  We develop the Akhiezer iteration, a generalization of the classical
Chebyshev iteration, for the inner product-free, iterative solution of
indefinite linear systems using orthogonal polynomials for measures supported
on multiple, disjoint intervals. The iteration applies to shifted linear solves
and can then be used for efficient matrix function approximation. Using the
asymptotics of orthogonal polynomials, error bounds are provided. A key
component in the efficiency of the method is the ability to compute the first
$k$ orthogonal polynomial recurrence coefficients and the first $k$ weighted
Stieltjes transforms of these orthogonal polynomials in $\mathrm{O}(k)$
complexity using a numerical Riemann--Hilbert approach. For a special class of
orthogonal polynomials, the Akhiezer polynomials, the method can be sped up
significantly, with the greatest speedup occurring in the two interval case
where important formulae of Akhiezer are employed and the Riemann--Hilbert
approach is bypassed.
