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Now, since the third term of the right hand side of this equation is to be neglected, one concludes that the removal of the last term does not cause any additional inaccuracy, if m is large enough. It is sufficient to choose m of the same order of magnitude as the largest eigenvalue of X squared or the reciprocal of /_ 1 y the smallest eigenvalue of A. Compute the matrix I X~2-A I by means of \ m / the algorithm (15) ; subtract X'1 from this result; and finally, multiply by m in order to obtain an approximation for the correction (11). Of course, this procedure may be repeated and thus set up a quadratically convergent algorithm which, moreover, is self-correcting. Summary. It has been proved that for any real (n X n)-matrix A with only positive eigenvalues the algorithm (1), with an initial matrix X(0) = &/, converges quadratically to the matrix A~* with positive eigenvalues. In a numerical case this algorithm, if continued indefinitely, may be divergent due to round-off errors, whose influence may increase in geometrical progression. This makes it necessary to stop the process as soon as the difference between two successive results no longer decreases ; of course, it is also desirable to have some additional accuracy in the numerical computation to take care of the round-off errors. Any approximation to A~* sufficiently accurate can be improved successively, the rate of convergence of the procedure being quadratic. Each step, however, involves either the solution of a system of n2 linear equations or the extraction of the square-root of a matrix, which may be achieved by a quadratically convergent iteration process.

On Gauss’ speeding up device in the theory of single step iteration.