Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 341 / Section A / 5 / c

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- Solution c

Solution

The three substeps have total length

2 α + (1 - 2 α) = 1

. Their leading cubic defects add, so cancellation requires

2 α^{3} + (1 - 2 α)^{3} = 0.

Taking the real cube root and solving gives

α = \frac{1}{2 - 2 ^{1/3}} .

This is the coefficient in the higher-order composition of a symmetric splitting; the middle substep

1 - 2 α

is negative.

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