slarfgp/dlarfgp: avoid overflow when computing 1/ALPHA#1290
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In the general case (xnorm > eps*|alpha|, beta >= 0), the Householder reflector formula computes ALPHA = -XNORM^2 / (alpha + beta) which can be subnormal even when beta itself is safely above SMLNUM. When this subnormal ALPHA was used in the subsequent CALL SSCAL( N-1, ONE / ALPHA, X, INCX ) the reciprocal overflowed (e.g. ALPHA = 2^-137 → ONE/ALPHA = 2^137 which exceeds SP max ~3.4e38). Fix: guard the SSCAL with ABS(ALPHA) < SMLNUM. When ALPHA is very small, scale X by ONE/SMLNUM then by SMLNUM/ALPHA — both safe since ONE/SMLNUM = BIGNUM is below the overflow threshold and SMLNUM/ALPHA ≤ MAX_EXPONENT. The complex variants (clarfgp/zlarfgp) are already safe because they compute 1/ALPHA via CLADIV/ZLADIV, which internally handles overflow/underflow without intermediate overflow. Fixes Reference-LAPACK#938
martin-frbg
approved these changes
Jun 11, 2026
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In the general case (xnorm > eps*|alpha|, beta >= 0), the Householder reflector formula computes
ALPHA = -XNORM^2 / (alpha + beta)
which can be subnormal even when beta itself is safely above SMLNUM. When this subnormal ALPHA was used in the subsequent
CALL SSCAL( N-1, ONE / ALPHA, X, INCX )
the reciprocal overflowed (e.g. ALPHA = 2^-137 → ONE/ALPHA = 2^137 which exceeds SP max ~3.4e38).
Fix: guard the SSCAL with ABS(ALPHA) < SMLNUM. When ALPHA is very small, scale X by ONE/SMLNUM then by SMLNUM/ALPHA — both safe since ONE/SMLNUM = BIGNUM is below the overflow threshold and SMLNUM/ALPHA ≤ MAX_EXPONENT.
The complex variants (clarfgp/zlarfgp) are already safe because they compute 1/ALPHA via CLADIV/ZLADIV, which internally handles overflow/underflow without intermediate overflow.
Fixes #938