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116 lines
4.1 KiB
Plaintext
116 lines
4.1 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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## 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: cfappr,v 29.1 1999/12/14 09:15:44 chongo Exp $
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## @(#) $Source: /usr/local/src/cmd/calc/help/RCS/cfappr,v $
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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://reality.sgi.com/chongo/
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## Share and enjoy! :-) http://reality.sgi.com/chongo/tech/comp/calc/
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