Efficient computation of the complex error function

W Gautschi - SIAM Journal on Numerical Analysis, 1970 - SIAM
W Gautschi
SIAM Journal on Numerical Analysis, 1970SIAM
The paper is concerned with the computation of w(z)=\exp(-z^2)erfc(-iz) for complex z=x+iy
in the first quadrant Q_1:x\geqq0,y\geqq0. Using Stieltjes–theory of continued fractions it is
first observed that the Laplace continued fraction for w(z), although divergent on the real
line, represents w(z) asymptotically for z→∞ in the sector S:-π/4<\argz<5π/4. Specifically,
the n th convergent approximates w(z) to within an error of O(z^-2n-11) as z→∞ in S. A
recursive procedure is then developed which permits evaluating w(z) to a prescribed …
The paper is concerned with the computation of for complex in the first quadrant . Using Stieltjes– theory of continued fractions it is first observed that the Laplace continued fraction for , although divergent on the real line, represents asymptotically for in the sector . Specifically, the nth convergent approximates to within an error of as in S. A recursive procedure is then developed which permits evaluating to a prescribed accuracy for any . The procedure has the property that as becomes sufficiently large, it automatically reduces to the evaluation of the Laplace continued fraction, or, equivalently, to Gauss–Hermite quadrature of .
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