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Table 4 The coefficients of method (2.2) for order 7

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

k 6
\(\alpha _{n - 1}\) \(\frac{ - 7200(1440 + s(3132 + s(2320 + s(775 + s(120 + 7s))))) + (1 + s)(7{,}796{,}104 + s(8{,}479{,}052 + s(3{,}336{,}768 + 7s(80{,}701 + 4973s))))\beta _{0})}{(60(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s))))))}\)
\(\alpha _{n - 2}\) \(\frac{ (300(720 + s(2106 + s(1844 + s(685 + s(114 + 7s))))) - 5(1 + s)(78{,}796 + s(91{,}175 + s(37{,}473 + s(6547 + 413s))))\beta _{0})}{(215{,}824 + 2s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))} \)
\(\alpha _{n - 3}\) \(\frac{ ( - 1200(480 + s(1524 + s(1488 + s(605 + s(108 + 7s))))) + 5(1 + s)(174{,}152 + s(230{,}788 + s(104{,}616 + 7s(2807 + 187s))))\beta _{0})}{(9(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s))))))} \)
\(\alpha _{n - 4}\) \(\frac{225(360 + s(1188 + s(1228 + s(535 + s(102 + 7s))))) - 20(1 + s)(5701 + s(8066 + s(3942 + s(793 + 56s))))\beta _{0}}{3(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))} \)
\(\alpha _{n - 5}\) \(\frac{ - 96(288 + s(972 + s(1040 + s(475 + s(96 + 7s))))) + 5(1 + s)(7496 + s(11{,}020 + s(5664 + 7s(173 + 13s))))\beta _{0}}{ 4(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))}\)
\(\alpha _{n - 6}\) \(\frac{300(240 + s(822 + s(900 + s(425 + s(90 + 7s))))) - (1 + s)(95{,}356 + s(143{,}753 + s(76{,}527 + s(17{,}173 + 1379s))))\beta _{0}}{ 90(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))}\)
\(\beta _{1}\) \(\frac{(60(720 + 7s(6 + s)(84 + s(6 + s)(17 + s(6 + s)))) - 10(1 + s)(2484 + s(4292 + s(2785 + s(855 + s(125 + 7s)))))\beta _{0})}{ (3s(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s))))))}\)
\(\beta _{s}\) \(\frac{360( - 40 + 23 \beta _{0})}{s(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))} \)