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Fix Gram-Schmidt bug in SelfAdjointEigenSolver::computeDirect and add small matrix benchmarks
Fix a bug in the 3x3 direct eigensolver's Gram-Schmidt orthogonalization for near-degenerate eigenvalues. The code was subtracting the projection onto eivecs.col(l) (itself) instead of onto eivecs.col(k): // Before (bug): subtracts scalar multiple of self — does nothing useful eivecs.col(l) -= eivecs.col(k).dot(eivecs.col(l)) * eivecs.col(l); // After (fix): removes component along eivecs.col(k) eivecs.col(l) -= eivecs.col(k).dot(eivecs.col(l)) * eivecs.col(k); This path is taken when two of three eigenvalues are nearly equal, which is common for covariance matrices of near-planar point clouds. Also add comprehensive small fixed-size matrix benchmarks covering the operations that dominate robotics/CV inner loops: matmul, matvec, inverse, determinant, LLT, LDLT, PartialPivLU, ColPivHouseholderQR, JacobiSVD, SelfAdjointEigenSolver (iterative and direct) for sizes 2x2 through 8x9. Note: the direct 3x3 eigensolver (computeDirect) is 3x faster than the iterative solver but has 5-6 orders of magnitude worse residuals for near-degenerate eigenvalues. This is inherent to the closed-form algorithm, not a consequence of the Gram-Schmidt bug. Users should prefer compute() when accuracy matters and computeDirect() only when speed is critical and eigenvalues are well-separated. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
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@@ -691,7 +691,7 @@ struct direct_selfadjoint_eigenvalues<SolverType, 3, false> {
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if (d0 <= 2 * Eigen::NumTraits<Scalar>::epsilon() * d1) {
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// If d0 is too small, then the two other eigenvalues are numerically the same,
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// and thus we only have to ortho-normalize the near orthogonal vector we saved above.
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eivecs.col(l) -= eivecs.col(k).dot(eivecs.col(l)) * eivecs.col(l);
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eivecs.col(l) -= eivecs.col(k).dot(eivecs.col(l)) * eivecs.col(k);
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eivecs.col(l).normalize();
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} else {
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tmp = scaledMat;
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