The convert(expr, confrac) command converts a number, series, rational function, or other algebraic expression to a continued-fraction approximation.

If expr is numeric then maxit (optional) is the maximum number of partial quotients to be computed, and cvgts (optional) will be assigned a list of the convergents. A list of the partial quotients is returned as the function value.

If expr is a series and no additional arguments are specified, a continued-fraction approximation (to the order of the series) is computed.  It is equivalent to either an $\left(n\,n\right)$ or $\left(n\,n-1\right)$ Pade approximant (depending on the parity of the order). By specifying 'subdiagonal' as an optional third argument, the continued-fraction computed will be equivalent to a $\left(n\,n\right)$ or $\left(n-1\,n\right)$ Pade approximant.

If expr is a ratpoly (quotient of polynomials) in x, the calling sequence is convert(expr, confrac, x). The rational form is converted into its associated continued-fraction form as required for efficient evaluation of numerical subroutines.

If expr is any other algebraic expression, the third argument specifies a variable and (optionally) the fourth argument specifies order. The series function is applied to the arguments to obtain a series and then case series applies.

By default, a rational polynomial is converted to a monic continued fraction, that is, one with monic polynomials in the non-fractional part of the denominator.  If the option regular or simple is specified then a regular or a simple continued fraction is returned, respectively.

Otherwise, `convert/confrac` is applied to each component of a non-algebraic structure.

For information on the inverse transformation, see NumberTheory[ContinuedFraction].

$\mathrm{convert}\left(2.3\,\mathrm{confrac}\right)$

$\left[2\,3\,3\right]$

$\mathrm{convert}\left(\frac{21}{13}\,\mathrm{confrac}\,'\mathrm{convergents}'\right)$

$\left[1\,1\,1\,1\,1\,2\right]$

$\mathrm{convergents}$

$\left[1\,2\,\frac{3}{2}\,\frac{5}{3}\,\frac{8}{5}\,\frac{21}{13}\right]$

$\mathrm{convert}\left(\exp \left(x\right)\,\mathrm{confrac}\,x\right)$

$1+\frac{x}{1+\frac{x}{-2+\frac{x}{-3+\frac{x}{2+\frac{x}{5}}}}}$

$\mathrm{convert}\left(\exp \left(x\right)\,\mathrm{confrac}\,x\,\mathrm{subdiagonal}\right)$

$\frac{1}{1+\frac{x}{-1+\frac{x}{-2+\frac{x}{3+\frac{x}{2-\frac{x}{5}}}}}}$

$r≔\frac{3x^3+10x^2+12}{3x^3-2x^2+12}$

$r≔\frac{3x^3+10x^2+12}{3x^3-2x^2+12}$

$\mathrm{convert}\left(r\,\mathrm{confrac}\,x\right)$

$1+\frac{4}{x-\frac{2}{3}+\frac{4}{x^2}}$

$\mathrm{convert}\left(r\,\mathrm{confrac}\,x\,\mathrm{regular}\right)$

$1+\frac{12}{3x-2+\frac{12}{x^2}}$

$\mathrm{convert}\left(r\,\mathrm{confrac}\,x\,\mathrm{simple}\right)$

$1+\frac{1}{\frac{x}{4}-\frac{1}{6}+\frac{1}{x^2}}$

The option subdiagonal can be used together with the optional argument var as of Maple 16.

The subdiagonal option was updated in Maple 16.