By Furui S., Sandhi M.M. (eds.)

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Extra resources for Advances in Speech Signal Processing

Example text

Repeated applica- tion of these transformations, moreover, c a u s e s the variables to converge rapidly to certain limiting values, for which the functions can be evaluated by elementary means. These invariance properties thus give 51 WALTER GAUTSCHI rise to interesting and powerful recursive algorithms for computing the functions in question. 1. 1. Elliptic integrals and Jacobian elliptic functions Definitions and special v a l u e s . The best known functions enjoying invariance properties of the type indicated are the elliptic i n t e grals of the first, second, and third kind.

Intrinsically different are repre- sentations of r in terms of continued fractions. There are many differ- ent types of continued fractions that can be used in this connection. We mention only the J-fractions (cf. + 1 ' r r ? n — . . —— , x+s 0 + x+s ' 2 n r. ^ 0 all k k and refer to Hart et al. [1968, p. 7 3ff] for others. (11) represents a rational function in F , . The continued fraction Conversely, a rational function in JR . can be represented in the form (11), u n l e s s certain n, n-1 ' determinants in the coefficients of p and q happen to vanish (Wall [1948, p.

Must converge, ' 57 n and a / t η' η both d e c r e a s e . Hence, ' WALTER GAUTSCHI t where T > M . n 1 T, ' n—« T The speed of convergence i s comparable to that of € , in the sense 2) t \ / n - T ~ -=r-€ T , n -+ » n' . 2(5), that wkw4) 2 Mr I }■ K«*"- from which ,2 t Since € n - t , = ^ - € n+1 T n + θ(€ x ) n' converges quactraticaliy, quadratically, in particular € . < c< , we easily obtain, for any p > 0 , ,2 t n - t , = ^ - € n+p+1 T n + θ(€ K ) , n' ' from which (2) follows by letting p -*■ oo .

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Advances in Speech Signal Processing by Furui S., Sandhi M.M. (eds.)

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