We present a new technique to solve complex sets of photochemical rate equations that is applicable to global modeling of the troposphere and stratosphere. The approach is based on the concept of families of species, whose chemical rate equations are tightly coupled. Variations of species concentrations within a family can be determined by inverting a linearized Jacobian matrix representing the family group. Since this group consists of a relatively small number of species the corresponding Jacobian has a low order (a minimatrix) compared to the Jacobian of the entire system. However, we go further and define a super-family that is the set of all families. The super-family is also solved by linearization and matrix inversion. The resulting Super-Family Matrix Inversion (SFMI) scheme is more stable and accurate than common family approaches. We discuss the numerical structure of the SFMI scheme and apply our algorithms to a comprehensive set of photochemical reactions. To evaluate performance, the SFMI scheme is compared with an optimized Gear solver. We find that the SFMI technique can be at least an order of magnitude more efficient than existing chemical solvers while maintaining relative errors in the calculations of 15% or less over a diurnal cycle. The largest SFMI errors arise at sunrise and sunset and during the evening when species concentrations may be very low. We show that sunrise/sunset errors can be further minimized through a careful treatment of photodissociation during these periods; the nighttime deviations are negligible from the point of view of acceptable computational accuracy. The stability and flexibility of the SFMI algorithm should be sufficient for most modeling applications until major improvements in other modeling factors are achieved. In addition, because of its balanced computational design, SFMI can easily be adapted to parallel computing architectures. SFMI thus should allow practical long-term integrations of global chemistry coupled to general circulation and climate models, studies of interannual and interdecadal compositional variability, simulations of past multidecadal chemical trends owing to anthropogenic emissions, long-term forecasting of changes in atmospheric constituents due to projected emissions, and sensitivity analyses for a wide range of physical and chemical parameters. ¿ 1999 American Geophysical Union