Least Squares Solution by Diagonal Partitioning

Abstract

A method has been developed in the Geodetic Survey Squadron to solve very large sets of weighted, linear observation equations on a computer with minimal core storage. The upper triangular portion of a symmetric matrix is partitioned and inverted to obtain the variance-covariance elements and the solution vector. Observation equations are stored externally with the zero elements eliminated. The normal equation partitions are formed while reading one observation equation at a time from an external storage device. The inversion technique requires storage of only two partitions in core with the remaining partitions residing on a random access device. A set of 25,000 observation equations and 2,003 unknowns was solved in 35 hours on an IBM 7094/MOD 1 using 20,000 words of core and external disk storage. Partition size was 91×91. An increase in partition size significantly reduces execution time. The solution can be interrupted and restarted to avoid long continuous computer runs.