# Tag Info

## Hot answers tagged conjugate-gradient

15 votes
Accepted

### Adaptive gradient descent step size when you can't do a line search

I'll begin with a general remark: first-order information (i.e., using only gradients, which encode slope) can only give you directional information: It can tell you that the function value decreases ...
14 votes
Accepted

### What are the differences between the different gradient-based numerical optimization methods?

First, you should forget about solving linear equations for the moment -- that's a different context from optimization, and trying to consider both on equal footing will only lead to confusion. Why ...
11 votes

### What is required of the objective function in order to use Gauss Newton method?

Just to clarify notation, I'll be discussing the Gauss-Newton method for the problem $\min \phi(x)=(1/2) \| F(x) \|_{2}^{2}$ with the search direction $p$ computed as the solution to the linear ...
10 votes

### Which C++ linear algebra library is probably the fastest on solving huge sparse [square matrix] linear system?

Eigen 3 is a nice C++ template library some of whose routines are parallelized. c.f. Eigen documentation The parallelization is OMP only, so if you intend to parallelise using MPI (and OMP) it is ...
• 341
10 votes
Accepted

### 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 ...
• 8,521
9 votes

### What is the worst case complexity of Conjugate Gradient?

The answer is a resounding yes. The convergence rate bound of $(\sqrt{\kappa}-1) / (\sqrt{\kappa}+1)$ is sharp over the set of symmetric positive definite matrices with condition number $\kappa$. In ...
• 2,435
9 votes
Accepted

You first have to make sure that your system is solvable: this happens iff the right-hand side $b$ is orthogonal to the kernel of $A$. If $A$ has a dimension-1 kernel spanned by $v$, you need to have $... • 9,666 7 votes ### BiCGSTAB convergence First of all, size 40 is pretty much microscopic for the sort of purposes that BiCGstab was invented. I assure you, this method is great for matrices of sizes in the millions and beyond. Then: even ... • 1,295 6 votes Accepted ### CG question: is symmetry always necessary? The CG method works by producing in every iteration a vector$x^k\in\mathbb{R}^n$that solves the minimization problem (assuming$x^0=0$for simplicity) $$\min_{x\in \mathcal{K}_k} \frac12(x-x^*)^TA(x-... 6 votes Accepted ### Caveats of Hessian free method The approximation error is \mathcal{O}(\varepsilon), where \varepsilon is the parameter in the finite difference calculation. It will be good to know how this approximation affects the number of ... 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 ... • 9,666 5 votes Accepted ### How to verify solution to pre-conditioned linear systems solver? You've started with a singular linear system of equations Ax=b. As a practical matter, it's unlikely that b lies exactly in the range of A, so at best you can find a least squares solution that ... 5 votes Accepted ### Relation between conjugate gradient method and finite elements method The methods are unrelated. The finite element method is a way to convert a partial differential equation into a finite dimensional problem ("discretization") so it can be solved on a computer. The ... • 52.2k 5 votes Accepted ### Why minimizing with respect to A-norm? In some sense it doesn't matter: In finite dimensions, all norms are equivalent, so if an algorithm conveniently has the property that proving convergence in one norm is easy, then that's what people ... • 52.2k 4 votes ### Why does conjugate gradient work with this nonsymmetric preconditioner? In short, orthogonalization of the Krylov vectors occurs with respect to the operator, but not with respect to the preconditioner. Alright, so say we want to solve Ax=b with preconditioner B. ... • 717 4 votes Accepted ### Hessian-free and Truncated Newton methods It looks like this paper is combining Hessian-free with truncated Newton method. Yes, it is. ...the approach is referred to as Hessian-free method. That is because the Hessian is never computed ... 4 votes ### Solving the Poisson equation with Neumann Boundary Conditions - Finite Difference, BiCGSTAB One of the standard ways to handle singular problems like Poisson with Neumann boundary conditions which lead to a sinuglar problem Ax=b is to make the obtained solution orthogonal to the kernel (... • 254 4 votes Accepted ### Why the iteration steps become twice if the step size reduces half for CG methods? Let's say the condition number of the original matrix is k_1, and the one after refining the mesh is k_2=4k_1. You then want to compare the number of iterations necessary to reach a tolerance \... • 52.2k 4 votes Accepted ### What's wrong with the **PCG and MINRES** in matlab? While very similar, each method is slightly different and you should definitely take this into account. The GMRES method is the simplest, it will construct an orthogonal basis for \mathcal{K}_k(A,b)... • 1,254 4 votes ### In which cases does the nonlinear conjugate gradient method take more than n steps? The nonlinear conjugate gradient method will converge for a quadratic function in N steps at most. You do not have a quadratic function, and have no guarantees on convergence. It can be helpful to ... • 2,037 3 votes Accepted ### The linear system in Quasi Newton method It is a well-known theorem that conjugate gradients is guaranteed to converge (in exact arithmetic) within r+1 steps for an identity-plus-rank-r matrix, regardless of its size. In finite precision,... • 2,435 3 votes ### Boundary conditions in conjugate gradient method for poisson's equation Neumann boundary conditions will show up in the right hand side of your problem, and so you can just start your iteration with any vector you want -- it doesn't have to satisfy any particular boundary ... • 52.2k 3 votes ### Would recalculating the residual in the conjugate gradient method help? This tutorial talks about recalculating the residual every 50 iterations to mitigate round-off errors. • 656 3 votes ### What happens when I use a conjugate gradient solver with a symmetric positive semi-definite matrix? To at least partially answer the question: You have to make sure the system is solvable in the first place. As mentined in the accepted answer, "the system Ax=b is solvable iff b \bot Ker(A)&... 3 votes ### Why does smoothed aggregation multigrid method used as preconditioner in conjugate gradient slows down the solution time? If I understand your question correctly you're solving a linear elasticity problem using conjugate gradient and it's preconditioned with a preconditioned AMG solver? It seems to me that this may be ... • 2,037 3 votes ### CG without division by 0 in a solution Either of p_k or r_k being zero implies exact convergence in exact arithmetic, so that is never a problem. Conjugate gradient was used as a direct solver for linear systems of equations much ... • 2,321 3 votes ### When will the Orthomin/CG iteration fails Your statement is not correct, I believe. The equivalent condition for the real case is that the iteration will fail if there is a vector x\in{\mathbb R}^n so that x^T A x = 0, which is equivalent ... • 52.2k 3 votes ### Reason for why apparent acceleration of algebraic multigrid solve by addition of positive definite diagonal matrix The computation in your update does most of the work towards a solution. You just need to note that \frac{\varepsilon_1}{\varepsilon_2} \leq \frac{\max D_{ii}}{\min D_{ii}} = \kappa(D), and that$$ ... • 9,666 2 votes ### Estimate extreme eigenvalues with CG This question is very related to another SE question on condition number estimates which contains relevant materials. As @Jack_Poulson mentioned, the following paper contains a detailed discussion on ... 2 votes ### Can this equation be solved with the conjugate gradient method? The trick is to view your equation as the first-order optimality conditions of an unconstrained optimization problem$\min_x f(x)$. Recall that the first-order optimality conditions are just$\nabla f(...
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