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https://issues.apache.org/jira/browse/MATH-241?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel
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Phil Steitz updated MATH-241:
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Attachment: binomialPatch.txt
First, thanks for reporting this. Due to log/exp rounding and double/long
conversion, the current code returns bad values for many long-representable
values, starting as low as n = 48. The returned value can be off by as much as
200,000. The error in binomial(66, 29) is 214,880. All b(n,k) for n < 48 are
exact.
Attached is a patch that ensures accuracy up to n = 200 (specified as a
constant) and allows the user to force exact computation for values beyond this
if desired. For n <= 200, the implementation works like an unwound recursive
implementation. I also improved the accuracy of the double-valued and log
versions. The latter perform better than the current implementations, but the
long-valued version is approximately 8x slower than the current version. I did
not benchmark the BigInteger version, but suspect that would be slower still.
The most accurate (for n <= 200) non-recursive formula that I could find is the
one that I implemented in the double version.
I also investigated overflow behavior and added tests to confirm correctness.
As stated in the API doc, overflows start at n = 67. For n = 200, values of k
less than 14 or greater than 186 can still be computed without overflow; but
all others throw ArithmeticException.
I would appreciate feedback on the patch and any better ideas on how to fix the
problem.
> MathUtils.binomialCoefficient(n,k) fails for large results
> ----------------------------------------------------------
>
> Key: MATH-241
> URL: https://issues.apache.org/jira/browse/MATH-241
> Project: Commons Math
> Issue Type: Bug
> Affects Versions: 2.0
> Reporter: Christian Semrau
> Assignee: Phil Steitz
> Fix For: 2.0
>
> Attachments: binomialPatch.txt
>
>
> Probably due to rounding errors, MathUtils.binomialCoefficient(n,k) fails for
> results near Long.MAX_VALUE.
> The existence of failures can be demonstrated by testing the recursive
> property:
> {noformat}
> assertEquals(MathUtils.binomialCoefficient(65,32) +
> MathUtils.binomialCoefficient(65,33),
> MathUtils.binomialCoefficient(66,33));
> {noformat}
> Or by directly using the (externally calculated and hopefully correct)
> expected value:
> {noformat}
> assertEquals(7219428434016265740L,
> MathUtils.binomialCoefficient(66,33));
> {noformat}
> I suggest a nonrecursive test implementation along the lines of
> {code:title=MathUtilsTest.java|borderStyle=solid}
> /**
> * Exact implementation using BigInteger and the explicit formula
> * (n, k) == ((k-1)*...*n) / (1*...*(n-k))
> */
> public static long binomialCoefficient(int n, int k) {
> if (k == 0 || k == n)
> return 1;
> BigInteger result = BigInteger.ONE;
> for (int i = k + 1; i <= n; i++) {
> result = result.multiply(BigInteger.valueOf(i));
> }
> for (int i = 1; i <= n - k; i++) {
> result = result.divide(BigInteger.valueOf(i));
> }
> if (result.compareTo(BigInteger.valueOf(Long.MAX_VALUE)) > 0) {
> throw new ArithmeticException(
> "Binomial coefficient overflow: " + n + ", "
> + k);
> }
> return result.longValue();
> }
> {code}
> Which would allow you to test the expected values directly:
> {noformat}
> assertEquals(binomialCoefficient(66,33),
> MathUtils.binomialCoefficient(66,33));
> {noformat}
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