Conjugate Gradient Solver Introduction

Linear solvers are super important as they are used in all major industries. Most engineering problems can be turned into one or more linear equation systems. Typically, the matrix formed in those systems is large in dimension and highly sparse in data pattern. Iteration methods such as the preconditioned conjugate gradient solver are a type of indirect solutions to these linear systems with very high efficiency.

Conjugate Gradient Algorithm

For linear system \(Ax=b\) with given preconditioner matrix \(M\), the preconditioned conjugate gradient method is shown in the following equations.

(1)\[\begin{split}x_0 &= 0 \\ r_0 &= b-Ax_0 \\ z_0 &= M^{-1}r_0 \\ \rho_0 &= r_0^Tz_0 \\ \beta_k &= 0 \\\end{split}\]

\[\begin{split}while\ k<maxIter\ &AND\ ||r_{k}|| > tol*||b|| \\ p_{k} &= z_{k} + \beta_{k-1}p_{k-1} \\ \alpha_k&=\frac{\rho_k}{p_k^TAp_k} \\ x_{k+1} &= x_k+\alpha_kp_k \\ r_{k+1} &= r_k+\alpha_kAp_k \\ z_{k+1} &= M^{-1}r_{k+1} \\ \rho_{k+1} &= r_{k+1}^Tz_{k+1} \\ \beta_k &= \frac{\rho_{k+1}}{\rho_k} \\ k &= k+ 1 \\\end{split}\]