Every essay
Grouped by field, below. Or come at it from what is taught and refused here — every claim settled by an assertion that would have failed the build; or from the fields themselves, the depth ladders, or the figure library, where every generator publishes what it checked before it drew.
The arithmetic underneath 19Two errors, and whose fault they are 18Elimination, and the swap 16Orthogonality, measured 16Least squares, and the road not to take 16Regularisation, and the answer that is chosen 15Eigenvalues, singular values, rank 32The eigenvalue problem that is not linear 20Reduction, and what a model is for 16Iterating, instead of factorising 39Sparsity, and what elimination costs 17Structure, and the solver that cannot see it 15Neither sparse nor dense 18When the index is a tuple 15The matrix that is a graph 19Exact arithmetic, and what it costs instead 15Randomised, and the guarantee that changes kind 16Methods that were designed apart 14When the problem arrives again 16Where the flop count stopped predicting the time 16The answer that depends on the machine 20The matrix a constraint makes 16
The arithmetic underneath
The representable numbers are spaced, and the spacing doubles at every power of two. Everything else follows from that: why subtracting two close numbers destroys the part you wanted, why the order of a sum changes its value, and why the mantissa is the one parameter on this site worth putting on a slider.
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.
Cancellation takes the answer, not a digit
Subtracting two nearly equal numbers is exact. That is what makes it dangerous — the subtraction introduces no error at all, it exposes error the operands were already carrying, and the exposure can consume every significant figure at once.
The order they are added in
Addition is associative in the algebra and is not associative in the arithmetic. The same million numbers, added in a different order, give answers that differ in the third significant figure — and the fix is not a wider float, it is a different order.
Buying the accuracy back
Factorise in single precision, then correct the answer using residuals computed in double, and the result is what a full double-precision solve would have given. Compute those residuals in single instead and the identical algorithm, at identical cost, recovers nothing.
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.
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.
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.
The direction the error leans
The size of one rounding error is set by the precision. How ten thousand of them combine is set by something else entirely — the rounding mode — and the fitted exponents are 0.47 for round-to-nearest and 1.01 for round-toward-infinity, on identical data at identical precision.
A coin flip that fixes the average
Add 0.1 to 256 a thousand times at eight significand bits and the answer is 256. Not approximately — the total never moves, not once, and no error bound says so. Round up one time in twenty instead of never, and it arrives at 348 against a true 356.
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.
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.
A bound that is proved
Every error statement on this site so far is a measurement of one run. Interval arithmetic makes a different kind of claim — the answer lies in this set, for this input, with no probability attached — and its failure mode is that it returns nothing at all. On a Hilbert system it proves a bound 23 times the error it bounds, and one size later it refuses.
Proving the answer is in the box
Every other method here computes a number and estimates how wrong it is. This one returns a verdict: there is exactly one solution in this box, or there is none, or — the honest third outcome — nothing can be said. Two of the three are proofs about infinitely many points from finitely many operations.
Where the box is cut
A branch-and-bound with an interval operator settles a whole square — two roots proved unique, forty-two regions proved empty, nothing left undecided, in 87 evaluations. Move the roots so one lands on the first bisection and it proves nothing at all, at any depth. Cutting at 0.485 instead of 0.5 finds both, in a quarter of the work.
Three walks and one bound
A left-to-right sum, a chain of three thousand rotations and a conjugate gradient residual recurrence share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against. All three come out at a half — 0.486, 0.554 and 0.507 — and nothing is rescaled.
The units that overflow before the answer does
A change of variable that is exact in the algebra requires γ² times a matrix to be a number the format can hold. In binary64 that is a bound nobody meets by accident. In binary32 it arrives at 10¹⁹ and in fp16 at 256, and past it there is no answer rather than a poor one.
Nine steps of pessimism
A proved bound is 8 to 26 times the error it bounds, at every precision from 16 to 40 significand bits. A carried interval is (√2)ᵐ times too wide after m re-enclosures. The two cross between eight and nine, so the method everybody warns against is the tighter of the two for a short computation.
Two errors, and whose fault they are
A good algorithm returns the exact answer to a nearby problem. So a wrong answer has two possible authors, and they can be measured separately: the backward error says how well the algorithm did, the condition number says how much the problem amplifies it, and only their product is what anyone sees.
The exact answer to a nearby problem
A good algorithm does not give an approximate answer to your problem. It gives the exact answer to a problem very close to yours — and once that is the definition, a wrong result has two possible authors and they can be measured apart.
A small residual is not a small error
Substituting the answer back and finding that it fits is the most natural check there is, and it verifies the wrong thing. A residual of 10⁻¹⁷ is entirely compatible with an answer whose second digit is wrong.
The condition number is an amplifier
κ is usually introduced as a definition and then quoted. It is a measurement: perturb the input by a known amount, look at how much the output moves, and the largest ratio you can find is the number.
An answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
The units the matrix is measured in
One linear system, written twice. The rows of the second are the rows of the first in different units, the solution is identical to the last bit, and the condition number has moved by eight orders of magnitude. One of those two numbers is a fact about the problem and the other is a fact about the notation.