Contents Previous Next Subchapters Current Chapters-> erf ierf erfc beta betai gamma gammln gammainc besselj bessely besseli besselk elliprf elliprd elliprj ellipf ellipe ellipi kolsmi psi airy isprime lentz Parent Chapters-> Omatrix6 special lentz Search Tools-> contents reference index search

Lentz's Method For Evaluating Continued Fractions
 Syntax `lentz(`a`, `b`)` See Also polval

Description
The i-th element of the return value is the continued fraction ```      b(i, 1) + a(i, 1)                -------                b(i, 2) + a(i, 2)                          -------                          b(i, 3) + ...                                        + a(i, n)                                         -----------                                          b(i, n + 1) ```where a and b are real, double-precision or complex matrices with the same row dimension. The value `n` in the expression above is the column dimension of a which must be one less than the column dimension of b. The return value is a column vector with the same row dimension as a and has the same type as results from coercion between the type of a and the type of b.

Example
The continued fraction representation of the tangent function is ```      tan(x) =  x               ---   2                1 - x                   ---   2                    3 - x                       ---   2                        5 - x                           ---                            7 - ... ```The tangent of `pi / 4` is one. We can compare this to the first five terms of the continued fraction as follows: ```      x = PI / 4d0      a = [   x, -x^2, -x^2, -x^2, -x^2]      b = [0, 1,   3,   5,   7,   9]      b = double(b)      print "lentz(a, b) =", lentz(a, b) ```
Reference
When the maximal difference between terms in the factional representation is less than machine epsilon times the maximum element in b, the expansion is assumed to have converged and the series is truncated.