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Table 1 The coefficients of method (2.2) for orders 2, 3, and 4

From: G-stability one-leg hybrid methods for solving DAEs

k 1 2 3
\(\alpha _{n - 1}\) −1 \(\frac{2( - 8 - 6s+ 3\beta _{0} + 3s\beta _{0})}{14 + 9s}\) \(\frac{ - 18(12 + s(15 + 4s)) + (1 + s)(119 + 46s)\beta _{0}}{2(85 + 90s+ 22s)} \)
\(\alpha _{n - 2}\) 0 \(\frac{2 + 3s- 6(1 + s)\beta _{0}}{14 + 9s}\) \(\frac{9(3 + 2s(3 + s)) - 4(1 + s)(17 + 7s)\beta _{0}}{85 + 90s+ 22s}\)
\(\alpha _{n - 3}\) 0 0 \(\frac{ - 2(4 + s(9 + 4s)) + (1 + s)(17 + 10s) \beta _{0}}{2(85 + 90s+ 22s)} \)
\(\beta _{1}\) \(\frac{1 + 2s - 2(1 + s)\beta _{0}}{2s}\) \(\frac{4 + 6s(2 + s) - (1 + s)(5 + 3s)}{s(14 + 9s)}\) \(\frac{ - 6(3 + 2s)(1 + s(3 + s)) + (1 + s)(17 + s(17 + 4s))\beta _{0}}{s(85 + 90s+ 22s)} \)
\(\beta _{s}\) \(- \frac{1 - 2\beta _{0}}{2s}\) \(\frac{ - 4 + 5\beta _{0}}{s(14 + 9s)}\) \(\frac{ - 18 + 17\beta _{0}}{s(85 + 90s+ 22s)} \)