Diagonal padé approximations to the exponential function and p-stability

Abstract

If P j (z)/P j (−z) is the (j,j)-diagonal Pade approximation to exp(z), we show that those numerical methods for y″ = f(t y) which, when applied to the test equation have associated characteristic polynomial are not P-stable in the sense of Lambert and Watson [1] except for j=1 (the trapezoidal rule). Further, we show that for j≧2 the size of the interval of periodicity of such a method does not exceed 12 and for j→ ∞ it decreases down to π2. Our present investigation has been motivated by a result noted to the contrary in [3].