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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!
112 lines
4.0 KiB
Plaintext
112 lines
4.0 KiB
Plaintext
NAME
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cfappr - approximate a real number using continued fractions
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SYNOPSIS
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cfappr(x [,eps [,rnd]]) or cfappr(x, n [,rnd])
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TYPES
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x real
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eps real with abs(eps) < 1, defaults to epsilon()
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n real with n >= 1
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rnd integer, defaults to config("cfappr")
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return real
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DESCRIPTION
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If x is an integer or eps is zero, either form returns x.
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If abs(eps) < 1, cfappr(x, eps) returns the smallest-denominator
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number in one of the three intervals, [x, x + abs(eps)],
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[x - abs(eps], x], [x - abs(eps)/2, x + abs(eps)/2].
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If n >= 1 and den(x) > n, cfappr(x, n) returns the nearest above,
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nearest below, or nearest, approximation to x with denominator less
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than or equal to n. If den(x) <= n, cfappr(x,n) returns x.
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In either case when the result v is not x, how v relates to x is
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determined by bits 0, 1, 2 and 4 of the argument rnd in the same way as
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these bits are used in the functions round() and appr(). In the
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following y is either eps or n.
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rnd sign of remainder x - v
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0 sgn(y)
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1 -sgn(y
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2 sgn(x), "rounding to zero"
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3 -sgn(x), "rounding from zero"
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4 +, "rounding down"
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5 -, "rounding up"
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6 sgn(x/y)
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7 -sgn(x/y)
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If bit 4 of rnd is set, the other bits are irrelevant for the eps case;
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thus for 16 <= rnd < 24, cfappr(x, eps, rnd) is the smallest-denominator
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number differing from x by at most abs(eps)/2.
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If bit 4 of rnd is set and den(x) > 2, the other bits are irrelevant for
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the bounded denominator case; in the case of two equally near nearest
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approximations with denominator less than n, cfappr(x, n, rnd)
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returns the number with smaller denominator. If den(x) = 2, bits
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0, 1 and 2 of rnd are used as described above.
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If -1 < eps < 1, cfappr(x, eps, 0) may be described as the smallest
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denominator number in the closed interval with end-points x and x - eps.
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It follows that if abs(a - b) < 1, cfappr(a, a - b, 0) gives the smallest
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denominator number in the interval with end-points a and b; the same
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result is returned by cfappr(b, b - a, 0) or cfappr(a, b - a, 1).
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If abs(eps) < 1 and v = cfappr(x, eps, rnd), then
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cfappr(x, sgn(eps) * den(v), rnd) = v.
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If 1 <= n < den(x), u = cfappr(x, n, 0) and v = cfappr(x, n, 1), then
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u < x < v, den(u) <= n, den(v) <= n, den(u) + den(v) > n, and
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v - u = 1/(den(u) * den(v)).
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If x is not zero, the nearest approximation with numerator not
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exceeding n is 1/cfappr(1/x, n, 16).
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EXAMPLE
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; c = config("mode", "frac")
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; x = 43/30; u = cfappr(x, 10, 0); v = cfappr(x, 10, 1);
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; print u, v, x - u, v - x, v - u, cfappr(x, 10, 16)
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10/7 13/9 1/210 1/90 1/63 10/7
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; pi = pi(1e-10)
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; print cfappr(pi, 100, 16), cfappr(pi, .01, 16), cfappr(pi, 1e-6, 16)
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311/99 22/7 355/113
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; x = 17/12; u = cfappr(x,4,0); v = cfappr(x,4,1);
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; print u, v, x - u, v - x, cfappr(x,4,16)
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4/3 3/2 1/12 1/12 3/2
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LIMITS
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none
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LINK LIBRARY
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NUMBER *qcfappr(NUMBER *q, NUMBER *epsilon, long R)
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SEE ALSO
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appr, cfsim
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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 01:23:59
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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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