Milestones in the Development of Iterative Solution Methods

Algorithm 2

Preconditioned conjugate gradient algorithm.

Given       𝐱⁽⁰⁾,𝜀         initial guess and

                  stopping tolerance

Set        𝐱⁽⁰⁾,𝐠=𝐴𝐱−𝐛,

           𝐡=[𝐵]⁻¹𝐠

    𝛿₀=𝐠^𝑇𝐡

    𝐝=−𝐡        initial search direction

Repeat    until convergence

    𝐡=𝐴𝐝

    𝜏=𝛿₀/(𝐝^𝑇𝐡)

    𝐱=𝐱+𝜏𝐝       new approximation

    𝐠=𝐠+𝜏𝐡       new (iterative) residual

    𝛿₁=𝐠^𝑇𝐠

    𝐡=[𝐵]⁻¹𝐠       new pseudoresidual

    𝛿₁=𝐠^𝑇𝐡

    if 𝛿₁≤𝜀 then stop

    𝛽=𝛿₁/𝛿₀,𝛿₀=𝛿₁

    𝐝=−𝐡+𝛽𝐝        new search direction