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Mathieu function with eigenvalue a for q is a periodic solution and can be expressed as a Fourier series.
The relationship between the coefficients is expressed using a tridiagonal matrix T.
Backward recursion is used as an approximate method to solve the coefficient sequence.
In backward recursion, terms may cancel each other and produce a value close to zero, although this is a rare case.
In this case, it must be solved in pieces as a linear problem because the digits drop out and subsequent values become inaccurate.
When q is small, the accuracy of calculation is extremely poor for ce, se(n=1), se(n=2), where the a - r0 is close to zero.
Therefore, it is necessary to separately prepare an approximation of zero-shifted eigenvalue function.