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 technieques 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.