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#| -*- Scheme -*-
Copyright (c) 1987, 1988, 1989, 1990, 1991, 1995, 1997, 1998,
1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006,
2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014,
2015, 2016, 2017, 2018, 2019, 2020
Massachusetts Institute of Technology
This file is part of MIT scmutils.
MIT scmutils is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or (at
your option) any later version.
MIT scmutils is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
General Public License for more details.
You should have received a copy of the GNU General Public License
along with MIT scmutils; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301,
USA.
|#
;;; Some processes, such as finding the roots of a polynomial, can
;;; benefit by heuristic rounding of results (to a nearby rational).
(declare (usual-integrations))
;;; Heuristic rounding will occur to a rational within
(define heuristic-rounding-tolerance 1.0e-13)
;;; that is expressible with a denominator less than the
(define heuristic-rounding-denominator 20)
;;; if such a rational exists.
;;; Complex numbers with one part small enough are forced to be real
;;; or imaginary.
(define heuristic-one-part-insignificant 1.0e-10)
;;; If both real part and imaginary part of a complex number are tiny,
;;; it is also set to zero. This is dangerous.
(define heuristic-rounding-tiny 1.0e-15)
;;; If the number is a small rational multiple of an important
;;; constant, we may substitute the symbolic constant:
(define heuristic-symbolize? #t)
(define (heuristic-canonicalize-real a #!optional symbolize? zero-threshold)
(if (default-object? symbolize?) (set! symbolize? heuristic-symbolize?))
(if (default-object? zero-threshold) (set! zero-threshold heuristic-rounding-tiny))
(h-c-r a symbolize? zero-threshold))
(define (heuristic-round-real x #!optional zero-threshold)
(if (default-object? zero-threshold)
(h-c-r x #f heuristic-rounding-tiny)
(h-c-r x #f zero-threshold)))
(define (h-c-r a symbolize? zero-threshold)
(if (and zero-threshold (< (abs a) zero-threshold))
0
(let lp ((ideas *important-numbers*))
(if (null? ideas)
a
(let* ((ag (/ a (caar ideas)))
(af (rationalize->exact ag heuristic-rounding-tolerance)))
(if (and (not (= af 0))
(< (denominator af) heuristic-rounding-denominator))
(if symbolize?
(symb:* af (cadar ideas)) ;symbolic version
(* af (caar ideas)))
(lp (cdr ideas))))))))
(define *important-numbers*
`((1 1)
(,pi :pi)
(,(/ 1 pi) (/ 1 :pi))
(,(exp 1) (exp 1))
(,(exp -1) (exp -1))
(,(sqrt 2) (sqrt 2))
(,(sqrt 3) (sqrt 3))
(,(sqrt 5) (sqrt 5))
(,(sqrt 7) (sqrt 7))
(,:euler :euler)
(,:phi :phi)
))
(define (heuristic-canonicalize-complex z #!optional symbolize? zero-threshold)
(if (default-object? symbolize?) (set! symbolize? heuristic-symbolize?))
(if (default-object? zero-threshold) (set! zero-threshold heuristic-rounding-tiny))
(h-c-c z symbolize? zero-threshold))
(define (heuristic-round-complex z #!optional zero-threshold)
(if (default-object? zero-threshold)
(h-c-c z #f heuristic-rounding-tiny)
(h-c-c z #f zero-threshold)))
(define (h-c-c z symbolize? zero-threshold)
(if (real? z)
(h-c-r (real-part z) symbolize? zero-threshold) ;gets 3.5 from 3.5+0.0i
(let ((r (real-part z)) (i (imag-part z)))
(let ((ar (abs r)) (ai (abs i)))
(cond ((and (< ar zero-threshold)
(< ai zero-threshold))
0)
((< ai (* heuristic-one-part-insignificant ar))
(h-c-r r symbolize? zero-threshold))
((< ar (* heuristic-one-part-insignificant ai))
(g:make-rectangular 0 (h-c-r i symbolize? zero-threshold)))
(else
(let* ((a (angle z))
(af (h-c-r a symbolize? zero-threshold)))
(if (or (not (number? af)) (not (= af a)))
(g:make-polar (h-c-r (magnitude z) symbolize? zero-threshold)
af)
(g:make-rectangular (h-c-r r symbolize? zero-threshold)
(h-c-r i symbolize? zero-threshold))))))))))
;;; (set! heuristic-number-canonicalizer heuristic-canonicalize-complex)
#|
;;; Watch out--symb:pi is now in numsymb.scm
(define (heuristic-round-angle a)
(let* ((ag (/ a :pi))
(af (heuristic-round-real ag)))
(if (< (abs (- ag af)) heuristic-rounding-tolerance)
(if heuristic-symbolize?
(g:* af symb:pi)
(* af :pi))
a)))
(define symb:pi ':pi)
(define symb:-pi (symb:- 0 symb:pi))
(define symb:pi/6 (symb:/ symb:pi 6))
(define symb:-pi/6 (symb:- 0 symb:pi/6))
(define symb:pi/3 (symb:/ symb:pi 3))
(define symb:-pi/3 (symb:- 0 symb:pi/3))
(define symb:pi/2 (symb:/ symb:pi 2))
(define symb:-pi/2 (symb:- 0 symb:pi/2))
(define symb:pi/4 (symb:/ symb:pi 4))
(define symb:-pi/4 (symb:- 0 symb:pi/4))
(define symb:2pi (symb:* 2 symb:pi))
(define symb:-2pi (symb:- 0 symb:2pi))
|#
|
