The numerical inversion of the characteristic equation with applications to positive quadratic forms in normal variables

When a random variable can only take on positive values, the distribution function of the variable can be determined from the inverse Laplace transform of the characteristic equation Murli and Rizzardi's (1990) implementation of Talbot's (1979) algorithm for the numerical inversion of Laplace transforms was used to calculate percentage points for quadratic forms in normal variables for several benchmark examples and a problem arising in monitoring process variability. The accuracy and time requirements of this algorithm are comparable to Davies (1980) algorithm. A two moment approximation initially proposed by Satterthwaite (1941) and later studied by Box (1954) was shown to provide high quality estimates for the percentage points of quadratic forms in normal variables.