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Conjugate gradient squared method

Algorithm for solving matrix-vector equations From Wikipedia, the free encyclopedia

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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]

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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]

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Algorithm

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The algorithm is as follows:[9]

  1. Choose an initial guess
  2. Compute the residual
  3. Choose
  4. For do:
    1. If , the method fails.
    2. If :
    3. Else:
    4. Solve , where is a pre-conditioner.
    5. Solve
    6. Check for convergence: if there is convergence, end the loop and return the result
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