TABLE OF CONTENTS The conjugate gradient iteration is the "original" Krylov subspace iteration, the most famous of these methods and one of the mainstays of scientific computing. Discovered by Hestenes and Stiefel in 1952, it solves symmetric positive definite systems of equations amazingly quickly if the eigenvalues are well distributed.
As in the last two lectures, let 
be real and symmetric, and suppose we wish to solve a nonsingular system of equations Ax = b, with exact solution x * = A ?1 b. Let 
denote the nth Krylov subspace (33.5) generated by b,
One approach based on this Krylov subspace would be to solve the system by GMRES. As described in Lecture 35, this would mean that at step n, x * is approximated by the vector 
that minimizes ? r n ? 2, where r n = b ? Ax n. Actually, the usual GMRES algorithm does more work than is necessary for minimizing ? r n ? 2. Since A is symmetric, faster algorithms are available based on three-term instead of ( n+1)-term recurrences at step n. One of these goes by the names of conjugate residuals or MINRES ("minimal residuals").
These methods, at least when constructed to apply to both definite and indefinite matrices, involve certain complications. Rather than describe them, we turn directly to the simpler and more important positive definite case.
