numtheory

compute simple continued fraction expansions for all real roots of a rational polynomial


Calling Sequence


cfracpol(pol, n)
cfracpol(pol)


Parameters


pol



rational polynomial

n



integer (n + 1 is the number of partial quotients)





Description


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The cfracpol function returns simple continued fraction expansions of all real roots of a rational polynomial pol. Each expansion is given in list form with at most $n\+1$ quotients. If the second argument n is not present, it defaults to 10.

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The command with(numtheory,cfracpol) allows the use of the abbreviated form of this command.



Examples


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$\mathrm{with}\left(\mathrm{numtheory}\right)\:$

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$\mathrm{cfracpol}\left({x}^{4}{x}^{3}4{x}^{2}\+4x\+1\,20\right)$

$\left[{}{2}{\,}{22}{\,}{1}{\,}{7}{\,}{2}{\,}{1}{\,}{1}{\,}{2}{\,}{1}{\,}{2}{\,}{1}{\,}{17}{\,}{4}{\,}{4}{\,}{1}{\,}{1}{\,}{4}{\,}{2}{\,}{18}{\,}{1}{\,}{10}{\,}{\mathrm{...}}\right]{\,}\left[{}{1}{\,}{1}{\,}{3}{\,}{1}{\,}{3}{\,}{1}{\,}{1}{\,}{1}{\,}{1}{\,}{1}{\,}{1}{\,}{4}{\,}{1}{\,}{1}{\,}{1}{\,}{4}{\,}{1}{\,}{2}{\,}{4}{\,}{5}{\,}{18}{\,}{\mathrm{...}}\right]{\,}\left[{1}{\,}{2}{\,}{1}{\,}{21}{\,}{1}{\,}{7}{\,}{2}{\,}{1}{\,}{1}{\,}{2}{\,}{1}{\,}{2}{\,}{1}{\,}{17}{\,}{4}{\,}{4}{\,}{1}{\,}{1}{\,}{4}{\,}{2}{\,}{18}{\,}{\mathrm{...}}\right]{\,}\left[{1}{\,}{1}{\,}{4}{\,}{1}{\,}{3}{\,}{1}{\,}{1}{\,}{1}{\,}{1}{\,}{1}{\,}{1}{\,}{4}{\,}{1}{\,}{1}{\,}{1}{\,}{4}{\,}{1}{\,}{2}{\,}{4}{\,}{5}{\,}{18}{\,}{\mathrm{...}}\right]$
 (1) 
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$\mathrm{cfracpol}\left({x}^{6}{x}^{5}6{x}^{4}\+6{x}^{3}\+8{x}^{2}8x\+1\right)$

$\left[{}{2}{\,}{44}{\,}{1}{\,}{3}{\,}{3}{\,}{1}{\,}{1}{\,}{1}{\,}{3}{\,}{2}{\,}{3}{\,}{\mathrm{...}}\right]{\,}\left[{}{2}{\,}{1}{\,}{1}{\,}{6}{\,}{1}{\,}{7}{\,}{34}{\,}{1}{\,}{12}{\,}{1}{\,}{5}{\,}{\mathrm{...}}\right]{\,}\left[{0}{\,}{6}{\,}{1}{\,}{2}{\,}{4}{\,}{3}{\,}{1}{\,}{1}{\,}{3}{\,}{1}{\,}{63}{\,}{\mathrm{...}}\right]{\,}\left[{0}{\,}{1}{\,}{2}{\,}{1}{\,}{2}{\,}{2}{\,}{16}{\,}{1}{\,}{1}{\,}{5}{\,}{11}{\,}{\mathrm{...}}\right]{\,}\left[{1}{\,}{1}{\,}{1}{\,}{1}{\,}{7}{\,}{6}{\,}{10}{\,}{2}{\,}{29}{\,}{20}{\,}{1}{\,}{\mathrm{...}}\right]{\,}\left[{1}{\,}{1}{\,}{10}{\,}{3}{\,}{1}{\,}{13}{\,}{1}{\,}{1}{\,}{3}{\,}{1}{\,}{4}{\,}{\mathrm{...}}\right]$
 (2) 
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$a\u2254117260219{x}^{6}\+139540883{x}^{5}\+17033080{x}^{4}\+800302{x}^{3}\+18628{x}^{2}\+216x\+1\:$

