Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Shared exponent — where it appears

The single scale factor a block of values holds between them, in place of one exponent per value. It is chosen as the power of two that places the block's largest magnitude inside the significand's integer range, so it is decided by one entry and applied to all of them.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

half a stepthe 32 values of one block, in the order they arrivestep 0.0625one scale, thirty-two valuesoctaves inside the block1.9entries rounded to zero0worst error over its bound131 levels either side of zerothe largest entry chose the 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.

arithmetic · Block formats entries rounded to zero, of 320block, as given310block, sorted by size22E4M3, either order0median entry's relative errorblock, as given1block, sorted0.00321E4M30.022the same numbers, three waysdeleted, as given310deleted, sorted22deleted, per-element08-bit significands, blocks of 32sorting is free and changes no value

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.

arithmetic · Block formats 0102030405060first barrier depth at which the far well is wrong by half, octavesfp16E4M3E5M2gradual underflow, from onegradual, from the topcounter, floor and ceiling9 of 9counter, window from the rates9 of 9gradual underflow, from onegradual, from the topcounter, floor and ceiling1 of 9counter, window from the rates8 of 9gradual underflow, from onegradual, from the topcounter, floor and ceiling1 of 9counter, window from the rates1 of 9ring: chains that keep the far well past sixty octavesthe window is the unbounded exponent

Room for the step, not the value

Starting a recurrence at the top of an eight-bit format's range spends its headroom once. An exponent counter beside the stored value was predicted to spend it at every step, remove every range failure and leave only the precision. Rescaled on the value — back to one whenever it nears the bottom or the top — it does not: E4M3 still loses its far well on seven or more chains of nine, because one climbing step multiplies the value by forty and E4M3 clips silently at 448. Rescaled into a window read off the chain's own rates, it reproduces the unbounded exponent bit for bit on all 531 runs of every format, and what is left of E4M3's limit is nothing.

arithmetic · Subnormals

Named alongside it

The objects these essays reach for when they reach for this one.

Dynamic rangeFp8SignificandBlock floating-pointExponent rangeOutliersQuantisationOverflowPermutationRelative errorScalingSilent failure

All concepts