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                <term xml:lang="en">quadratic forms of Gaussian random variables</term>
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              <p>Quadratic forms in Gaussian random variables yield generalized noncentral chisquare distributions which appear in many test statistics. Classical cdf methods (e.g., characteristic-function inversion, saddlepoint approximations) can be effective but are not designed for reliable finite-precision computation and offer limited structural or complexity insight. We study finite-precision cdf evaluation and its binary complexity.&lt;/p&gt;&lt;p&gt;Building on earlier low-dimensional results, we note that relevant modified Laplace transforms of the cdf are available in closed-form and are D-finite functions. This reduces the general d dimensional cdf evaluation problem to evaluating a truncated holonomic power series with positive coefficients, together with explicit tail bounds. Next, we develop a reduced-rounding-error evaluation by switching from the standard alternating-sign scalar recurrence to an equivalent coupled recurrence in which all update coefficients are nonnegative. This "positivity feature" supports a sharper forward-error analysis and improves robustness of the finiteprecision evaluation.&lt;/p&gt;&lt;p&gt;Finally, we propose an accelerated coefficient-generation algorithm. Using a transposed multipoint-evaluation primitive, it computes collectively-via a transposed Vandermonde/remainder-tree construction-the truncated contributions in-duced by the rational factors of the holonomic equation. The remaining step, which assembles the solution by exponentiating a truncated series formed from these aggregated transforms, is carried out with FFT-based power-series exponentiation. Overall, the method achieves quasi-linear complexity in the maximum between the number of Gaussian random variables and the truncation order.</p>
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