Schwinger-Lanczos algorithm for calculation of off-shell \(T\)-matrix elements and Wynn's epsilon algorithm (Q1587742)

\( {\mathbf V} \) being an interaction potential and \( {\mathbf G}(0) \) a Green's function, stationary values of the elements of a matrix valued functional depending upon \( {\mathbf V} \) and \( {\mathbf G}(0) \) are expressed as convergents of continued fractions whose coefficients may be obtained by use of a recursive process. It is shown that these convergents are equivalent to those of the continued fraction associated with the Born series \( \{ \sum a(k) \lambda^{k} \mid ( k \geq 0) \} \) when \( \lambda = 1 \) where, the \( \varphi \) depending upon the way in which the series is being used, \[ a(k) = \langle \varphi(i) |{\mathbf V}{\mathbf G}(0) {\mathbf V}{\mathbf G}(0) \ldots {\mathbf G}(0) {\mathbf V} |\varphi(j)\rangle \] \( {\mathbf G}(0) \) and \( {\mathbf V} \) occurring \( k \) and \( k+1 \) times, respectively. The authors compare numerical results obtained by use of the recursive process and of the epsilon algorithm in the calculation of values of the convergents. The reviewer suggests that the absence of any independent knowledge of the exact solution of the variational problem being considered naturally vitiates any comparison of methods of approximation. Furthermore, there is no error analysis at all. The vector epsilon-algorithm offers a method for the simultaneous calculation of an entire row of variational matrix elements; the matrix epsilon-algorithm offers a method for the direct computation of the entire matrix. Possibly there are vector and matrix extensions of the recursive process proposed in this paper.