On preconditioning a nonsingular matrix

For a nonsingular n by n matrix A, a diagonal matrix D^* is derived which minimizes an upper bound on the spectral condition number of DA. Replacement of the linear system Ax=c with the prescaled system D^*Ax=D^*c requires about 3n² operations for dense matrices and fewer for sparse, banded matrices and is recommended for the conjugate gradient and other methods of solution. Examples are given showing the advantageous effect of prescaling on condition number, and a simple computational algorithm is presented. The extension to nondiagonal scaling matrices is discussed.