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

67 questions
Filter by
Sorted by
Tagged with
45 views

### 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 ...
67 views

857 views

95 views

### The linear system in Quasi Newton method

I have implemented a Quasi Newton method for my problem, where I use the Hessian matrix approximation based approach. Hence, there is a linear system solve in every iteration. I solve the linear ...
276 views

109 views

### Can this equation be solved with the conjugate gradient method?

Let $A$ be positive-definite and $C$ diagonal positive-definite, consider the problem of solving the following equation for $\bf x$ A{\bf x}+C\begin{bmatrix} e^{x_1} \\ \vdots \\ e^{x_n} \end{...
1k views

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

Let $A\in \mathbb{R}^{n\times n}$, symmetric and positive definite. Suppose it takes $m$ units of work to multiply a vector by $A$. It is well known that performing the CG algorithm on $A$ with ...
261 views

### Why does conjugate gradient work with this nonsymmetric preconditioner?

In this previous thread the following multiplicative way to combine symmetric preconditioners $P_1$ and $P_2$ for the symmetric system $Ax=b$ was suggested: \begin{align} P_\text{combo}^{-1} :=& ...
105 views

### Conjugate gradient: the 1-norm of the residual

I am trying to solve $Ax=b$ using the conjugate gradient method. However, it is important to me to obtain a bound not only on the usual residual $||b-Ax_k||_2$ but also on the quantity $||b-Ax_k||_1$. ...
126 views

### Solve FEM matrix from coupled system

I'm developing an FEM solver for a coupled system. I have diffusion and potential equations which result in positive definite matrices for each equation, but the coupling makes the overall system ...
89 views

159 views

### Would recalculating the residual in the conjugate gradient method help?

The conjugate gradient method suffers from an accumulation of errors as it continues. For this reason it is unwise to use it as a direct solver. My question is, would it help to recalculate the ...
6k views

### How to solve this system with conjugate gradient algorithm in matlab

CG Algorithm https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!386&v=3 System of equations, the question and the example https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!387&v=3 <...
74 views

### Which non-linear conjugate gradient possess finite termination property

There are many variants of non-linear conjugate gradient method available ( Flatcher-Reeves, Polak-Rebiere, Dai-Yuan). In case of minimization of quadratic function when first search direction is ...