All linearly iterative algorithms can be transformed into the form
x=f(x) for some value
x and some function
f(x). This system provides a way to perform an iterative algorithm as a process of finding the solution of this equation.
Furthermore, many existing implementations provide very limited control over how iterations are performed: what if the value is bound and the iterative step takes it out the boundaries? what if one wants to log the progress of computation? what if the computation needs to be stopped after some iterations?..
CL-ITERATIVE system provides the refined control over the progress of iterations by means of special structures called (unsurprisingly) controls.
The library was tested with SBCL (1.2.14) and ECL (15.3.7) on Linux (Ubuntu 15.10 x86-64).
The key idea of the method is to perform the computation only if
ITERATOR:ITERATOR object is in state
CONTINUE and stop otherwise (for category theory lovers and purists:
ITERATOR:ITERATOR is amalgamated
Either monad on top of another
- successfully stops the computation if certain condition on computation value is reached.
- stops the computation with a failure if certain condition on computation value is reached.
- stops the computation with a failure if number of iterations exceeds the limit.
- logs the progress of iterative computation, can be viewed as a probe.
- successfully stops the computation if the value has converged in some sense (using user-specified predicate of closeness
CLOSE-P; for mathematical purists: user specifies the topology of the computation space).
- simplified version of
CONVERGED-VALUEfor numbers where closeness is defined as
- general control that changes the value according to specified function. This control can be used, for example, to keep the value bound.
CL-ITERATIVE-EX provides the extension for some of these controls to add extra info to the
ITERATOR:ITERATOR (or rather to extended
ITERATOREX:ITERATOREX) computation object (useful to identify why computation had stoped).
A control, in general, is any object on which two methods
APPLY-CONTROL are specialized. This way the library of available controls can be extended by a user. A sequence of controls can be combined together into a single control using
FIXED-POINTprovide the entry point into iterative algorithms.
FIXED-POINTis a bit more end-user oriented. It accepts as arguments:
- the implementation of
- initial approximation.
- the control applied before the iterative algorithm starts.
- a combined control that is applied after each update of the value by
f(x). This control should contain the way to stop the computation.
- final treat of the computation value after all the iterations are finished.
=ITERATE= is similar, except it accepts the initial computation object instead of
INIT-VALUE and iterations are defined by controls only (in fact,
FUNCTION wrapped into the control
Consider the problem of computing the square root of a number
S using Heron's method:
--- x = \/ S : 1 / S \ x = - |x + --- | n+1 2 \ n x / n x = 1 0
The following function implements it:
(defun sqrt-heron (s) (flet ((improve (x) (* 0.5d0 (+ x (/ s x))))) (multiple-value-bind (final-x successful-p info) (fixed-point #'improve 1d0 :pre-treat (add-info) ; add stopping info :controls (combine-controls (converged-number-with-id) ; converge with default precision (limit-iterations-with-id 20))) ; limit to 20 iterations ;; Just in case did not converge: shouldn't happen for any reasonable S > 0 ;; due to quadratic convergence of the algorithm (assert successful-p () "Could not find the square root of S = ~A" s) ;; Just for illustrative purposes: return extra info - why computation ;; was stopped? (values final-x info))))
If want to find square root of 4,
> (sqrt-heron 4d0) 2.0d0 ((:CONVERGED-NUMBER))
If we want to peek into how the computation proceeds, we can add the logging function:
(defun sqrt-heron (s) (flet ((improve (x) (* 0.5d0 (+ x (/ s x)))) (log-function (indicator x) ; log computation (if (eq indicator :init) (format t "~&INIT: x = ~A~%" x) (format t "~&x = ~A~%" x)))) (multiple-value-bind (final-x successful-p info) (fixed-point #'improve 1d0 :pre-treat (add-info) :controls (combine-controls (log-computation #'log-function) ; add it before convergence test (converged-number-with-id) (limit-iterations-with-id 20))) (assert successful-p () "Could not find the square root of S = ~A" s) (values final-x info))))
Then, the output and the result will look as follows:
> (sqrt-heron 4d0) INIT: x = 1.0d0 x = 2.5d0 x = 2.05d0 x = 2.000609756097561d0 x = 2.0000000929222947d0 x = 2.000000000000002d0 x = 2.0d0 2.0d0 ((:CONVERGED-NUMBER))
Check the system
CL-ITERATIVE-TESTS for more examples.
Copyright (c) 2016 Alexey Cherkaev
Distributed under LGPLv3 license.
- Alexey Cherkaev (mobius-eng), Alexey Cherkaev (mobiuseng)
- LGPLv3, GPLv3