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abscissas accuracy algebraic equations analogous application approximate solution Benster bilinear boundary conditions boundary problems calculation complex numbers contour convergence of Newton's corresponding defined denote determination element elliptic integrals equa error estimate exact solution example fulfilled Gauss formula given gives Hartree Hence inequality initial value integral equation integrand interpolation interval Kantorovich Let us consider linear algebraic linear operation matrix method of lines modification Newton's method Newton's process nonlinear norm obtain odd function odd polynomial operation mapping operation sending ordinary ordinary differential equations orthogonal Ostrowski partial differential equations particular solution periodic functions points polynomial of degree proof proper numbers proper value proper vectors quadratic form quadrature formula quantities region relaxation method replaced residual forces satisfy second derivative solution of system Southwell Southwell's space sphere successive approximations symmetric Theorem tion transformation trigonometric polynomial unknowns variable Vi(x Vl(x whence