Certain higher order iterative procedures for the numerical solution of ordinary differential equations require a separate procedure for determining starting values. Such is the case in the Adams and Milne methods. The work involved in programming the computation of starting values may be of the same magnitude as that required in programming the method itself. This paper shows that the three-point Adams formula leads to a method which may be considered “self-starting” in the sense that very minor modifications of the method provide starting values of sufficient accuracy for continuing the solution. Let the equation under solution be y′ = ƒ( y, t ), and let the values y′ ( t 0 ) = y 0 ′, y′ ( t 0 + h ) = y 1′, y ( t 0 ) = y 0 , y ( t 0 + h ) = y 1, be given. The general three-point predictor-corrector process consists of estimating (in some unspecified way) a value y 2 ′ of y 2 ′ computing a first estimate y 2 by means of a closed three-point integration formula; obtaining the (presumably) better value y 2 ′ = ƒ( y 2 , t 0 + 2 h ); and then repeating the process until some convergence criterion is satisfied, either for y 2 ′ or for y 2 . The process could have started by first estimating y 2 , rather than y 2 ′. It is important to note that as long as the step size h and the first estimate y 2 ′ of y 2 ′ result in a converging process, the values to which there is convergence are independent of the method of prediction. The Adams three-point formula is y n +1 = y n + h /12[5ƒ( t n +1 , y n +1 ) + 8ƒ( t n , y n ) - ƒ( t n -1 , y n -1)]. (1) T n , the single step truncation error, is given by T n = - h 4 /24ƒ′′′( ξ ), t n -1 ≦ ξ ≦ t n +1 . The quantities y n , y ′ n +1 , y n ′, y ′ n -1 are assumed exact. Let the initial conditions y 0 ′, y 0 be specified for y ′ = ƒ( y, t ). As a first approximation, let y (0) -1 ′ = y 0 ′, y (0) +1 ′ = y 0 ′, y (0) -1 = y 0 , y (0) +1 = y 0 . The superscript zero in parentheses indicates that these are the initial estimates of y , y ′. The starting procedure consists of performing corrective iterations in the forward (+1) and backward (-1) direction as follows: y ( i ) +1 = y 0 + h /12[5ƒ( t +1 , y ( i -1) +1 ) + 8ƒ( t 0 , y 0 ) - ƒ( t -1 , y ( j ) -1 )], y ( j ) -1 = y 0 - h /12[5ƒ( t -1 , y ( j -1) -1 ) + 8ƒ( t 0 , <