[Scipy-tickets] [SciPy] #683: scipy.special.erf has significant errors just within its switchover radius

SciPy Trac scipy-tickets@scipy....
Sun Jun 20 17:43:33 CDT 2010

#683: scipy.special.erf has significant errors just within its switchover radius
  Reporter:  tardis         |       Owner:  somebody   
      Type:  defect         |      Status:  closed     
  Priority:  normal         |   Milestone:  0.8.0      
 Component:  scipy.special  |     Version:             
Resolution:  fixed          |    Keywords:  erf complex
Changes (by pv):

  * status:  new => closed
  * resolution:  => fixed


 Fixed in r6548

 The current complex-valued algorithm is actually a bit lousy, as the
 global maximum accuracy is theoretically limited to about 1e-8 due to
 finite precision, even with an optimal cutoff choice. We inevitably lose
 half of the digits at some points.

 The asymptotic expansion error can be estimated by the magnitude of the
 smallest term in the asymptotic series (R=|z|),
 err ~ Gamma(2 R**2 + 2) / ((2 R)**(R**2 + 1/2) Gamma(R**2 + 3/2) 2**(R**2
 + 1/2))
 and the roundoff error by the largest term in the Taylor series multiplied
 by machine epsilon and number of terms,
 err ~ EPS R**(2 R**2) / (2 Gamma(R**2 + 1/2))
 This gives an "optimal" crossover point R = 4.36, pretty close to what you
 got, with the best estimated relative accuracy around 2e-8.

 I guess the previous cutoff R=5.8 was found by noting that at this point,
 the asymptotic series (disregarding finite-precision arithmetic) gives
 machine epsilon accuracy.

Ticket URL: <http://projects.scipy.org/scipy/ticket/683#comment:2>
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