Conjugate gradient squared method

In numerical linear algebra, the conjugate gradient squared method (CGS) is an iterative algorithm for solving systems of linear equations of the form ${\displaystyle Ax=b}$, particularly in cases where calculating ${\displaystyle A^{T}}$ is impractical.^[1] The CGS method was developed as an improvement to the Biconjugate gradient method.^[2][3]

The Algorithm

The algorithm is thus:^[4]

1. Choose an initial guess ${\displaystyle x_{0}}$ 
2. Compute the residual ${\displaystyle r_{0}=b-Ax_{0}}$ 
3. Choose another residual ${\displaystyle {\tilde {r}}_{0}=r_{0}}$ 
4. Repeat the following until convergence is reached, or a maximum number of iterations is exceeded:
   1. ${\displaystyle \rho ={\tilde {r}}^{T}r}$ 

See Also

• Biconjugate gradient method
• Biconjugate gradient stabilized method

References

1. Wolfram Mathworld. "Conjugate Gradient Squared Method".
2. Mathworks. "cgs".
3. Henk van der Vorst (2003). "Bi-Conjugate Gradients". Iterative Krylov Methods for Large Linear Systems. ISBN 0-521-81828-1.
4. R. Barrett; M. Berry; T. F. Chan; J. Demmel; J. Donato; J. Dongarra; V. Eijkhout; R. Pozo; C. Romine; H. Van der Vorst (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition. SIAM.