So far, we have described preconditioners in only one of two classes: those that approximate the coefficient matrix, and where linear systems with the preconditioner as coefficient matrix are easier to solve than the original system. Polynomial preconditioners can be considered as members of the second class of preconditioners: direct approximations of the inverse of the coefficient matrix.
Suppose that the coefficient matrix of the linear system is normalized to the form , and that the spectral radius of is less than one. Using the Neumann series, we can write the inverse of as , so an approximation may be derived by truncating this infinite series. Since the iterative methods we are considering are already based on the idea of applying polynomials in the coefficient matrix to the initial residual, there are analytic connections between the basic method and polynomially accelerated one.
Dubois, Greenbaum and Rodrigue  investigated the relationship between a basic method using a splitting , and a polynomially preconditioned method with
The basic result is that for classical methods, steps of the polynomially preconditioned method are exactly equivalent to steps of the original method; for accelerated methods, specifically the Chebyshev method, the preconditioned iteration can improve the number of iterations by at most a factor of .
Although there is no gain in the number of times the coefficient matrix is applied, polynomial preconditioning does eliminate a large fraction of the inner products and update operations, so there may be an overall increase in efficiency.
Let us define a polynomial preconditioner more abstractly as any polynomial normalized to . Now the choice of the best polynomial preconditioner becomes that of choosing the best polynomial that minimizes . For the choice of the infinity norm we thus obtain Chebyshev polynomials, and they require estimates of both a lower and upper bound on the spectrum of . These estimates may be derived from the conjugate gradient iteration itself; see §.
Since an accurate lower bound on the spectrum of may be hard to obtain, Johnson, Micchelli and Paul  and Saad  propose least squares polynomials based on several weight functions. These functions only require an upper bound and this is easily computed, using for instance the ``Gerschgorin bound'' ; see [.4]Va:book. Experiments comparing Chebyshev and least squares polynomials can be found in Ashby, Manteuffel and Otto .
Application of polynomial preconditioning to symmetric indefinite problems is described by Ashby, Manteuffel and Saylor . There the polynomial is chosen so that it transforms the system into a definite one.