Conjugate gradient and sparse matrix techniques are utilized in the solution of a geomagnetic inverse problem. Global crustal data sets collected from low-earth orbit are quickly inverted (using a design matrix approach) or continued to a common altitude (using a normal matrix approach) even when using parameterizations of 10000 or more dipoles. The sparsity results from the rapid decay of the magnetic field with distance from the dipole. Iterative techniques such as the conjugate gradient save computer time and space when compared to more direct approaches using the Householder transformation, thus allowing problems that were intractable to all but the largest supercomputers to be performed on workstations of only moderate power. ¿ American Geophysical Union 1996