15.5 Kettenbrüche

Function: cf (expr) ¶

Converts expr into a continued fraction. expr is an expression comprising continued fractions and square roots of integers. Operands in the expression may be combined with arithmetic operators. Aside from continued fractions and square roots, factors in the expression must be integer or rational numbers. Maxima does not know about operations on continued fractions outside of cf.

cf evaluates its arguments after binding listarith to false. cf returns a continued fraction, represented as a list.

A continued fraction a + 1/(b + 1/(c + ...)) is represented by the list [a, b, c, ...]. The list elements a, b, c, … must evaluate to integers. expr may also contain sqrt (n) where n is an integer. In this case cf will give as many terms of the continued fraction as the value of the variable cflength times the period.

A continued fraction can be evaluated to a number by evaluating the arithmetic representation returned by cfdisrep. See also cfexpand for another way to evaluate a continued fraction.

See also cfdisrep, cfexpand, and cflength.

Examples:

• expr is an expression comprising continued fractions and square roots of integers.

  (%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]);
  (%o1)               [59, 17, 2, 1, 1, 1, 27]
  (%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13));
  (%o2)        [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]
  

• cflength controls how many periods of the continued fraction are computed for algebraic, irrational numbers.

  (%i1) cflength: 1$
  (%i2) cf ((1 + sqrt(5))/2);
  (%o2)                    [1, 1, 1, 1, 2]
  (%i3) cflength: 2$
  (%i4) cf ((1 + sqrt(5))/2);
  (%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
  (%i5) cflength: 3$
  (%i6) cf ((1 + sqrt(5))/2);
  (%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
  

• A continued fraction can be evaluated by evaluating the arithmetic representation returned by cfdisrep.

  (%i1) cflength: 3$
  (%i2) cfdisrep (cf (sqrt (3)))$
  (%i3) ev (%, numer);
  (%o3)                   1.731707317073171
  

• Maxima does not know about operations on continued fractions outside of cf.

  (%i1) cf ([1,1,1,1,1,2] * 3);
  (%o1)                     [4, 1, 5, 2]
  (%i2) cf ([1,1,1,1,1,2]) * 3;
  (%o2)                  [3, 3, 3, 3, 3, 6]
  

Function: cfdisrep (list) ¶

Constructs and returns an ordinary arithmetic expression of the form a + 1/(b + 1/(c + ...)) from the list representation of a continued fraction [a, b, c, ...].

(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1)                     [1, 1, 1, 2]
(%i2) cfdisrep (%);
                                  1
(%o2)                     1 + ---------
                                    1
                              1 + -----
                                      1
                                  1 + -
                                      2

Function: cfexpand (x) ¶

Returns a matrix of the numerators and denominators of the last (column 1) and next-to-last (column 2) convergents of the continued fraction x.

(%i1) cf (rat (ev (%pi, numer)));

`rat' replaced 3.141592653589793 by 103993/33102 =3.141592653011902
(%o1)                  [3, 7, 15, 1, 292]
(%i2) cfexpand (%); 
                         [ 103993  355 ]
(%o2)                    [             ]
                         [ 33102   113 ]
(%i3) %[1,1]/%[2,1], numer;
(%o3)                   3.141592653011902

Option variable: cflength ¶

Default value: 1

cflength controls the number of terms of the continued fraction the function cf will give, as the value cflength times the period. Thus the default is to give one period.

(%i1) cflength: 1$
(%i2) cf ((1 + sqrt(5))/2);
(%o2)                    [1, 1, 1, 1, 2]
(%i3) cflength: 2$
(%i4) cf ((1 + sqrt(5))/2);
(%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
(%i5) cflength: 3$
(%i6) cf ((1 + sqrt(5))/2);
(%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]