Any Nonincreasing Convergence Curve is Possible for GMRES

Given a nonincreasing positive sequence f ( 0 ) ≥ f ( 1 ) ≥ ⋯ ≥ f ( n - 1 ) > 0, it is shown that there exists an n by n matrix A and a vector r⁰ with \| r⁰ \| = f ( 0 ) such that f ( k ) = \| rᵏ \|,\,k = 1, ⋯ ,n - 1, where rᵏ is the residual at step k of the GMRES algorithm applied to the linear system $Ax = b$, with initial residual r⁰ = b - Ax⁰. Moreover, the matrix A can be chosen to have any desired eigenvalues.