## Computing Special Functions By Using Quadrature Rules (2002)

Citations: | 7 - 0 self |

### BibTeX

@MISC{Gil02computingspecial,

author = {A. Gil and J. Segura and N. M. Temme and Amparo Gil and Javier Segura and Nico M. Temme},

title = {Computing Special Functions By Using Quadrature Rules},

year = {2002}

}

### OpenURL

### Abstract

The usual tools for computing special functions are power series, asymptotic expansions, continued fractions, di#erential equations, recursions, and so on. Rather seldom are methods based on quadrature of integrals. Selecting suitable integral representations of special functions, using principles from asymptotic analysis, we develop reliable algorithms which are valid for large domains of real or complex parameters. Our present investigations include Airy functions, Bessel functions and parabolic cylinder functions. In the case of Airy functions we have improvements in both accuracy and speed for some parts of Amos's code for Bessel functions. 2000 Mathematics Subject Classification: 65D20, 65D32, 33C10, 33F05, 41A60.

### Citations

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Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables
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(Show Context)
Citation Context ...efined by # = 2 3 z 3/2 .Also,for|ph z|s2 3 #, Ai(-z) # 1 # # z - 1 4 # cos # # - 1 4 # # +O(1/#) # , Bi(-z) #- 1 # # z - 1 4 # sin # # - 1 4 # # +O(1/#) # . (2.3) One integral representation is (see =-=[1]-=-, (p. 447)) Ai(x)= 1 2# # # -# cos # 1 3 t 3 + xt # dt, x # IR . (2.4) This integral is not a suitable representation for numerical quadrature, in particular when x is positive. In that case Ai(x) is ... |

385 |
Methods of numerical integration
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- 1984
(Show Context)
Citation Context ...a period, then |R n |# constant n k . (3.4) In the case of the Bessel function, we can take any k in this theorem, and we infer that now the error is exponentially small. For more details we refer to =-=[3]-=-. In [13] (Vol. II, p. 218) this Bessel function integral is considered in detail, and from this reference and [12] we derive an upper bound for R n : |R n |#2e x/2 (x/2) 2n (2n)! , (3.5) which is qui... |

99 |
Special Functions: An Introduction to the Classical
- Temme
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(Show Context)
Citation Context ...listic for the value of x we chose. Another example is the Bessel function integral for general order J # (x)= 1 2#i # C e -x sinh t+#t dt, (3.6) where C starts at -#- i# and terminates at #+ i#; see =-=[15]-=- (p. 222). On this contour oscillations will occur, but we will select a special contour that is free of oscillations for the case x # #. We wr i te # = x cosh ,s# 0. The saddle point of -x sinh t + #... |

63 |
The Special Functions And Their Approximations, Vol
- Luke
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(Show Context)
Citation Context ..., then |R n |# constant n k . (3.4) In the case of the Bessel function, we can take any k in this theorem, and we infer that now the error is exponentially small. For more details we refer to [3]. In =-=[13]-=- (Vol. II, p. 218) this Bessel function integral is considered in detail, and from this reference and [12] we derive an upper bound for R n : |R n |#2e x/2 (x/2) 2n (2n)! , (3.5) which is quite realis... |

18 |
Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order
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(Show Context)
Citation Context ... be used: Ai(z)=-e -2#i/3 Ai # e -2#i/3 z # - e 2#i/3 Ai # e 2#i/3 z # . (2.10) For large values of z we use the asymptotic expansion and for small values Taylor series. Comparing this algorithm with =-=[2]-=-, an important package for complex Bessel functions, we notice several improvements. In some cases Amos's code is less accurate than ours, and in some cases we also have a faster code. More details ar... |

5 |
Algorithm 819: AIZ, BIZ: two Fortran 77 routines for the computation of complex Airy functions
- Gil, Segura, et al.
- 2002
(Show Context)
Citation Context ... for complex Bessel functions, we notice several improvements. In some cases Amos's code is less accurate than ours, and in some cases we also have a faster code. More details are given in the papers =-=[8]-=- and [10]. 3. The trapezoidal rule Gauss quadrature is one example of evaluating integrals. It has a very good performance for various types of integrals over real intervals. One of the drawbacks is t... |

5 | Computing complex Airy functions by numerical quadrature
- Gil, Segura, et al.
(Show Context)
Citation Context ...plex Bessel functions, we notice several improvements. In some cases Amos's code is less accurate than ours, and in some cases we also have a faster code. More details are given in the papers [8] and =-=[10]-=-. 3. The trapezoidal rule Gauss quadrature is one example of evaluating integrals. It has a very good performance for various types of integrals over real intervals. One of the drawbacks is that it is... |

