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Some folks might think: “you still use RCS”?!? And we will say, hey, at least we switched from SCCS to RCS back in … I think it was around 1994 ... at least we are keeping up! :-) :-) :-) Logs say that SCCS version 18 became RCS version 19 on 1994 March 18. RCS served us well. But now it is time to move on. And so we are switching to git. Calc releases produce a lot of file changes. In the 125 releases of calc since 1996, when I started managing calc releases, there have been 15473 file mods!
137 lines
4.8 KiB
Plaintext
137 lines
4.8 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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## 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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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://www.isthe.com/chongo/
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## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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