In some cases the previous algorithm would not return the closest
approximation. This would happen when a semi-convergent was the
closest, as the previous algorithm would only consider convergents.
As an example, consider an initial value of 5/4, and trying to find the
closest approximation with a maximum of 4 for numerator and denominator.
The previous algorithm would return 1/1 as the closest approximation,
while this version will return the correct answer of 4/3.
To do this, the main loop performs effectively the same operations as it
did before. It must now keep track of the last three approximations,
n2/d2 .. n0/d0, while before it only needed the last two.
If an exact answer is not found, the algorithm will now calculate the
best semi-convergent term, t, which is a single expression with two
divisions:
min((max_numerator - n0) / n1, (max_denominator - d0) / d1)
This will be used if it is better than previous convergent. The test
for this is generally a simple comparison, 2*t > a. But in an edge
case, where the convergent's final term is even and the best allowable
semi-convergent has a final term of exactly half the convergent's final
term, the more complex comparison (d0*dp > d1*d) is used.
I also wrote some comments explaining the code. While one still needs
to look up the math elsewhere, they should help a lot to follow how the
code relates to that math.
Cc: Oskar Schirmer <[email protected]>
Signed-off-by: Trent Piepho <[email protected]>
---
lib/rational.c | 63 +++++++++++++++++++++++++++++++++++++++-----------
1 file changed, 50 insertions(+), 13 deletions(-)
diff --git a/lib/rational.c b/lib/rational.c
index ba7443677c90..31fb27db2deb 100644
--- a/lib/rational.c
+++ b/lib/rational.c
@@ -3,6 +3,7 @@
* rational fractions
*
* Copyright (C) 2009 emlix GmbH, Oskar Schirmer <[email protected]>
+ * Copyright (C) 2019 Trent Piepho <[email protected]>
*
* helper functions when coping with rational numbers
*/
@@ -10,6 +11,7 @@
#include <linux/rational.h>
#include <linux/compiler.h>
#include <linux/export.h>
+#include <linux/kernel.h>
/*
* calculate best rational approximation for a given fraction
@@ -33,30 +35,65 @@ void rational_best_approximation(
unsigned long max_numerator, unsigned long max_denominator,
unsigned long *best_numerator, unsigned long *best_denominator)
{
- unsigned long n, d, n0, d0, n1, d1;
+ /* n/d is the starting rational, which is continually
+ * decreased each iteration using the Euclidean algorithm.
+ *
+ * dp is the value of d from the prior iteration.
+ *
+ * n2/d2, n1/d1, and n0/d0 are our successively more accurate
+ * approximations of the rational. They are, respectively,
+ * the current, previous, and two prior iterations of it.
+ *
+ * a is current term of the continued fraction.
+ */
+ unsigned long n, d, n0, d0, n1, d1, n2, d2;
n = given_numerator;
d = given_denominator;
n0 = d1 = 0;
n1 = d0 = 1;
+
for (;;) {
- unsigned long t, a;
- if ((n1 > max_numerator) || (d1 > max_denominator)) {
- n1 = n0;
- d1 = d0;
- break;
- }
+ unsigned long dp, a;
+
if (d == 0)
break;
- t = d;
+ /* Find next term in continued fraction, 'a', via
+ * Euclidean algorithm.
+ */
+ dp = d;
a = n / d;
d = n % d;
- n = t;
- t = n0 + a * n1;
+ n = dp;
+
+ /* Calculate the current rational approximation (aka
+ * convergent), n2/d2, using the term just found and
+ * the two prior approximations.
+ */
+ n2 = n0 + a * n1;
+ d2 = d0 + a * d1;
+
+ /* If the current convergent exceeds the maxes, then
+ * return either the previous convergent or the
+ * largest semi-convergent, the final term of which is
+ * found below as 't'.
+ */
+ if ((n2 > max_numerator) || (d2 > max_denominator)) {
+ unsigned long t = min((max_numerator - n0) / n1,
+ (max_denominator - d0) / d1);
+
+ /* This tests if the semi-convergent is closer
+ * than the previous convergent.
+ */
+ if (2u * t > a || (2u * t == a && d0 * dp > d1 * d)) {
+ n1 = n0 + t * n1;
+ d1 = d0 + t * d1;
+ }
+ break;
+ }
n0 = n1;
- n1 = t;
- t = d0 + a * d1;
+ n1 = n2;
d0 = d1;
- d1 = t;
+ d1 = d2;
}
*best_numerator = n1;
*best_denominator = d1;
--
2.20.1
