Using parity to accelerate Hermite function computations: Zeros of truncated Hermite series, Gaussian quadrature and Clenshaw summation

Although Hermite functions have been studied for over a century and have been useful for analytical and numerical solutions in a myriad of areas, the theory of Hermite functions has gaps. This article is a unified treatment of all the operations—quadrature, summation, differentiation, and rootfinding—that can be accelerated by exploiting parity. Any function u(y) can be decomposed into its parts that are symmetric and antisymmetric with respect to the origin. Suppose that the quadrature on y∈[−∞,∞] is symmetric in the sense that if yj is a quadrature abscissa with weight wj, then −yj is also a quadrature point with weight wj. The number of multiplications is halved by evaluating the quadrature as ∫−∞∞u(y)dy≈w0u(0)+∑n=1Mwnu(yn)+u(−yn). Parity is equally useful in computing the zeros, maxima and minima of a truncated Hermite series of degree N for the important special cases that the series terms are all either symmetric or antisymmetric. The zeros and critical points are the eigenvalues of a companion matrix whose dimension is N/2 instead of N. In addition, for Hermite functions, we show that parity exploitation halves the dimension of the Jacobi matrix (a special case of the companion matrix) whose eigenvalues are the abcissas of Hermite–Gauss quadrature. The number of floating point operations for the recursion for the weights can likewise be halved. Lastly, the same is true for Chenshaw summation of a Hermite series.