A direct method for solving linear algebraic systems using a reduced Gram-Schmidt process

An elimination procedure is presented, where, at each step, one equation of the reduced system has its coefficients kept at their original values and the vectors constructed with the coefficients of the other equations are made orthogonal to the vector of the coefficients in one of them. A simple backward substitution is applied to determine the unknowns of the original system of equations. The solution method seems to be more stable than the classical Gaussian elimination as tested on some benchmark electric and magnetic field problems relative to multi-body configurations involving truncation of infinite systems of equations. The number of arithmetic operations required for large systems is only two thirds of that in the full Gram-Schmidt orthogonalization.