Abstract: According to the Fluctuationlessness Theorem, once the fluctuation contributions are set aside, the matrix that represents an algebraic operator acting by multiplication with a scalar function of a single variable coincides with the result of evaluating that same function at the matrix of the independent variable, both taken over the identical subspace and through the identical basis set. Extending this idea to the remainder of a Taylor series yields an approximation with considerable flexibility. Because the Euler scheme and the higher-order Taylor schemes both lean on the Taylor expansion when solving initial value problems, we transfer the Fluctuationlessness Theorem to the integral form of the Taylor remainder and, by widening its scope, fit it to the Euler method. In this revised version, we make explicit that the fluctuationless approximation of the univariate remainder integral coincides exactly with the Gauss–Jacobi quadrature rule associated with the weight of that remainder; the eigenvalues of the truncated universal matrix are the Gauss nodes and the squared first components of its eigenvectors are the Gauss weights. On this basis we supply a complete local and global error analysis and prove that the scheme has order k+1, and we discuss its computational cost. We then characterise the scheme sharply: for every autonomous problem (in particular the linear test equation) it coincides exactly with the Taylor method of order k+1, and its region of absolute stability is that of the corresponding truncated exponential — so any improvement over the Taylor method is a genuinely non-autonomous effect. Building on this understanding, we introduce a predicted-state generalisation, in which the interior states entering the remainder integral are supplied by a degree-q Taylor predictor instead of being frozen; the resulting one-step scheme has order k+q+1, so that already the choice k=1, q=2 gives a fourth-order method that uses only first and second derivatives and decisively outperforms the second-order Taylor method on both non-autonomous and autonomous problems. Numerical experiments report the exact solutions, a convergence study, the stability regions, and the order of every member of the family.

Keywords: Fluctuationlessness, Matrix representation, Approximation, Taylor expansion, Univariate functions, Euler method, Higher-order Taylor methods, Gauss–Jacobi quadrature, Error analysis, Convergence, Numerical solutions of I.V.P.

Pages: 299-310