Release calc version 2.10.2t30

help/cfappr

@@ -0,0 +1,89 @@
+NAME
+    cfappr - approximate a real number using continued fractions
+
+SYNOPSIS
+    cfappr(x [,eps [,rnd]]) or cfappr(x, n [,rnd])
+
+TYPES
+    x		real
+    eps 	real with abs(eps) < 1, defaults to epsilon()
+    n		real with n >= 1
+    rnd		integer, defaults to config("cfappr")
+
+    return	real
+
+DESCRIPTION
+    If x is an integer or eps is zero, either form returns x.
+
+    If abs(eps) < 1, cfappr(x, eps) returns the smallest-denominator
+    number in one of the three intervals, [x, x + abs(eps)],
+    [x - abs(eps], x], [x - abs(eps)/2, x + abs(eps)/2].
+
+    If n >= 1 and den(x) > n, cfappr(x, n) returns the nearest above,
+    nearest below, or nearest, approximation to x with denominator less
+    than or equal to n.  If den(x) <= n, cfappr(x,n) returns x.
+
+    In either case when the result v is not x, how v relates to x is
+    determined by bits 0, 1, 2 and 4 of the argument rnd in the same way as
+    these bits are used in the functions round() and appr().  In the
+    following y is either eps or n.
+
+		rnd	sign of remainder x - v
+
+		0		sgn(y)
+		1		-sgn(y
+		2		sgn(x), "rounding to zero"
+		3		-sgn(x), "rounding from zero"
+		4		+, "rounding down"
+		5		-, "rounding up"
+		6		sgn(x/y)
+		7		-sgn(x/y)
+
+    If bit 4 of rnd is set, the other bits are irrelevant for the eps case;
+    thus for 16 <= rnd < 24, cfappr(x, eps, rnd) is the smallest-denominator
+    number differing from x by at most abs(eps)/2.
+
+    If bit 4 of rnd is set and den(x) > 2, the other bits are irrelevant for
+    the bounded denominator case; in the case of two equally near nearest
+    approximations with denominator less than n, cfappr(x, n, rnd)
+    returns the number with smaller denominator.   If den(x) = 2,  bits
+    0, 1 and 2 of rnd are used as described above.
+
+    If -1 < eps < 1, cfappr(x, eps, 0) may be described as the smallest
+    denominator number in the closed interval with end-points x and x - eps.
+    It follows that if abs(a - b) < 1, cfappr(a, a - b, 0) gives the smallest
+    denominator number in the interval with end-points a and b; the same
+    result is returned by cfappr(b, b - a, 0) or cfappr(a, b - a, 1).
+
+    If abs(eps) < 1 and v = cfappr(x, eps, rnd), then
+    cfappr(x, sgn(eps) * den(v), rnd) = v.
+
+    If 1 <= n < den(x), u = cfappr(x, n, 0) and v = cfappr(x, n, 1), then
+    u < x < v, den(u) <= n, den(v) <= n, den(u) + den(v) > n, and
+    v - u = 1/(den(u) * den(v)).
+
+    If x is not zero, the nearest approximation with numerator not
+    exceeding n is 1/cfappr(1/x, n, 16).
+
+EXAMPLE
+    > c = config("mode", "frac")
+    > x = 43/30; u = cfappr(x, 10, 0); v = cfappr(x, 10, 1);
+    > print u, v, x - u, v - x, v - u, cfappr(x, 10, 16)
+    10/7 13/9 1/210 1/90 1/63 10/7
+
+    > pi = pi(1e-10)
+    > print cfappr(pi, 100, 16), cfappr(pi, .01, 16), cfappr(pi, 1e-6, 16)
+    311/99 22/7 355/113
+
+    > x = 17/12; u = cfappr(x,4,0); v = cfappr(x,4,1);
+    > print u, v, x - u, v - x, cfappr(x,4,16)
+    4/3 3/2 1/12 1/12 3/2
+
+LIMITS
+    none
+
+LIBRARY
+    NUMBER *qcfappr(NUMBER *q, NUMBER *epsilon, long R)
+
+SEE ALSO
+    appr, cfsim