In 2000 we described a proposal for a logarithmic arithmetic unit, which we suggested would offer a faster, more accurate alternative to floating-point procedures. Would it in fact do so, and could it feasibly be integrated into a microprocessor so that the intended benefits might be realized? Here, we describe the European logarithmic microprocessor, a device designed around that unit, and compare its performance with that of a commercial superscalar pipelined floating-point processor. We conclude that the experiment has been successful, and that for 32-bit work, logarithmic arithmetic may now be the technique of choice.

The process of matrix iteration to solve the nth order linear matrix equation Ax + u = 0 for x is considered.The matrix C = I --A, determining the convergence of the process, is assumed to be convergent and to be known without error.A quadratic measure of convergence a(C) = [[ C liE~tin is introduced.For the special case of symmetric C and Tr (C) = 0, a(C) is the standard deviation of the eigenvalues of C. The measure a(C) is the basis for the analysis of the relative error in the iterative vector x(m) for m ~ ~, due to roundoff errors generated at each iteration.If the generated errors are assumed to be independent random variables with mean zero and standard deviation erda, then the errors do not accumulate but stay at a constant level.In fact, if x(m) = (I -~ Am)x8 with x, ffi --A-~u, then the estimated value 8(A) of a(A~) for m --~ ¢¢ is shown to be 8(A) = [~-0 o's(C~)]~'erms > [1 ~(CC)]~.e ....This reduces to 8(A) = erms/(1 --~09 t if cTc = o'0~I and is of the order of e~. for almost all a0 < 1.So the solution obtained by constant iteration is the best possible solution, using s-digit arithmetic and 2s-digit inner product evaluation plus roundoff.This minimum error level is reached after approximately 2 log b(A)/log a-(C) iterations, which is almost proportional to s.The error is proportional to (R 2 --1)½/2 In R if R is the radix of the number system used, which follows from an estimation of e~. in normalized floating point arithmetic.Computer tests on many random systems are included and show good agreement with the proposed error analysis.