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Merge pull request #27347 from MaximSmolskiy:improve-solveCubic-accuracy

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Improve solveCubic accuracy #27347

### Pull Request Readiness Checklist

Fix #27323 

```
2e-13 * x^3 + x^2 - 2 * x + 1 = 0 -> x^3 + 5e12 * x^2 - 1e13 * x + 5e12 = 0
```

The problem that coefficients have quite big magnitudes and current calculations are subject to round-off error

```
Q = (a1 * a1 - 3 * a2) * (1./9)
R = (2 * a1 * a1 * a1 - 9 * a1 * a2 + 27 * a3) * (1./54)
Qcubed = Q * Q * Q = a1^6/729 - (a1^4 a2)/81 + (a1^2 a2^2)/27 - a2^3/27
R * R = R^2 = a1^6/729 - (a1^4 a2)/81 + (a1^2 a2^2)/36 + (a1^3 a3)/27 - (a1 a2 a3)/6 + a3^2/4
d = Qcubed - R * R
```

Let `a1`, `a2`, `a3` have quite big same magnitudes, then we see that `Qcubed` and `R * R` have same terms `a1^6/729` and `-(a1^4 a2)/81` (which will be reduced in `d`), but they level out the other terms (these terms have `6`th and `5`th degree and other terms - less or equal than `4`th degree).
So, if these terms will participate in the calculation, this will lead to a huge round-off error.
But if we expand the expression, then round-off error should be less
```
d = Qcubed - R * R = 1/108 (a1^2 a2^2 - 4 a2^3 - 4 a1^3 a3 + 18 a1 a2 a3 - 27 a3^2)
```

See details at https://github.com/opencv/opencv/wiki/How_to_contribute#making-a-good-pull-request

- [x] I agree to contribute to the project under Apache 2 License.
- [x] To the best of my knowledge, the proposed patch is not based on a code under GPL or another license that is incompatible with OpenCV
- [x] The PR is proposed to the proper branch
- [x] There is a reference to the original bug report and related work
- [x] There is accuracy test, performance test and test data in opencv_extra repository, if applicable
      Patch to opencv_extra has the same branch name.
- [x] The feature is well documented and sample code can be built with the project CMake
This commit is contained in:
Maxim Smolskiy
2025-05-24 09:56:48 +03:00
committed by GitHub
parent b8099d3cc2
commit 023d14ecc4
2 changed files with 32 additions and 4 deletions
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modules/core/test/test_math.cpp
+23 -2
View File
@@ -2522,7 +2522,7 @@ TEST(Core_SolveCubicCubic, accuracy)
const auto num_roots = solveCubic(coeffs, roots);
EXPECT_EQ(num_roots, 1);
EXPECT_EQ(roots[0], 1.);
EXPECT_NEAR(roots[0], 1., 1e-8);
}
{
@@ -2568,7 +2568,7 @@ TEST(Core_SolveCubicNormalizedCubic, accuracy)
const auto num_roots = solveCubic(coeffs, roots);
EXPECT_EQ(num_roots, 1);
EXPECT_EQ(roots[0], 1.);
EXPECT_NEAR(roots[0], 1., 1e-8);
}
{
@@ -2597,6 +2597,27 @@ TEST(Core_SolveCubicNormalizedCubic, accuracy)
}
}
TEST(Core_SolveCubic, regression_27323)
{
{
const std::vector<double> coeffs{2e-13, 1, -2, 1};
std::vector<double> roots;
const auto num_roots = solveCubic(coeffs, roots);
EXPECT_EQ(num_roots, 1);
EXPECT_EQ(roots[0], -5e12 - 2.);
}
{
const std::vector<double> coeffs{5e12, -1e13, 5e12};
std::vector<double> roots;
const auto num_roots = solveCubic(coeffs, roots);
EXPECT_EQ(num_roots, 1);
EXPECT_EQ(roots[0], -5e12 - 2.);
}
}
TEST(Core_SolvePoly, regression_5599)
{
// x^4 - x^2 = 0, roots: 1, -1, 0, 0