Performance limitations of an analog method for solving simultaneous linear equations

The limitations inherent in a recently proposed analog method for solving simultaneous linear equations are examined, and methods for overcoming some of these limitations are discussed. In its original form the method requires that all eigenvalues of the matrix of coefficients lie in a unit circle centered on (1,0) in the complex plane. Proper scaling of the matrix and the data vector extends this region to the entire right half of the complex plane (neglecting the effects of noise). A modification of the algorithm is described that allows the region to be further extended to the entire complex plane. If the product of the gains in the forward and feedback branches is not unity, the solution produced by the algorithm is shown to be in error. Finally, the effects of noise, which is inevitably significant in any analog realization of the algorithm, are examined. Noise is found to produce a limiting mean square error of the solution, thus preventing perfect convergence to the ideal solution vector. A procedure for determination of when to stop the iteration is proposed.