Bilinear Fractional Transformations

A bilinear fractional transformation (BiLFT) is a two-argument function of the form

(lambda (x y)
  (/ (+ (* A x y) (* B x) (* C y) D)
     (+ (* E x y) (* F x) (* G y) H)))

William Gosper figured out how to use BiLFTs to perform linear fractional operations on infinite compositions of LFTs. It isn’t as hard as it might seem.

BiLFTs do not form a group under functional composition with LFTs, but they have this interesting property: if you hold either input to the BiLFT constant, the BiLFT becomes a LFT on the other input. So if we consider just one of the inputs at a time (or the output), we can apply some group-like operations.

You can compose LFTs with BiLFTs in three ways. First, you can run the output of a BiLFT into the input of a LFT. Second and third, you can run the output of a LFT into the either the x or y input of a BiLFT. Composing a LFT with a BiLFT in any of these ways produces another BiLFT, so there is some notion of closure under composition. Composing the identity LFT with a BiLFT in any of these ways does not change anything. For any composition of a LFT with a BiLFT, you can compose the inverse of the LFT to undo the composition.

BiLFTs can also form infinite compositions, but since there are two inputs, each input can be infinite. We cannot use a Scheme stream, but we can create a two-tailed stream-like object called a binary-expression. A binary-expression is a composition of two infinite compositions. We represent a binary-expression as an object with two delayed cdrs.

(defclass binary-expression ()
  ((generation :initarg :generation
               :initform 0)
   (bilft :initarg :bilft
          :initform (error "Required initarg :bilft was omitted."))
   (delayed-left :initarg :delayed-left
                 :initform (error "Required initarg :delayed-left was omitted."))
   (delayed-right :initarg :delayed-right
                  :initform (error "Required initarg :delayed-right was omitted."))))

Like a LFT, we use binary-expressions by trying to operate on the BiLFT, but forcing one of the tails and refining the binary-expression when necessary.

(defun refine-left (binary-expression)
  (let ((delayed-left (slot-value binary-expression 'delayed-left)))
    (if (null delayed-left)
        (error "Attempt to refine-left on empty stream.")
        (let ((left (force delayed-left)))
          (make-instance 'binary-expression
                         :generation (+ (slot-value binary-expression 'generation) 1)
                         :bilft (compose-bilft-lft-x
                                 (slot-value binary-expression 'bilft)
                                 (if (empty-stream? left)
                                     (make-lft 1 1
                                               0 0)
                                     (stream-car left)))
                         :delayed-left (if (empty-stream? left)
                                           nil
                                           (stream-delayed-cdr left))
                         :delayed-right (slot-value binary-expression 'delayed-right))))))

(defun refine-right (binary-expression)
  (let ((delayed-right (slot-value binary-expression 'delayed-right)))
    (if (null delayed-right)
        (error "Attempt to refine-right on empty stream.")
        (let ((right (force delayed-right)))
          (make-instance 'binary-expression
                         :generation (+ (slot-value binary-expression 'generation) 1)
                         :bilft (compose-bilft-lft-y
                                 (slot-value binary-expression 'bilft)
                                 (if (empty-stream? right)
                                     (make-lft 1 1
                                               0 0)
                                     (stream-car right)))
                         :delayed-left (slot-value binary-expression 'delayed-left)
                         :delayed-right (if (empty-stream? right)
                                            nil
                                            (stream-delayed-cdr right)))))))

refine-fairly alternates forcing the delayed-left or delayed-right tail while refine-disjoint looks at the overlap in the ranges of the BiLFT.

(defun refine-fairly (binary-expression)
  (if (zerop (mod (slot-value binary-expression 'generation) 2))
      (if (null (slot-value binary-expression 'delayed-left))
          (refine-right binary-expression)
          (refine-left binary-expression))
      (if (null (slot-value binary-expression 'delayed-right))
          (refine-left binary-expression)
          (refine-right binary-expression))))

(defun refine-disjoint (binary-expression)
  (if (bilft-disjoint? (slot-value binary-expression 'bilft))
      (refine-right binary-expression)
      (refine-left binary-expression)))

You can do arithmetic on infinite compositions by using the appropriate BiLFT:

(defparameter bilft-add 
  (make-bilft 0 1 1 0
              0 0 0 1))

(defparameter bilft-subtract 
  (make-bilft 0 1 -1 0
              0 0 0 1))

(defparameter bilft-multiply 
  (make-bilft 1 0 0 0
              0 0 0 1))

(defparameter bilft-divide 
  (make-bilft 0 1 0 0
              0 0 1 0))

Converting a binary-expression to a LFT stream

binary-expressions can be operated on in an analagous way to an infinite composition of LFTs, but in order to nest binary expressions, we need to be able to turn one into an infinite composition of LFTs so it can become the left or right input of the other. The way to do this is to generate a stream of LFTs and their inverses. The inverses are composed into the output of the binary-expression while the LFTs are composed downstream into one of the inputs of the next binary-expression.

