Summary form only given, as follows. Presents a robust and flexible algorithm for the general calculation of the off-axis expansion solution to Laplace's equation. The limiting factor in application of the technique is shown to be a series truncation error and not errors in calculating the numerical derivatives as previously assumed. This remains true even for single precision calculations. Application of the algorithm to the accurate computation of off-axis values of arbitrary magnetic fields from on-axis or coil data is presented. For a single ideal wire, magnetic field accuracies of better than 0.01% of the exact elliptic integral solution can be obtained out to approximately 70-80% of the wire radius. Accuracy improves dramatically (usually by many orders of magnitude) for radii closer to the axis. Results are also shown for thin current disks, thin solenoids and thick coils. The accuracy of field calculations for these sources is substantially the same as for the ideal wire. With these basic building blocks it is possible to construct equivalents to many magnetic systems, including systems containing permanent magnets. Extension of the algorithm to multi-source magnetic systems and its use in both the synthesis and optimization of such systems are discussed and results presented. The utility of this technique is not as a replacement for more general codes, but as a rapid and flexible adjunct. The compactness of the method makes it ideal for direct incorporation into other programs.