Thursday, September 18, 2014

A useful, if somewhat pointless, trick with homographic functions

In my previous posts I showed that if you are applying a homographic function to a continued fraction, you can partly evaluate the answer before you completely apply the function. Instead of representing homographic functions as lambda expressions, I'll represent them as a list of the coefficients a, b, c, and d in (lambda (t) (/ (+ (* a t) b) (+ (* c t) d))). I'll represent a simple continued fraction as a stream of the integer terms in the denominators.
Here is how you partly apply a homographic function to a continued fraction:
(define (partly-apply hf cf)
  (let ((a (first  hf))
        (b (second hf))
        (c (third  hf))
        (d (fourth hf)))
    (if (empty-stream? cf)
        (values (list a a
                      c c)
        (let ((term (head cf)))
          (values (list (+ (* a term) b) a
                        (+ (* c term) d) c)
                  (tail cf))))))
Partly evaluating a homographic function involves looking at the limits of the function as t starts at 1 and goes to infinity:
(define (partly-evaluate hf)
  (let ((a (first hf))
        (b (second hf))
        (c (third hf))
        (d (fourth hf)))

    (if (and (same-sign? c (+ c d))
             (let ((limit1 (quotient      a       c))
                   (limit2 (quotient (+ a b) (+ c d))))
               (= limit1 limit2)))
        (let ((term (quotient a c)))
          (let ((new-c (- a (* c term)))
                (new-d (- b (* d term))))
            (values term (list c d new-c new-d))))
        (values #f #f))))
We can combine these two steps and make something useful. For example, we can print the value of applying a homographic function to a continued fraction incrementally, printing the most significant digits before computing further digits.
(define (print-hf-cf hf cf)
  (call-with-values (lambda () (partly-evaluate hf))
    (lambda (term hf*)
      (if (not term)
          (call-with-values (lambda () (partly-apply hf cf))
            (display term) 
            ;; take reciprocal and multiply by 10
            (let ((a (first hf*))
                  (b (second hf*))
                  (c (third hf*))
                  (d (fourth hf*)))
              (print-hf-cf (list (* c 10) (* d 10)
                                 a        b)
But how often are you going to apply a homographic function to a continued fraction? Fortunately, the identity function is homographic (coefficients are 1 0 0 1), so applying it to a continued fraction doesn't change the value. The square root of 2 is a simple continued fraction with coefficients [1 2 2 2 ...] where the 2s repeat forever. We apply the identity homographic function to it and print the results:
(printcf (list 1 0 0 1) sqrt-two)
; Quit!
As you can see, we start printing the square root of two and we don't stop printing digits until the user interrupts.

A fancier version could truncate the output and print a decimal point after the first iteration.