The math works such that we can select any LFT that has an inverse, but we want our binary-expression to represent a narrowing transformation, so we try composing a few different LFTs and their inverses and see if we still have a narrowing transformation. If we find one, we proceed with the computation, but if we cannot find a LFT and its inverse that preserves the narrowing, we refine the binary-expression by forcing one of the two delayed inputs and composing it.

(defun decompose (left composed)
  (values left (funcall (inverse-lft left) composed)))

(defun decompose-range? (left composed if-success if-failure)
  (multiple-value-bind (left right) (decompose left composed)
    (if (range? right)  ;; does it still narrow?
        (funcall if-success left right)
        (funcall if-failure))))

(defun try-decompose-digit (lft if-success if-failure)
  (decompose-range?
   (make-lft 2 1 0 1) lft
   if-success
   (lambda ()
     (decompose-range?
      (make-lft 1 0 1 2) lft
      if-success
      (lambda ()
        (decompose-range?
         (make-lft 3 1 1 3) lft
         if-success
         if-failure))))))
    
(defun binary-expression-decompose-digit (binary-expression)
  (try-decompose-digit
   (slot-value binary-expression 'bilft)
   (lambda (digit bilft)
     (values digit (make-instance 'binary-expression
                                  :generation (slot-value binary-expression 'generation)
                                  :bilft bilft
                                  :delayed-left (slot-value binary-expression 'delayed-left)
                                  :delayed-right (slot-value binary-expression 'delayed-right))))
   (lambda ()
     (binary-expression-decompose-digit (refine-disjoint binary-expression)))))

(defun binary-expression->lft-stream (binary-expression)
  (multiple-value-bind (digit remainder) (binary-expression-decompose-digit binary-expression)
    (cons-lft-stream digit 
                     (binary-expression->lft-stream remainder))))

Now we have the machinery we need to do arithmetic on pairs of LFT streams.

Infinite Expression Trees

You can only go so far with a finite number of arithmetic operations, but you can go much further with an infinite number. For example, you can create converging infinite series. We can recursively generate an infinite tree of binary expressions. We unfold the tree and call a generating function at each recursive level.

(defun unfold-expression-tree-1 (left generate counter)
  (funcall (funcall generate counter)
           left
           (delay-lft-stream (unfold-expression-tree-1 left generate (+ counter 1)))))

Often the root of the tree is special, so the usual way we call this is

(defun unfold-expression-tree (root-bilft left generate)
  (funcall root-bilft
           left
           (delay-lft-stream (unfold-expression-tree-1 left generate 1))))

Square Root of Infinite Composition

To compute the square root of an infinite composition, we create an infinite tree. Each level of the tree refines the estimate of the square root of the estimate from the next level down in the tree.

(defun sqrt-lft-stream (lft-stream)
  (unfold-expression-tree-1
    lft-stream
    (constantly (make-bilft 1 2 1 0
                            0 1 2 1))
    0))

but we need not construct an infinite tree if each level is identical. We just re-use the level:

(defun sqrt-lft-stream (lft-stream)
  (let ((stream nil))
    (setq stream
          (funcall (make-bilft 1 2 1 0
                               0 1 2 1)
                   lft-stream
                   (delay-lft-stream stream)))
    stream))

This feeds back the square root into its own computation. (These ideas are from Peter Potts and Reinhold Heckmann.)

e^x and log(x)

Peter Potts gives these formulas for e^x and log(x) where x is an infinite composition of LFTs:

(defun %exp-lft-stream (lft-stream)
  (unfold-expression-tree-1
   (lambda (n)
       (make-bilft (+ (* 2 n) 2) (+ (* 2 n) 1) (* 2 n) (+ (* 2 n) 1)
                   (+ (* 2 n) 1) (* 2 n) (+ (* 2 n) 1) (+ (* 2 n) 2)))
   0
   (funcall (inverse-lft (make-lft 1 -1 1 1)) lft-stream)))

(defun %log-lft-stream (lft-stream)
  (unfold-expression-tree
   (make-bilft 1 1 -1 -1
               0 1  1  0)
   (lambda (n)
     (make-bilft n (+ (* n 2) 1)       (+ n 1) 0
                 0       (+ n 1) (+ (* n 2) 1) n))
   lft-stream))

These infinite expression trees converge only on a limited range, so we need to use identities on the arguments and results to extend the range to cover a wide set of inputs.

