dlasd2/slasd2: Increase deflation tolerance to match dlasd7/slasd7#1286
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dlasd2/slasd2: Increase deflation tolerance to match dlasd7/slasd7#1286jschueller wants to merge 1 commit into
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Fix DBDSDC/SBDSDC returning non-orthogonal U/V for bidiagonal matrices with many nearly-equal singular values (e.g., all singular values ≈ 1). The divide-and-conquer bidiagonal SVD has two code paths: - Full vector path (DLASD2/SLASD2): deflation tolerance = 8 * EPS - Compact path (DLASD7/SLASD7): deflation tolerance = 64 * EPS With the weaker tolerance (8*EPS), singular values differing by only ~10*EPS (e.g., ~2e-15 for double precision) escape deflation. The subsequent Z computation in DLASD3 then suffers catastrophic cancellation from denominators (σ_i - σ_j) that are tiny, polluting the singular vectors and causing loss of orthogonality. Raise DLASD2 and SLASD2 to 64*EPS, matching DLASD7 and SLASD7, so more close singular values are deflated and the singular-vector computation remains stable. Closes Reference-LAPACK#255
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Fix DBDSDC/SBDSDC returning non-orthogonal U/V for bidiagonal matrices with many nearly-equal singular values (e.g., all singular values ≈ 1).
The divide-and-conquer bidiagonal SVD has two code paths:
With the weaker tolerance (8EPS), singular values differing by only ~10EPS (e.g., ~2e-15 for double precision) escape deflation. The subsequent Z computation in DLASD3 then suffers catastrophic cancellation from denominators (σ_i - σ_j) that are tiny, polluting the singular vectors and causing loss of orthogonality.
Raise DLASD2 and SLASD2 to 64*EPS, matching DLASD7 and SLASD7, so more close singular values are deflated and the singular-vector computation remains stable.
Closes #255