TABLE OF CONTENTS In the previous section we derived the CG algorithm as a special case of FOM for SPD matrices. Similarly, a new algorithm can be derived from GMRES for the particular case where A is Hermitian. In this case, the residual vectors should be A-orthogonal, i.e., conjugate. In addition, the vectors Ap i, i = 0, 1, , are orthogonal. When looking for an algorithm with the same structure as CG, but satisfying these conditions, we find the conjugate residual (CR) algorithm. Notice that the residual vectors are now conjugate to each other, hence the name of the algorithm.
Compute r 0 := b ? Ax 0, p 0 := r 0
For j = 0, 1, , until convergence, Do
? j := ( r j, Ar j)/( Ap j, Ap j)
x j+1 := x j + ? jp j
r j+1 := r j ? ? jAp j
? j := ( r j+1, Ar j+1)/( r j, Ar j)
p j+1 := r j+1 + j p j
Compute Ap j+1 = Ar j+1 + ? j Ap j
EndDo
Line 8 in the above algorithm computes Ap j+1 from Ar j+1 without an additional...
