We know that if the matrix $A$ is symmetric positive definite, the FOM (full orthogonalization method) and the GMRES are theoretically equivalent to the CG (conjugate gradient) and the CR (conjugate residual).

Can you introduce some good references in the CR, especially for the convergence properties of the CR?

  • $\begingroup$ [This link][1] can help you. All the best. [1]: cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf $\endgroup$
    – Shainath
    Commented Aug 2, 2013 at 5:08
  • 1
    $\begingroup$ I always like Saad's book, although chapter 6.8. on the CR method is quite short. Quote from it: "Since the two methods exhibit typically similar convergence, the Conjugate Gradient method is often preferred." $\endgroup$
    – davidhigh
    Commented Apr 5, 2017 at 0:17

1 Answer 1


I found a quite extensive technical report by David Chin-Lung Fong and Michael Saunders, "CG versus MINRES: an empirical comparison". In this report, they say (Sec. 2.2) that for symmetric positive definite matrices, MINRES generates the same iterates as CR, and a large portion of the report is devoted to CR and analyses of its properties.

The report also mentions several another references for CR - which is in Deutch:

  • E. Stiefel, "Relaxationsmethoden bester strategie zur lösung linearer gleichungssysteme," Comm. Math. Helv., 29 (1955), pp. 157–179.

and some of CR properties are discussed and proven in

where the extension of CR to indefinite systems is also discussed.


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