6.3.90.900
4 Custom Continued Fractions
4.1 Creation Concepts
4.1.1 Creation Introduction
There are two standard types of continued fractions. Simple continued fractions
are of the form
a0 + 1/(a1 + 1/(a2 + ...)). General continued fractions are of the form
d0 + n0/(d1 + n1/(d2 + ...)). This library offers some procedures, sequence utilities, and sequences
to help you make your own continued fractions.
4.1.2 Creation Caveats
As mentioned in the main introduction,
continued fractions in this library generally only allow the first non-zero term to be negative. Short
of enumerating all the terms of a sequence, there is no way to guarantee this. Thus, hand-crafted
continued fractions for exploratory purposes may violate some assumptions of this library.
This library operates under the additional assumption that all generated terms of continued fractions are exact
integers. One may always transform an arbitrary continued fraction with rational terms to one with
integer terms. For instance, in the continued fraction 1 + (3/2)/(1 + ...) the denominator
of 2 can be eliminated by multiplying this portion of the continued fraction by 2/2, giving
1 + 3/(2 + (2* ...)).
4.2 Provided Sequences
The forms provided by
racket/sequence are
recommended but not provided. In particular, many continued fractions have initial terms that do not follow a pattern,
while subsequent terms do, so sequence-append is helpful. Additionally, some specific sequences are given.
4.2.1 Sequence Utilities
Like
map, but for sequences. If any sequence dies,
then this sequence dies, since it can no longer be guaranteed to succeed in applying the procedure argument.
Examples:
| > (require racket/sequence) |
|
|
'(0 3 6 9 12) |
Omits every other term.
Takes a term from each sequence in argument order before continuing any sequence.
If any sequence dies it is culled from the interleaving action and the main sequence continues until there are no inner sequences left
producing terms.
Example:
|
'(0 10 100 1 11 101 2 12 102 3 13 4 14 5 6 7 8 9) |
4.2.2 Specific Sequences
Endless values of the argument.
Sequences of even and odd numbers, respectively.
Examples:
|
'(10 12 14 16 18 20 22 24 26 28) |
|
'(9 11 13 15 17 19 21 23 25 27) |
An endless sequence of squares, starting with the first square that is
greater than or equal to the argument.
Examples:
|
'(9 16 25 36 49 64 81 100 121 144) |
|
'(9 16 25 36 49 64 81 100 121 144) |
Like sqaures.
Example:
|
'(0 1 8 27 64 125 216 343 512 729) |
A sequence of triangle numbers. Triangle numbers are of the form
(n*(n+1))/2.
A sequence of triangle numbers multiplied by 2.
Examples:
|
'(0 1 3 6 10 15 21 28 36 45) |
|
#t |
The first term is init. The second term is init plus the first term of the
base-sequence. The third term is the second term plus the second term of the
base-sequence. And so on. The optional argument, if present, will automatically go through
the sequence until the term is greater than or equal to strip.
4.3 Continued Fraction Creation
4.3.1 Continued Fraction Procedures
Two main forms are provided for continued fractions.
Creates the continued fraction
a0 + 1/(a1 + 1/(a2 + ...)) from the
given sequence. For the keyword argument, see
Forcing.
Creates the continued fraction
d0 + n0/(d1 + n1/(d2 + ...)) from
the
d and
n sequences. For the keyword argument, see
Forcing.
4.3.2 Forcing
Well-behaved continued fractions have terms which are always positive or always negative. Unfortunately this
is not always the case. Sometimes there is a crossover where some terms switch from being positive to being
negative. A sequence provided may at some point emit a zero at a known location.
Such terms can cause havoc with continued fraction arithmetic. For that reason, the #:force
keyword is provided. It allows the user to internally force the consumption of the specified number of
terms before any arithmetic or other output is attempted. This overrides any precision or
consume-limit parameters.
As an example, the continued fraction (-3 -2 -1 0 1 2 3 ...) is equivalant to
(-3 -2 0 2 3 ...) which is eventually just (0 4 5 6 ...). Since
internal algorithms assume that no zeros will appear, emission of a term from a
continued fraction sequence may occur prematurely if #:force is not used.
Such emission does not affect the accuracy of continued fraction arithmetic, but it does invalidate
guarantees of
precision-emit or the correctness of terms given by
base-emit.
Examples:
|
|
| > (sequence->list bad-cf) |
'(-6 1 3 3 0 -4 1 3 5 6 7 8 9 10 11 12 13 14) |
|
|
| > (sequence->list good-cf) |
'(0 6 7 8 9 10 11 12 13 14) |
| ; The overall accuracy of arithmetic is not affected: |
| ; these are both valid representations of the same rational. |
|
#t |
|
'(-5 -2 536 2 8 5 6 2 5 7 6 5 8 9 0 6 5 1 0 5) |
|
'(0 1 6 2 8 5 6 2 5 7 6 5 8 9 0 6 5 1 0 5) |