## #1062 continued fractions: sign of sqrt terms ignored

open
nobody
5
2012-11-18
2007-01-14
Robert Dodier
No

cflength : 4;
cf(sqrt(2) - 1);
=> [0, 2, 2, 2, 2] /* OK */
cf(sqrt(2) - 2);
=> [0, - 1, - 1, - 2, - 2, - 2] /* OK (but not canonicalized) */
cf(1 - sqrt(2));
=> [2, 2, 2, 2, 2] /* WRONG */
cf(1 + sqrt(2));
=> [2, 2, 2, 2, 2] /* OK */
cf(2 - sqrt(2));
=> [3, 2, 2, 2, 2] /* WRONG */
cf(2 + sqrt(2));
=> [3, 2, 2, 2, 2] /* OK */

cf ignores the minus sign on sqrt (i.e. treats -sqrt(foo) the same as +sqrt(foo)), as shown in the above examples. cf thinks minus signs are OK if not attached to sqrt.

## Discussion

(%i27) cflength:3\$
(%i28) cf((16*(119+48*sqrt(6)))/337);
(%o28) [11, 4, 3, 4, 4, 5, 1, 27, 1, 6]
period not visible ! And I claim the "5" is incorrect, should be "4"
Though one can check that in all intermediate steps the sign of the sqrt(6) term is positive.
Indeed :
(%i29) float((16*(119+48*sqrt(6)))/337);
(%o29) 11.23207157999252
(%i30) float(cfdisrep([11, 4, 3, 4, 4]));
(%o30) 11.23206751054852
This term is below the true value, so next should be above :
(%i31) float(cfdisrep([11, 4, 3, 4, 4, 4]));
(%o31) 11.23207171314741
Correct _ is above
(%i32) float(cfdisrep([11, 4, 3, 4, 4, 5]));
(%o32) 11.232070910556
Wrong : is below.

JF Mertens