Milestones in the Development of Iterative Solution Methods

Algorithm 1

Standard conjugate gradient algorithm.

Given     𝐱⁽⁰⁾,𝜀              initial guess and absolute or

                        relative stopping tolerance

Set      𝐱⁽⁰⁾,𝐠=𝐴𝐱−𝐛,

       𝛿₀=𝐠^𝑇𝐠

       𝐝=−𝐠      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

       if 𝛿₁≤𝜀 then stop,  otherwise

       𝛽=𝛿₁/𝛿₀,𝛿₀=𝛿₁

       𝐝=−𝐠+𝛽𝐝      new search direction