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141 lines
5.0 KiB
Plaintext
141 lines
5.0 KiB
Plaintext
NAME
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cfsim - simplify a value using continued fractions
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SYNOPSIS
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cfsim(x [,rnd])
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TYPES
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x real
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rnd integer, defaults to config("cfsim")
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return real
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DESCRIPTION
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If x is not an integer, cfsim(x, rnd) returns either the nearest
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above x, or the nearest below x, number with denominator less than
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den(x). If x is an integer, cfsim(x, rnd) returns x + 1, x - 1, or 0.
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Which of the possible results is returned is controlled
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by bits 0, 1, 3 and 4 of the parameter rnd.
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For 0 <= rnd < 4, the sign of the remainder x - cfsim(x, rnd) is
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as follows:
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rnd sign of x - cfsim(x, rnd)
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0 +, as if rounding down
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1 -. as if rounding up
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2 sgn(x), as if rounding to zero
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3 -sgn(x), as if rounding from zero
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This corresponds to the use of rnd for functions like round(x, n, rnd).
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If bit 3 or 4 of rnd is set, the lower order bits are ignored; bit 3
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is ignored if bit 4 is set. Thusi, for rnd > 3, it sufficient to
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consider the two cases rnd = 8 and rnd = 16.
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If den(x) > 2, cfsim(x, 8) returns the value of the penultimate simple
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continued-fraction approximant to x, i.e. if:
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x = a_0 + 1/(a_1 + 1/(a_2 + ... + 1/a_n) ...)),
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where a_0 is an integer, a_1, ..., a_n are positive integers,
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and a_n >= 2, the value returned is that of the continued fraction
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obtained by dropping the last quotient 1/a_n.
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If den(x) > 2, cfsim(x, 16) returns the nearest number to x with
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denominator less than den(x). In the continued-fraction representation
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of x described above, this is given by replacing a_n by a_n - 1.
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If den(x) = 2, the definition adopted is to round towards zero for the
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approximant case (rnd = 8) and from zero for the "nearest" case (rnd = 16).
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For integral x, cfsim(x, 8) returns zero, cfsim(x,16) returns x - sgn(x).
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In summary, for cfsim(x, rnd) when rnd = 8 or 16, the results are:
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rnd integer x half-integer x den(x) > 2
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8 0 x - sgn(x)/2 approximant
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16 x - sgn(x) x + sgn(x)/2 nearest
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From either cfsim(x, 0) and cfsim(x, 1), the other is easily
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determined: if one of them has value w, the other has value
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(num(x) - num(w))/(den(x) - den(w)). From x and w one may find
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other optimal rational numbers near x; for example, the smallest-
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denominator number between x and w is (num(x) + num(w))/(den(x) + den(w)).
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If x = n/d and cfsim(x, 8) = u/v, then for k * v < d, the k-th member of
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the sequence of nearest approximations to x with decreasing denominators
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on the other side of x is (n - k * u)/(d - k * v). This is nearer
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to or further from x than u/v according as 2 * k * v < or > d.
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Iteration of cfsim(x,8) until an integer is obtained gives a sequence of
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"good" approximations to x with decreasing denominators and
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correspondingly decreasing accuracy; each denominator is less than half
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the preceding denominator. (Unlike the "forward" sequence of
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continued-fraction approximants these are not necessarily alternately
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greater than and less than x.)
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Some other properties:
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For rnd = 0 or 1 and any x, or rnd = 8 or 16 and x with den(x) > 2:
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cfsim(n + x, rnd) = n + cfsim(x, rnd).
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This equation also holds for the other values of rnd if n + x and x
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have the same sign.
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For rnd = 2, 3, 8 or 16, and any x:
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cfsim(-x, rnd) = -cfsim(x, rnd).
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If rnd = 8 or 16, except for integer x or 1/x for rnd = 8, and
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zero x for rnd = 16:
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cfsim(1/x, rnd) = 1/cfsim(x, rnd).
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EXAMPLE
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> c = config("mode", "frac");
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> print cfsim(43/30, 0), cfsim(43/30, 1), cfsim(43/30, 8), cfsim(43/30,16)
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10/7 33/23 10/7 33/23
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> x = pi(1e-20); c = config("mode", "frac");
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> while (!isint(x)) {x = cfsim(x,8); if (den(x) < 1e6) print x,:;}
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1146408/364913 312689/99532 104348/33215 355/113 22/7 3
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LIMITS
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none
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LINK LIBRARY
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NUMBER *qcfsim(NUMBER *x, long rnd)
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SEE ALSO
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cfappr
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## Copyright (C) 1999 Landon Curt Noll
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##
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## Calc is open software; you can redistribute it and/or modify it under
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## the terms of the version 2.1 of the GNU Lesser General Public License
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## as published by the Free Software Foundation.
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##
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## Calc is distributed in the hope that it will be useful, but WITHOUT
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## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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## Public License for more details.
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##
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## A copy of version 2.1 of the GNU Lesser General Public License is
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## distributed with calc under the filename COPYING-LGPL. You should have
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## received a copy with calc; if not, write to Free Software Foundation, Inc.
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## 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
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##
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## @(#) $Revision: 29.1 $
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## @(#) $Id: cfsim,v 29.1 1999/12/14 09:15:44 chongo Exp $
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## @(#) $Source: /usr/local/src/cmd/calc/help/RCS/cfsim,v $
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##
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## Under source code control: 1994/09/30 01:29:45
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## File existed as early as: 1994
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##
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## chongo <was here> /\oo/\ http://reality.sgi.com/chongo/
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## Share and enjoy! :-) http://reality.sgi.com/chongo/tech/comp/calc/
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