Cg-method:
select pi such that
ri* rk = 0 i ≠ k orthogonal and
pi*A pk= 0 for all i,k i ≠ k
pi conjugate gradients
select pi such that
ri* rk = 0 i ≠ k orthogonal and
pi*A pk= 0 for all i,k i ≠ k
pi conjugate gradients
1.7
| Stiefel
called this method initially „n-Schrittverfahren“ and the n vectors p0 ..,
pn-1.weight vectors. They are conjugate, (pi*
Apk) = 0 i ≠ k.
According to the annual report of IAM for the "Studienjahr 1950/51"
Stiefel elaborated this method during that period (October 1950 to July
1951). Late in June or early in July 1951, M. Hestenes with the cooperation
of J. B. Rosser, G. Forsythe and L. Paige of INA devised what he called a
Conjugate Gradient Method for solving systems of linear equations. He
formulated three versions of this routine. Stiefel was invited about that
time to give a talk on solving systems of linear equations at an INA
symposium which took place August 23-25, 1951 . The librarian gave Stiefel on his arrival at
INA a paper describing Hestenes' work. According to Hestenes' recollection of
the events shortly thereafter Stiefel came to his office with this paper in
hand and said, "Look! This is my talk." Stiefel had invented the
same algorithm from a different point of view which I have given here. He
looked upon it as a relaxation method whereas Hestenes viewed it as a
gradient routine on conjugate subspaces, a topic on which he had worked
already in 1936 when he developed an algorithm for constructing a set of
mutually conjugate directions in Euclidian space for the purpose of studying
quadric surfaces. Because they had devised the same routine independently at
about the same time, Stiefel and Hestenes decided to write a joint paper
describing the routine and its properties. |