Advantages of cg method
•Theoretically, the solution is found in a finite number of steps, and, if stopped early, may already give a useful approximate solution
•It is a strategical procedure taking into account in each step the information obtained in all the previous steps
•At each step, the value of the error function f(x) is diminished. So also is the distance of the estimate xi from the solution x
•The method allows to take advantage of parallel computing
1.8
In his invited address at the 1954 International Mathematical Congress in Amsterdam, Stiefel had concluded that: ,,Among all scalar iteration algorithms, the method of conjugate gradients is the best strategy,“
In the beginning the method attracted a lot of interest. In the 1960s some unsuccessful applications on large problems due to an unsophisticated use almost caused its passing into oblivion. Work by John Reid in the early 1970s renewed attention to the algorithm. As the rich program of this symposium shows it has since then been the subject of many research efforts. Furthermore it is nowadays together with incomplete Cholesky as preconditioner  a standard algorithm for solving linear systems involving large, sparse,  positive definite matrices in universities and industry.
Its importance has been recognized  about two years ago by selecting Krylov Subspace methods (CG is the mother of all of them)  as one of the "Top 10 Algorithms of the (twentieth) Century" by Computing in Science & Engineering (CS&E) magazine, a joint publication of the IEEE Computer Society and the American Institute of Physics.