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We consider the dissipative spin-orbit problem in Celestial Mechanics, whichdescribes the rotational motion of a triaxial satellite moving on a Keplerianorbit subject to tidal forcing and "drift".  Our goal is to construct quasi-periodic solutions with fixed frequency,satisfying appropriate conditions.  With the goal of applying rigorous KAM theory, we compute such quasi-periodicsolution with very high precision. To this end, we have developed a veryefficient algorithm. The first step is to compute very accurately the returnmap to a surface of section (using a high order Taylor's method with extendedprecision). Then, we find an invariant curve for the return map using recentalgorithms that take advantage of the geometric features of the problem. Thismethod is based on a rapidly convergent Newton's method which is guaranteed toconverge if the initial error is small enough. So, it is very suitable for acontinuation algorithm.  The resulting algorithm is quite efficient. We only need to deal with a onedimensional function. If this function is discretized in $N$ points, thealgorithm requires $O(N \log N) $ operations and $O(N) $ storage. The mostcostly step (the numerical integration of the equation along a turn) is trivialto parallelize.  The main goal of the paper is to present the algorithms, implementationdetails and several sample results of runs.  We also present both a rigorous and a numerical comparison of the results ofaveraged and not averaged models.

Efficient and accurate KAM tori construction for the dissipative spin-orbit problem using a map reduction