A popular krylov subspace method for solving linear systems of equations, particularly those that exhibit symmetric positive definiteness.

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### BFGS vs. Conjugate Gradient Method

What considerations should I be making when choosing between BFGS and conjugate gradient for optimization? The function I am trying to fit with these variables are exponential functions; however, the ...
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### What is the principle behind the convergence of Krylov subspace methods for solving linear systems of equations?

As I understand it, there are two major categories of iterative methods for solving linear systems of equations: Stationary Methods (Jacobi, Gauss-Seidel, SOR, Multigrid) Krylov Subspace methods (...
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### Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method. In general, CG is preferable choice in many cases of SPD (symmetric-positive-definite) because it requires less ...
811 views

### Calculating determinant while solving $Ax=b$ using CG

I am solving $Ax=b$ for a huge sparse positive definite matrix $A$ using the conjugate gradient (CG) method. It is possible to compute the determinant of $A$ using the information produced during the ...
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For a project, I have to implement these two methods and compare how they perform on different functions. It looks like the conjugate gradient method is meant to solve systems of linear equations of ...
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### 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 $x_*$ ...
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### Hessian-free and Truncated Newton methods

In this paper on Deep Learning for Machine Learning, the approach is referred to as Hessian-free method. That is because the Hessian is never computed explicitly. Instead, the product of the Hessian ...
441 views

### PETSc Krylov Subspace and nullspace

How is the nullspace correction implemented in the PETSc conjugate gradient solver? I searched the source code and documentation, but could not find references on the actual implementation. For ...
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### Usage of DifferentiableMultivariateFunction and NonLinearConjugateGradientOptimizer

I try to use the matlab fminunc functionality in java and found the optimisation functions in commons-math. I have absolutely no clue how to use them right because the example tells nothing about the ...
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### Blowup of error in Conjugate Gradient method with periodic Dirichlet Poisson matrix

My problem is that the L2-Norm of the residual for the periodic Poisson matrix $P$ is initially decreasing but starts to blow up after a certain number of iterations. The blowup happens earlier the ...
68 views