Let \(\boldsymbol{X}=(X_1,\ldots,X_n)'\) be a column random vector which follows a multidimensional normal law with mean vector \(\boldsymbol{0}\) and non-singular covariance matrix \(\boldsymbol{\Sigma}\).
Let \(\boldsymbol{\mu}=(\mu_1,\ldots,\mu_n)'\) be a constant vector, and consider the quadratic form
$$Q=(\boldsymbol{x}+\boldsymbol{\mu})'\boldsymbol{A}(\boldsymbol{x}+\boldsymbol{\mu})=\sum_{r=1}^m\lambda_r\chi^2_{h_r;\delta_r}.$$
The function imhof computes \(P[Q>q]\).
The \(\lambda_r\)'s are the distinct non-zero characteristic roots of
\(A\Sigma\), the \(h_r\)'s their respective orders of
multiplicity, the \(\delta_r\)'s are certain linear combinations
of \(\mu_1,\ldots,\mu_n\) and the
\(\chi^2_{h_r;\delta_r}\) are independent
\(\chi^2\)-variables with \(h_r\) degrees of freedom and
non-centrality parameter \(\delta_r\). The variable
\(\chi^2_{h,\delta}\) is defined here by the
relation \(\chi^2_{h,\delta}=(X_1 +
\delta)^2+\sum_{i=2}^hX_i^2\), where \(X_1,\ldots,X_h\) are
independent unit normal deviates.