dbbcsd/sbbcsd/cbbcsd/zbbcsd: fix insufficiently accurate U2 in CS dec…#1293
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dbbcsd/sbbcsd/cbbcsd/zbbcsd: fix insufficiently accurate U2 in CS dec…#1293jschueller wants to merge 1 commit into
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…omposition The Golub-Reinsch-SVD-style iteration in xBBCSD used an absolute convergence threshold THRESH ~ 90*EPS to decide when off-diagonal bulges are negligible and can be skipped. For rows where the diagonal entries are small (tiny singular values in the ratio), this absolute threshold is too large relative to the diagonal, causing the algorithm to stop chasing bulges prematurely and producing inaccurate singular vectors (||X21 - U2 D2 V^*|| up to 50*EPS ||X21||). Fix: change all bulge-convergence checks (RESTART flags in the inner loop, initial bulge-chase decisions, and the IMAX-1 cleanup) from absolute to relative by scaling THRESH by the adjacent diagonal entries: RESTART11 = |B11E|^2+|BULGE|^2 <= (THRESH * MAX(|B11D(I-1)|,|B11D(I)|,UNFL))^2 This mirrors DBDSQR's relative convergence check |E| <= TOL*|D|. Applied to all 4 bidiagonal blocks (B11/B21/B12/B22) at all 3 check-points in the iteration. Fixes Reference-LAPACK#965
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…omposition
The Golub-Reinsch-SVD-style iteration in xBBCSD used an absolute convergence threshold THRESH ~ 90EPS to decide when off-diagonal bulges are negligible and can be skipped. For rows where the diagonal entries are small (tiny singular values in the ratio), this absolute threshold is too large relative to the diagonal, causing the algorithm to stop chasing bulges prematurely and producing inaccurate singular vectors (||X21 - U2 D2 V^|| up to 50*EPS ||X21||).
Fix: change all bulge-convergence checks (RESTART flags in the inner loop, initial bulge-chase decisions, and the IMAX-1 cleanup) from absolute to relative by scaling THRESH by the adjacent diagonal entries:
RESTART11 = |B11E|^2+|BULGE|^2 <= (THRESH * MAX(|B11D(I-1)|,|B11D(I)|,UNFL))^2
This mirrors DBDSQR's relative convergence check |E| <= TOL*|D|. Applied to all 4 bidiagonal blocks (B11/B21/B12/B22) at all 3 check-points in the iteration.
Fixes #965