Dynamic range — where it appears
Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.
Where the hardware went
bfloat16 carries eight mantissa bits, which puts its refinement threshold at a condition number of 256. That is not an exotic matrix. It is an ordinary one, and past it the method still improves the answer by a factor of four hundred while getting nowhere near a usable one.
The other half of a format
fp16 and tf32 have the same eleven significand bits and their largest numbers are 65,504 and 3.4·10³⁸. For two phases this site simulated the significand alone, so it was obliged to report them as the same format — which is a claim, and a false one.
A norm that overflows before it is a norm
The vector of sixteen thousands has a Euclidean norm of 4,000, which fp16 represents exactly. Written as the square root of the sum of squares it returns infinity, because squaring doubles the exponent — and the expression costs half the format's range on the one computation every iterative method performs at every step.
One exponent for thirty-two numbers
Share the exponent across a block and the cost per value drops from eight bits to 6.25, and the accuracy improves — up to about three octaves of spread inside a block. Past that a single outlier deletes the thirty-one values beside it, and the 2-norm barely notices.
The definition asks for more of what defeats it
The rank of a p × p Hankel matrix of Markov parameters resolves a degree of p, and it needs 2p parameters to do it. Those parameters grow like the norm of the state matrix raised to their index, so the count that buys resolution is the same count that buys dynamic range. One model, four run lengths, and a spread that runs from 10¹¹ to 10¹⁰².
A bit buys an octave
The outlier a block survives is exactly two raised to its significand width — 8 at three bits, 32 at five, 128 at seven, 512 at nine. Each extra bit doubles the range the block tolerates and halves the ordinary entry's error. Reordering the same numbers buys every octave at once and costs nothing.
Named alongside it
The objects these essays reach for when they reach for this one.
SignificandOverflowbfloat16Half-precisionBlock floating-pointExponent rangeFp8OutliersQuantisationShared exponentUnit roundoffCatastrophic cancellation