Recurrence relations of the form O-rPr+1 = brpr + CrPr-1 are examined in two cases: (A) oscillatory systems, for which b% + 4arcr < 0; (B) monotonie systems, for which bl¡. + 4arcr > 0. In both cases, a posteriori methods are supplied for constructing strict and realistic error bounds in 0(r) arithmetic operations. A priori bounds, also requiring 0(r) arithmetic operations, are supplied in Case B. Several illustrative numerical examples are included.