Any Nonincreasing Convergence Curve is Possible for GMRES

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