5 |
Steepest descent paths for integrals defining the modified Bessel functions of imaginary order
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(Show Context)
Citation Context ...ical evaluations it seems better to use integral representations. Again, saddle point methods give a standard form, and we can apply the trapezoidal rule for infinite integrals. See also [6], [7] and =-=[14]-=-. 8 4. Parabolic cylinder functions. One example is U(a, z)= e 1 4 z 2 i # 2# # c+i# c-i# e -zs+ 1 2 s 2 s -a1 2 ds, c > 0. (5.3) These functions are related with confluent hypergeometric functions (K... |

3 | 2002, Computation of Bessel and Airy functions and of related Gaussian quadrature formulae
- Gautschi
(Show Context)
Citation Context ...os( 1 3 t 3 + xt), x = -16, with stationary points at t = # -x (top) and with x = 16 (bottom), with no stationary points inside the shown interval. A much better representation is used by Gautschi in =-=[4]-=-: Ai(z)= z - 1 4 e -# # ##( 5 6 ) # # 0 # 1+ t 2# # - 1 6 t - 1 6 e -t dt, (2.5) 4 where # = 2 3 z 3 2 , |ph #|s=#|ph z|s2 3 #. (2.6) Now the dominant term e -# is in front of the integral; the integr... |

3 | On non-oscillating integrals for computing inhomogeneous Airy functions
- Gil, Segura, et al.
- 2001
(Show Context)
Citation Context ...ics are used to bring the integral in a standard from suitable for applying the trapezoidal rule. 2. Inhomogeneous Airy functions, that is, solutions of the equation d 2 w dx 2 - xw = 1 # . (5.1) See =-=[5]-=- for the construction of non-oscillating integral representations for these functions. In [9] numerical algorithms based on these integrals are given. 3. Modified Bessel functions of imaginary order. ... |

1 |
Temme (2001). Evaluation of the modified Bessel function of the third kind of imaginary orders
- Gil, Segura, et al.
(Show Context)
Citation Context ...but for numerical evaluations it seems better to use integral representations. Again, saddle point methods give a standard form, and we can apply the trapezoidal rule for infinite integrals. See also =-=[6]-=-, [7] and [14]. 8 4. Parabolic cylinder functions. One example is U(a, z)= e 1 4 z 2 i # 2# # c+i# c-i# e -zs+ 1 2 s 2 s -a1 2 ds, c > 0. (5.3) These functions are related with confluent hypergeometri... |

1 |
Temme (2001). Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. Accepted for publication in
- Gil, Segura, et al.
(Show Context)
Citation Context ...or numerical evaluations it seems better to use integral representations. Again, saddle point methods give a standard form, and we can apply the trapezoidal rule for infinite integrals. See also [6], =-=[7]-=- and [14]. 8 4. Parabolic cylinder functions. One example is U(a, z)= e 1 4 z 2 i # 2# # c+i# c-i# e -zs+ 1 2 s 2 s -a1 2 ds, c > 0. (5.3) These functions are related with confluent hypergeometric fun... |

1 | GIZ, HIZ: Two Fortran 77 routines for the computation of complex Scorer functions. Accepted for publication in
- Gil, Segura, et al.
- 2002
(Show Context)
Citation Context ...ule. 2. Inhomogeneous Airy functions, that is, solutions of the equation d 2 w dx 2 - xw = 1 # . (5.1) See [5] for the construction of non-oscillating integral representations for these functions. In =-=[9]-=- numerical algorithms based on these integrals are given. 3. Modified Bessel functions of imaginary order. The prototype is K ia (x)= # # 0 e -x cosh t cos at dt. (5.2) for a # 0,x > 0. These integral... |

1 |
Error estimates for Luke's approximation formulas for Bessel and Hankel functions
- Krumhaar
- 1965
(Show Context)
Citation Context ... and we infer that now the error is exponentially small. For more details we refer to [3]. In [13] (Vol. II, p. 218) this Bessel function integral is considered in detail, and from this reference and =-=[12]-=- we derive an upper bound for R n : |R n |#2e x/2 (x/2) 2n (2n)! , (3.5) which is quite realistic for the value of x we chose. Another example is the Bessel function integral for general order J # (x)... |