG solver solve a non positive definite sparse matrix

I highly recommend the following read: J.R. Shewchuk, "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" In short, if the matrix is non-positive definite, there is no ...
Anton Menshov's user avatar
  • 8,702
10 votes
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Is there an iterative solver for dense matrices with possible zero diagonal entries?

Iterative Krylov-subspace solvers generally only require matrix-vector products and don't care whether or where there are zeros in the matrix. In your case, unless you have other information about the ...
Wolfgang Bangerth's user avatar
9 votes
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Solve linear system with Newton-Raphson method

Yes you can do this, and it will converge in one iteration regardless of the starting value. This is because each step of Newton's method involves solving a linear system with the Jacobian of the ...
Reid.Atcheson's user avatar
9 votes
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Why MATLAB chooses the Householder in its built-in function gmres.m?

If you're using GMRES, typically you have a large stiff system. The extra work done for the householder algorithm is negligible compared to the expense of GMRES and the preconditionder. As such, we ...
EMP's user avatar
  • 2,089
8 votes
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Finite Elements: using preconditioned conjugate gradients with incomplete cholesky decomposition

While preconditioned CG with incomplete Cholesky (ICC) is reasonably straightforward to formulate mathematically, writing an efficient implementation is a non-trivial matter. Here's some of the ...
GoHokies's user avatar
  • 2,216
8 votes
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Efficient implementation of preconditioners for iterative solvers

I don't particularly care for the notation $M^{-1}$ precisely because of the confusion you find yourself in. I (and others) simply call the preconditioner $P$. The point, however, is that for ...
Wolfgang Bangerth's user avatar
8 votes

Operation count for GMRES

Performing $k$ steps of GMRES uses $O(n k^2)$ time and $O(n k)$ memory. In other words, the algorithm gets more and more expensive with each additional iteration. In theory, the algorithm terminates ...
Richard Zhang's user avatar
8 votes
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Optimality of block-Jacobi preconditioner

Consider a $2\times 2$ matrix $A$: $$ A=\left(\begin{array}{cc} 1 & 0\\ 2 & 1 \end{array}\right) $$ Singular values of $A$ are: $$ \sigma_1 = \sqrt{2}+1,\quad \sigma_2=\sqrt{2}-1 $$ resulting ...
Anton Menshov's user avatar
  • 8,702
8 votes
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What is the **contraction factor and convergence factor** of a iteration method?

These concepts are related. Let $A = M - N$ and consider the iteration $$ M x_{k+1} = N x_k + b. $$ We can write this as a mapping $\Phi : \mathbb{R}^n \to \mathbb{R}^n$ defined by $$ \Phi(y) = M^{-1}(...
Will P.'s user avatar
  • 831
7 votes
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Why do not we choose the error solution norm as an iterative method's criterion?

We can't use the criterion you show last in practice because it requires us to know what $\kappa(A)$ is. But computing the condition number is, in general, more expensive than solving a linear system. ...
Wolfgang Bangerth's user avatar
6 votes
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Fastest way to solve a sparse unsymmetric system many times

Generally speaking, for many right-hand side (RHS) problems, a direct solver is a more feasible solution for several reasons: Major computations are performed during the factorization step (which is ...
Anton Menshov's user avatar
  • 8,702
6 votes
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Set of linear ordinary differential equations with a mass matrix

B is known as a mass matrix. There are many methods which can directly solve with mass matrices. The main ones are implicit methods: implicit Runge-Kutta methods ...
Chris Rackauckas's user avatar
6 votes

For which problems Krylov subspace methods are preferred over multigrid methods?

The answer depends somewhat on the discretization; for example, some boundary integral discretizations result in very well-conditioned matrices, for which a Krylov solver works just fine without ...
Jesse Chan's user avatar
  • 3,152
6 votes

Why my parallel code using MPI is much slower than the serial one?

The first thing you need to ask yourself: is your problem big enough that the overhead of MPI messaging is less than the work that you save. Your problem size is 10k which is small, but on the other ...
Victor Eijkhout's user avatar
6 votes
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Iterative linear solver for "ugly" saddle point system

You should stick with GMRES, it is the only method that is essentially guaranteed to get a solution here. The real problem appears to be you need a better preconditioner. You could try sticking with ...
EMP's user avatar
  • 2,089
6 votes
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Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular

It's a complicated question, which is why I've recorded a whole bunch of video lectures on the topic :-) Take a look at lectures 34 and following here: https://www.math.colostate.edu/~bangerth/videos....
Wolfgang Bangerth's user avatar
6 votes

Why minimizing with respect to A-norm?

Wolfgang Bangerth's answer already says almost everything, but another subtle detail is that GMRES/MINRES minimize the norm of the residual, i.e., $\|Ax_k-b\|$, while CG minimizes the (A-)norm of the ...
Federico Poloni's user avatar
6 votes

Eigenvectors of Laplacian

They're on Wikipedia, for instance, in a page with the slightly unclear name of "Eigenvalues and eigenvectors of the second derivative".
Federico Poloni's user avatar
6 votes

Is there a way to generate a matrix-free decomposition for a matrix-free operator?

This is not efficient, primarily because (i) the factors of sparse matrices are, in general, not sparse themselves, and (ii) when computing matrix factorizations, you don't just need $A_{ij}$ but you ...
Wolfgang Bangerth's user avatar
6 votes

Approximately, at any given time, what proportion of the world's total HPC resources are dedicated towards inverting matrices?

The number is almost certainly unverifiable because every HPC system's overall user community is different and because there are so many systems around. I think that the number is vastly smaller than ...
Wolfgang Bangerth's user avatar
5 votes

Regarding impractical usage of direct solvers of linear systems

The "100,000 unknowns" rule-of-thumb applies to sparse matrices rather than dense ones. A naive direct solver, which doesn't take advantage of sparsity at all, could in principle have $O(n^3)$ ...
Daniel Shapero's user avatar
5 votes
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Stopping criteria in iterative methods for solving nonlinear equations

It can happen in principle that the condition on $f$ will never be satisfied, if, for example, $|f'(x_*)|u>\mathit{tolfun}$, where $u$ is the unit roundoff. When I tried your example with $K=10^8,...
Kirill's user avatar
  • 11.4k
5 votes

MATLAB: code for restarted gmres

Technically, the template Matlab code provided at Netlib already calculates the obtained residual at every iteration. It's just not recorded in the way that this is an output of the subroutine. After ...
Anton Menshov's user avatar
  • 8,702
5 votes

Is it necessary to invert precondition matrix for iterative solver?

Based on your comments, it looks like your main difficulty is how having an LU factorisation for a given matrix $M$ makes it easy to find a vector $\mathbf{y}$ such that $M\mathbf{y}=\mathbf{b}$ for a ...
origimbo's user avatar
  • 2,259

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