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Abstract.A superlinear procedure for finding a multiple root is presented. In it the secant method is applied to the given function divided by a divided difference whose increment shrinks toward zero as the root is approached. Two function evaluations per step are required, but no derivatives need be calculated.CR Category: 5.15.

Equation-solving methods that utilize alternate function and derivative values are developed. The procedures are similar to the secant or to Muller's method, and are especially competitive when the derivative is simpler to obtain than the function itself. Related hybrid methods are also found to be attractive.

Abstract.A method of Regula Falsi type for finding a simple root of a non-linear equation is presented. It is similar to the Pegasus procedure, but exhibits a higher asymp. tobic convergence efficiency. This is accomplished by eliminating one or two of the slower Pegasus substeps.CR Category: 5.1.

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