# September 1963 - Vol. 6 No. 9

## Features

A formula for numerical integration is prepared, which involves an exponential term. This formula is compared to two standard integration methods, and it is shown that for a large class of differential equations, the exponential formula has superior stability properties for large step sizes. Thus this formula may be used with a large step size to decrease the total computing time for a solution significantly, particularly in those engineering problems where high accuracy is not needed.
Some computer operating systems have a bad habit of inserting tape marks into the output stream willy-nilly. This often results in loss of output when a tape is scratched after its first file has been printed. Other operating systems, notably those recently distributed by IBM for use with the 7090, put tape marks on print tapes only on command, so that frequently a print tape is removed from the computer without any tape mark, and extraneous material is printed.
M. L. Pei [Comm. ACM 5, 10 (Oct. 1962)] gave an explicit inverse for a matrix of the form M + &dgr;I, where M is an n-square matrix of ones and &dgr; is a nonzero parameter. The eigenvalues of the Pei matrix were given by W. S. LaSor [Comm. ACM 6, 3 (Mar. 1963)]. The eigenvectors may be obtained by considering the system (M+&dgrI)x = &lgr;x, the jth equation of which is S + &dgr;xj = &lgr;xj , (1) where S denotes ∑ni=1 xi. On summing the equations for j = 1, 2, ··· , n, we obtain nS + &dgr;S = &lgr;S. From this we conclude that (a) S = 0 or (b) &lgr; = n + &dgr;.
The technique of obtaining arctangents by inverse interpolation [1] is a relatively long process not suitable for a subroutine. The Taylor series expansion for arguments less than unity converges rather slowly for those near unity. The method of small increments of the argument [2] is again inconvenient for a subroutine. A more rapid series expansion in terms of Chebyshev polynomial [3] is given in terms of a new argument, which is less than 0.1989. This method requires the storage of &pgr;, √2 - 1 and seven coefficients and is perhaps widely used. However, for multiple precision not only the coefficients have to be evaluated to the precision desired, but more must be used. Therefore the following alternative method may prove to be convenient and efficient.