Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Significand — where it appears

The digits of a floating-point number, as distinct from its exponent. Their count sets the relative accuracy of every operation, and it is the one parameter on this site that is genuinely a knob.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

[½, 1)[1, 2)[2, 4)0.5124gap 0.125gap 0.25 — twice as wide8 values per octavespacing doubles at each power of two

What a float can hold

The representable numbers are not a fine fuzz spread evenly over the line. They are evenly spaced inside each power-of-two interval and twice as far apart in the next one up, and almost everything else in this subject is a consequence of that one fact.

arithmetic · Floating-point 10¹10²10³10⁴10⁵10⁶10⁷10⁸10⁹10¹⁰110⁴10⁸10¹²10¹⁶condition number κ(A)× short of a double solvebf16fp16fp32bf16fp16fp32the reference is soundreference solve, worst backward error10⁻¹⁶bfloat16 threshold κ256fp32 threshold κ1.7·10⁷eight refinement steps, residual always in doubleflat at 1 means it reached double

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.

arithmetic · Mixed-precision 3456789101100.250.50.751bits in the exponent fieldeach curve as a fraction of its own maximumbfloat16fp16rangeprecisionto 617 decadesto 3.9 digitswhat the split buysbfloat16: largest number3.4·10³⁸fp16: largest number6.6·10⁴a bit of exponent doubles the rangea bit of significand adds a third of a digit

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.

arithmetic · Floating-point -47-37-27-17-7313233301234log₁₀ of the vector's normfp1611 bitsbfloat168 bitstf3211 bitsbinary3224 bitspale: the format's range · blue: √(Σ(xᵢ/m)²)·m · red: √(Σxᵢ²)fp16 and tf32 have the same eleven significand bitsand their bars do not overlap

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.

arithmetic · Overflow 10⁻⁷10⁻⁵10⁻³10⁻¹⁰10⁻⁸10⁻⁶magnitudespacing to the next numberthe smallest normalgradualflush to zerosmallest normal6.1·10⁻⁵smallest subnormal6·10⁻⁸octaves of subnormals10pairs that lie under FTZ10the spacing stops halving and stays putwhich is what makes x − y = 0 mean x = y

The numbers below the smallest one

Below the smallest normal number the spacing stops halving and stays put, all the way to zero. That is what gradual underflow is, and the thing it buys is the sentence every algorithm assumes without being told — x minus y is zero only when x equals y.

arithmetic · Subnormals 2022242628303210⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²k, where the entries are near 2ᵏrelative error of the determinantas writtenfused: exactproducts need 54 bitsone rounding, the whole answertrue determinant1naive, k = 300fused, k = 301first wrong at k27sizes returning 06both forms conformand the source does not say which

One multiply the compiler removed

A determinant whose value is exactly 1, computed as exactly 0 by the expression that is written down, and exactly 1 by the same expression with the multiply and the add fused. Both forms conform to IEEE-754, both are legal compilations of the same source, and nothing in the program says which one you have.

machine · Fma contraction 10⁻³10⁻¹10¹10³01magnitude448NaN — no ∞0.0156 — the smallest normaldrawn from the format's own rulespositive finite values126largest finite value448worst round-trip error0the subnormals are the evenly spaced ticks at the lefteverything a byte can be

Eight bits, and a format that breaks the rules

E4M3 reuses the exponent code IEEE reserves for infinities, so it reaches 448 where the same bits under IEEE's rules would reach 240 — and has no infinity left to signal an overflow with. The same computation is a NaN on one conforming device and 448 on another.

arithmetic · Floating-point 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

Named alongside it

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

Dynamic rangeHalf-precisionOverflowIEEE 754Unit roundoffbfloat16Exponent rangeFp8Block floating-pointCatastrophic cancellationOutliersQuantisation

All concepts