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$\mathrm{cfracpol}\left(a\right)$

$\left[{}{1}{\,}{1}{\,}{41}{\,}{7}{\,}{1}{\,}{7}{\,}{34}{\,}{1}{\,}{12}{\,}{1}{\,}{5}{\,}{\mathrm{...}}\right]{\,}\left[{}{1}{\,}{1}{\,}{42}{\,}{1}{\,}{1}{\,}{6}{\,}{1}{\,}{2}{\,}{4}{\,}{3}{\,}{1}{\,}{\mathrm{...}}\right]{\,}\left[{}{1}{\,}{1}{\,}{42}{\,}{1}{\,}{1}{\,}{1}{\,}{2}{\,}{1}{\,}{2}{\,}{2}{\,}{16}{\,}{\mathrm{...}}\right]{\,}\left[{}{1}{\,}{1}{\,}{42}{\,}{1}{\,}{2}{\,}{1}{\,}{1}{\,}{1}{\,}{7}{\,}{6}{\,}{10}{\,}{\mathrm{...}}\right]{\,}\left[{}{1}{\,}{1}{\,}{42}{\,}{1}{\,}{2}{\,}{1}{\,}{10}{\,}{3}{\,}{1}{\,}{13}{\,}{1}{\,}{\mathrm{...}}\right]{\,}\left[{1}{\,}{3}{\,}{3}{\,}{1}{\,}{1}{\,}{1}{\,}{3}{\,}{2}{\,}{3}{\,}{4}{\,}{1}{\,}{\mathrm{...}}\right]$
 (3) 
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$\mathrm{cfracpol}\left(\left(232x\+543\right)\left({x}^{6}{x}^{5}6{x}^{4}\+6{x}^{3}\+8{x}^{2}8x\+1\right)\,10\right)$

$\left[{}{3}{\,}{1}{\,}{1}{\,}{1}{\,}{14}{\,}{1}{\,}{4}\right]{\,}\left[{}{2}{\,}{44}{\,}{1}{\,}{3}{\,}{3}{\,}{1}{\,}{1}{\,}{1}{\,}{3}{\,}{2}{\,}{3}{\,}{\mathrm{...}}\right]{\,}\left[{}{2}{\,}{1}{\,}{1}{\,}{6}{\,}{1}{\,}{7}{\,}{34}{\,}{1}{\,}{12}{\,}{1}{\,}{5}{\,}{\mathrm{...}}\right]{\,}\left[{0}{\,}{6}{\,}{1}{\,}{2}{\,}{4}{\,}{3}{\,}{1}{\,}{1}{\,}{3}{\,}{1}{\,}{63}{\,}{\mathrm{...}}\right]{\,}\left[{0}{\,}{1}{\,}{2}{\,}{1}{\,}{2}{\,}{2}{\,}{16}{\,}{1}{\,}{1}{\,}{5}{\,}{11}{\,}{\mathrm{...}}\right]{\,}\left[{1}{\,}{1}{\,}{1}{\,}{1}{\,}{7}{\,}{6}{\,}{10}{\,}{2}{\,}{29}{\,}{20}{\,}{1}{\,}{\mathrm{...}}\right]{\,}\left[{1}{\,}{1}{\,}{10}{\,}{3}{\,}{1}{\,}{13}{\,}{1}{\,}{1}{\,}{3}{\,}{1}{\,}{4}{\,}{\mathrm{...}}\right]$
 (4) 
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$\mathrm{cfracpol}\left({x}^{6}3{x}^{5}\+5{x}^{3}3x\+1\right)$







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