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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!
175 lines
5.6 KiB
Plaintext
175 lines
5.6 KiB
Plaintext
NAME
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appr - approximate numbers by multiples of a specified number
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SYNOPSIS
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appr(x [,y [,z]])
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TYPES
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x real, complex, matrix, list
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y real
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z integer
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return same type as x except that complex x may return a real number
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DESCRIPTION
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Return the approximate value of x as specified by a specific error
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(epsilon) and config ("appr") value.
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The default value for y is epsilon(). The default value for z is
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the current value of the "appr" configuration parameter.
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If y is zero or x is a multiple of y, appr(x,y,z) returns x. I.e.,
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there is no "approximation" - the result represents x exactly.
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In the following it is assumed y is nonzero and x is not a multiple of y.
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For real x:
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appr(x,y,z) is either the nearest multiple of y greater
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than x or the nearest multiple of y less than x. Thus, if
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we write a = appr(x,y,z) and r = x - a, then a/y is an integer
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and abs(r) < abs(y). If r > 0, we say x has been "rounded down"
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to a; if r < 0, the rounding is "up". For particular x and y,
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whether the rounding is down or up is determined by z.
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Only the 5 lowest bits of z are used, so we may assume z has been
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replaced by its value modulo 32. The type of rounding depends on
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z as follows:
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z = 0 round down or up according as y is positive or negative,
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sgn(r) = sgn(y)
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z = 1 round up or down according as y is positive or negative,
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sgn(r) = -sgn(y)
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z = 2 round towards zero, sgn(r) = sgn(x)
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z = 3 round away from zero, sgn(r) = -sgn(x)
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z = 4 round down, r > 0
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z = 5 round up, r < 0
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z = 6 round towards or from zero according as y is positive or
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negative, sgn(r) = sgn(x/y)
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z = 7 round from or towards zero according as y is positive or
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negative, sgn(r) = -sgn(x/y)
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z = 8 a/y is even
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z = 9 a/y is odd
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z = 10 a/y is even or odd according as x/y is positive or negative
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z = 11 a/y is odd or even according as x/y is positive or negative
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z = 12 a/y is even or odd according as y is positive or negative
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z = 13 a/y is odd or even according as y is positive or negative
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z = 14 a/y is even or odd according as x is positive or negative
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z = 15 a/y is odd or even according as x is positive or negative
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z = 16 to 31 abs(r) <= abs(y)/2; if there is a unique multiple
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of y that is nearest x, appr(x,y,z) is that multiple of y
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and then abs(r) < abs(y)/2. If x is midway between
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successive multiples of y, then abs(r) = abs(y)/2 and
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the value of a is as given by appr(x, y, z-16).
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Matrix or List x:
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appr(x,y,z) returns the matrix or list indexed in the same way as x,
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in which each element t has been replaced by appr(t,y,z).
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Complex x:
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Returns appr(re(x), y, z) + appr(im(x), y, z) * 1i
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PROPERTIES
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If appr(x,y,z) != x, then abs(x - appr(x,y,z)) < abs(y).
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If appr(x,y,z) != x and 16 <= z <= 31, abs(x - appr(x,y,z)) <= abs(y)/2.
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For z = 0, 1, 4, 5, 16, 17, 20 or 21, and any integer n,
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appr(x + n*y, y, z) = appr(x, y, z) + n * y.
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If y is nonzero, appr(x,y,8)/y = an odd integer n only if x = n * y.
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EXAMPLES
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; print appr(-5.44,0.1,0), appr(5.44,0.1,0), appr(5.7,1,0), appr(-5.7,1,0)
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-5.5 5.4 5 -6
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; print appr(-5.44,-.1,0), appr(5.44,-.1,0), appr(5.7,-1,0), appr(-5.7,-1,0)
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-5.4 5.5 6 -5
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; print appr(-5.44,0.1,3), appr(5.44,0.1,3), appr(5.7,1,3), appr(-5.7,1,3)
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-5.5 5.5 6 -6
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; print appr(-5.44,0.1,4), appr(5.44,0.1,4), appr(5.7,1,4), appr(-5.7,1,4)
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-5.5 5.4 5 -6
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; print appr(-5.44,0.1,6), appr(5.44,0.1,6), appr(5.7,1,6), appr(-5.7,1,6)
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-5.4 5.4 6 -5
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; print appr(-5.44,-.1,6), appr(5.44,-.1,6), appr(5.7,-1,6), appr(-5.7,-1,6)
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-5.5 5.5 6 -6
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; print appr(-5.44,0.1,9), appr(5.44,0.1,9), appr(5.7,1,9), appr(-5.7,1,9)
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-5.5 5.5 5 -5
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; print appr(-.44,0.1,11), appr(.44,0.1,11), appr(5.7,1,11), appr(-5.7,1,11)
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-.4 .5 5 -6
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; print appr(-.44,-.1,11),appr(.44,-.1,11),appr(5.7,-1,11),appr(-5.7,-1,11)
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-.5 .4 6 -5
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; print appr(-.44,0.1,12), appr(.44,0.1,12), appr(5.7,1,12), appr(-5.7,1,12)
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-.4 .5 5 -6
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; print appr(-.44,-.1,12),appr(.44,-.1,12),appr(5.7,-1,12),appr(-5.7,-1,12)
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-.5 .4 6 -5
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; print appr(-.44,0.1,15), appr(.44,0.1,15), appr(5.7,1,15), appr(-5.7,1,15)
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-.4 .5 5 -6
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; print appr(-.44,-.1,15),appr(.44,-.1,15),appr(5.7,-1,15),appr(-5.7,-1,15)
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-.4 .5 5 -6
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; x = sqrt(7-3i, 1e-20)
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; print appr(x,1e-5,0), appr(x,1e-5,1), appr(x,1e-5,2), appr(x,1e-6,3)
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2.70331-.55488i 2.70332-.55487i 2.70331-.55487i 2.70332-.55488i
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LIMITS
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none
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LINK LIBRARY
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NUMBER *qmappr(NUMBER *q, NUMBER *e, long R);
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LIST *listappr(LIST *oldlp, VALUE *v2, VALUE *v3);
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MATRIX *matappr(MATRIX *m, VALUE *v2, VALUE *v3);
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SEE ALSO
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round, bround, cfappr, 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: 1994/09/25 17:18:21
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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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