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 , particularly in cases where computing the transpose is impractical.[1] The CGS method was developed as an improvement to the biconjugate gradient method.[2][3][4]
Background
A system of linear equations consists of a known matrix and a known vector . To solve the system is to find the value of the unknown vector .[3][5] A direct method for solving a system of linear equations is to take the inverse of the matrix , then calculate . However, computing the inverse is computationally expensive. Hence, iterative methods are commonly used. Iterative methods begin with a guess , and on each iteration the guess is improved. Once the difference between successive guesses is sufficiently small, the method has converged to a solution.[6][7]
As with the conjugate gradient method, biconjugate gradient method, and similar iterative methods for solving systems of linear equations, the CGS method can be used to find solutions to multi-variable optimisation problems, such as power-flow analysis, hyperparameter optimisation, and facial recognition.[8]
Algorithm
The algorithm is as follows:[9]
- Choose an initial guess
- Compute the residual
- Choose
- For do:
- If , the method fails.
- If :
- Else:
- Solve , where is a pre-conditioner.
- Solve
- Check for convergence: if there is convergence, end the loop and return the result
See also
- Biconjugate gradient method
- Biconjugate gradient stabilized method
- Generalized minimal residual method
References
- ^ Noel Black; Shirley Moore. "Conjugate Gradient Squared Method". Wolfram Mathworld.
- ^ Mathworks. "cgs". Matlab documentation.
- ^ a b Henk van der Vorst (2003). "Bi-Conjugate Gradients". Iterative Krylov Methods for Large Linear Systems. Cambridge University Press. ISBN 0-521-81828-1.
- ^ Peter Sonneveld (1989). "CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems". SIAM Journal on Scientific and Statistical Computing. 10 (1): 36–52. doi:10.1137/0910004. ProQuest 921988114.
- ^ "Linear equations" (PDF), Matrix Analysis and Applied Linear Algebra, Philadelphia, PA: SIAM, pp. 1–40, doi:10.1137/1.9780898719512.ch1 (inactive 31 January 2024), archived from the original (PDF) on 2004-06-10, retrieved 2023-12-18
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: CS1 maint: DOI inactive as of January 2024 (link) - ^ "Iterative Methods for Linear Systems". Mathworks.
- ^ Jean Gallier. "Iterative Methods for Solving Linear Systems" (PDF). UPenn.
- ^ Alexandra Roberts; Anye Shi; Yue Sun. "Conjugate gradient methods". Cornell University. Retrieved 2023-12-26.
- ^ 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.