Generic iterative algorithm with multiple controls

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Alexey Cherkaev (mobius-eng), Alexey Cherkaev (mobiuseng)




All linearly iterative algorithms can be transformed into the formx=f(x) for some value x and some function f(x). This systemprovides a way to perform an iterative algorithm as a process offinding 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).

2Modus operandi

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 Either monad).

The state of computation can be changed by means of controls. This system provides the following controls:

successfully stops the computation if certaincondition on computation value is reached.
stops the computation with a failure if certaincondition on computation value is reached.
stops the computation with a failure if numberof iterations exceeds the limit.
logs the progress of iterative computation, canbe viewed as a probe.
successfully stops the computation if the valuehas converged in some sense (using user-specified predicate ofcloseness CLOSE-P; for mathematical purists: user specifies thetopology of the computation space).
simplified version of CONVERGED-VALUE fornumbers where closeness is defined as |x-y|<eps.
general control that changes the value according tospecified function. This control can be used, for example, tokeep the value bound.

Packaged 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 INIT-CONTROL and 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 COMBINE-CONTROLS.

Functions ITERATE and FIXED-POINT provide the entry point into iterative algorithms. FIXED-POINT is a bit more end-user oriented. It accepts as arguments:

the implementation of f(x).
initial approximation.
the control applied before the iterative algorithmstarts.
a combined control that is applied after each updateof the value by f(x). This control should contain the way tostop the computation.
final treat of the computation value after all theiterations 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, FIXED-POINT calls ITERATE with FUNCTION wrapped into the control (ALTER-VALUE FUNCTION)).


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   /

    x  = 1

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)
           #'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)

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)
           #'improve 1d0
           :pre-treat (add-info)
           :controls (combine-controls
                      (log-computation #'log-function) ; add it before convergence test
                      (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

Check the system CL-ITERATIVE-TESTS for more examples.


Copyright (c) 2016 Alexey Cherkaev

Distributed under LGPLv3 license.

Dependencies (3)

  • alexandria
  • fiveam
  • optima

Dependents (0)

    • GitHub
    • Quicklisp