Infinite Composition

If you like functional composition, you’ll like linear fractional transformations (LFTs). They behave very nicely when you compose them (in fact, they are a group under functional composition). If you compose two LFTs, you get another LFT. You can keep composing as long as you wish. Let’s compose an infinite number of LFTs.

I suppose I should argue that composing an infinite number of LFTs will give you anything at all. Consider those LFTs with non-negative coefficients. When given an input in the range [0,∞], they will output a number in the range [A/C, B/D]. The output range is narrower than, and contained within, the input range. If we compose two such LFTs, the range of output of the outermost LFT will be even narrower. As we compose more and more of these narrowing LFTs, the range of output of the outermost LFT will become narrower and narrower. If we compose an infinite number of narrowing LFTs, the output range will narrow to a single point. Curiously, the limits on the range of output of a finite composition of LFTs are always rational numbers, yet the output from infinite composition of LFTs can be an irrational real number.

There are many ways to represent an infinite number of LFTs. A generator or a co-routine, for example, but I like the stream abstraction from Scheme. This is easily implemented in Common Lisp. A delay macro creates promises:

(defclass promise ()
  ((forced? :initform nil)
   (values-or-thunk :initarg :thunk
                    :initform (error "Required initarg :thunk omitted.")))
  (:documentation "A simple call-by-need thunk."))

(defmacro delay (expression)
  “Delays evaluation of an expression and returns a promise.”
  ‘(make-instance ’promise :thunk (lambda () ,expression)))

and the force function forces evaluation:

(defun force (promise)
  “Returns the values of a promise, forcing it if necessary.”
  (check-type promise promise)
  ;; Ensure the values have been memoized.
  (unless (slot-value promise ’forced?)
    (let ((values (multiple-value-list (funcall (slot-value promise ’values-or-thunk)))))
      ;; If this is not a recursive call, memoize the result.
      ;; If this is a recursive call, the result is discarded.
      (unless (slot-value promise ’forced?)
        (setf (slot-value promise ’values-or-thunk) values)
        (setf (slot-value promise ’forced?) t))))
   ;; Return the memoized values.
   (values-list (slot-value promise ’values-or-thunk)))

A stream is a cons of an element and a promise to produce the rest of the stream:

(defclass lft-stream ()
  ((car :initarg :car
        :initform (error "Required initarg :car omitted.")
        :reader stream-car)
   (delayed-cdr :initarg :delayed-cdr
                :initform (error "Required initarg :delayed-cdr omitted.")
                :reader stream-delayed-cdr)))

(defmacro cons-lft-stream (car cdr)
  ‘(make-instance ’stream :car ,car :delayed-cdr (delay ,cdr)))

(defun stream-cdr (stream)
  (force (stream-delayed-cdr stream)))

A stream of LFTs will represent an infinite composition. The first LFT is the outermost LFT in the composition. To incrementally compose the LFTs, we force the delayed cdr of the stream to get the second LFT. We compose the first LFT and the second LFT to get a new stream car, and the cdr of the delayed cdr is the new stream cdr. That is, we start with the stream {F …}, we force the tail, getting {F G …}, then we compose the first and second elements, getting {(F∘G) …}.

(defun refine-lft-stream (lft-stream)
  (let ((tail (stream-cdr lft-stream))) ;; forces cdr
    (make-instance ’stream
                   :car (compose (stream-car lft-stream) (stream-car tail))
                   :delayed-cdr (stream-delayed-cdr tail))))

so we can now write code that operates on what we have composed so far (in the car of the LFT stream), or, if we need to incrementally compose more LFTs, we repeatedly call refine-lft-stream until we have composed enough.

For example, we can compute the nearest float to an infinite composition as follows:

(defun nearest-single (lft-stream)
  (let ((f0 (funcall (stream-car lft-stream) 0))
        (finf (funcall (stream-car lft-stream) ’infinity)))
    (if (and (numberp f0)
             (numberp finf)
             (= (coerce f0 ’single-float)
                (coerce finf ’single-float)))
        (coerce f0 ’single-float)
        (nearest-single (refine-lft-stream lft-stream)))))

The problem with infinite compositions is that they may never narrow enough to proceed with a computation. For example, suppose an infinite composition converged to a number exactly in between two adjacent floating point numbers. The upper limit of the narrowing range will always round to the float that is above, while the lower limit will always round to the float that is below. The floats will never be equal no matter how many terms of the infinite composition are composed, so functions like nearest-single will never return a value. This is a tradeoff. We either continue computing, perhaps forever, with more and more precise values, or we stop at some point, perhaps giving an incorrect answer.

Operating on Infinite Compositions

You can compose a LFT with an infinite composition of LFTs. If we compose the LFT F with the infinite composition {G H …}, we can represent this one of two ways. Either we just tack the F on to the front, getting {F G H …}, or we continue further and eagerly compose the first two elements {(F∘G) H …}. If F is, for example, a LFT that adds 10 or multiplies by 7, the effect is to add 10 to or multiply by 7 the infinite composition. In this way, we can do arithmetic involving rational numbers and infinite compositions.

Sources of Infinite Compositions

It isn’t likely that you have a lot of ad hoc LFT streams you want to compose. Instead, we want some sources of infinite compositions. There are a few useful functions of finite arguments that return infinite compositions. I got these from Peter Potts's thesis. While most of these have limited ranges of arguments, you can use identity operations to divide down the input to an acceptable range and multiply up the output to the correct answer.

√(p/q)

Reinhold Heckmann came up with this one:

(defun %sqrt-rat (p q)
  (let ((diff (- p q)))
    (labels ((rollover (num den)
               (let ((d (+ (* 2 (- den num)) diff)))
                 (if (> d 0)
                     (cons-lft-stream (make-lft 1 0 1 2) (rollover (* num 4) d))
                     (cons-lft-stream (make-lft 2 1 0 1) (rollover (- d) (* den 4)))))))
       (rollover p q))))

The LFTs that this returns are curious in that when you compose them with a range, they divide the range in two and select one or other (high or low) segment. The outermost LFT in the infinite composition will represent a range that contains the square root, and the subsequent LFTs will narrow it down by repeatedly bifurcating it and selecting the top or bottom segment.

e^x, where x is rational and 1/2 < x ≤ 2

(defun %exp-rat (x)
  (check-type x (rational (1/2) 2))
  (cons-lft-stream
    (make-lft (+ 2 x) x
              (- 2 x) x)
    (lft-stream-map (lambda (n)
                      (make-lft (+ (* 4 n) 2) x
                                x             0))
                    (naturals))))

log(x), where x is rational and x > 1

(defun %log-rat (x)
  (check-type x (rational (1) *))
  (lft-stream-map
    (lambda (n)
      (funcall (make-lft 0             (- x 1)
                         (- x 1) (+ (* n 2) 1))
               (make-lft 0       (+ n 1)
                         (+ n 1)       2)))
    (integers)))

x^y, where x and y are rational and x > 1 and 0 < y < 1

(defun %rat-pow (x y)
  (check-type x (rational (1) *))
  (check-type y (rational (0) (1)))
  (cons-lft-stream
    (make-lft y 1
              0 1)
    (lft-stream-map
     (lambda (n)
       (funcall (make-lft 0             (- x 1)
                          (- x 1) (- (* n 2) 1))
                (make-lft 0       (- n y)
                          (+ n y)       2)))
     (naturals))))

tan(x), where x is rational and 0 < x ≤ 1

(defun %rat-tan (x)
  (check-type x (rational (0) 1))
  (lft-stream-map
   (lambda (n)
     (funcall (make-lft 0 x
                        x (+ (* n 4) 1))
              (make-lft 0 x
                        x (- (* n -4) 3))))
   (integers)))

tan^-1(x), where x is rational and 0 < x ≤ 1

(defun big-k-stream (numerators denominators)
  (cons-lft-stream (make-lft 0 (stream-car numerators)
                             1 (stream-car denominators))
                   (big-k-stream (stream-cdr numerators) (stream-cdr denominators))))

(defun %rat-atan (z)
  (check-type z (rational (0) 1))
  (let ((z-squared (square z)))
    (cons-lft-stream (make-lft 0 z
                               1 1)
                     (big-k-stream (stream-map (lambda (square)
                                                 (* z-squared square))
                                               (squares))
                                   (stream-cdr (odds))))))

These functions have limited applicability. In my next couple of posts, I’ll show some functions that transform LFT streams into other LFT streams, which can be used to increase the range of inputs and outputs of these primitive sources.