Exact arithmetic, and what it costs instead
An answer with no error in it
An integer matrix eliminated over the rationals rounds nothing, so the forward error is zero, the residual is the zero vector, and the identity this site is built on has no terms left. The cost does not vanish with the error. It moves into the length of the numbers, where three correct routes differ by four orders of magnitude.
Every intermediate is a minor
Fraction-free elimination divides by the previous pivot at every step and the division is always exact. Not usually, not for these entries — always, because the number being divided is a determinant with that pivot as a factor, which is a theorem and is checked here against the minors themselves.
The answer is longer than the question
An exact solution of an integer system is a vector of fractions, each of them a ratio of two determinants. So the output carries 2n long integers where the input carried n² short ones, and no algorithm can write it down more cheaply — the length of the answer is a floor under every exact solver rather than a property of one.
How many primes the answer needs
Work modulo a word-sized prime and no intermediate can exceed twenty-six bits, whatever the matrix does. The catch is that the answer must be reassembled from several such computations, and the number of them has to be fixed before the first one runs — by a theorem about how large a determinant can be, not by trying more until it settles.
A prime that divides the answer
A modular elimination reports a singular matrix and is telling the truth — over the field with p elements the matrix is singular. Over the rationals it is not. Nothing in the residue distinguishes the two cases, no quantity is small enough to be suspicious, and the wrong answer is a correct computation of a different question.
A fraction recovered from one remainder
A solution over the rationals can be computed modulo a prime power and then recovered — the residue determines the fraction uniquely, but only once the modulus is twice the square of the fraction's longer part. Below that there is no partial credit: the algorithm returns a different fraction with the same residue, and it is a perfectly good one.
The rank depends on the ring
A floating-point rank is a decision about a threshold. Remove the arithmetic error entirely and the threshold goes away — and the answer still is not a property of the array of numbers, because one integer matrix has rank six over the rationals, five modulo three and four modulo two, with nothing rounded and nothing decided.
What a determinant does not determine
Two integer matrices can have the same determinant, the same rank and the same size, and define genuinely different maps. What separates them is a list of integers each dividing the next — computed here twice, once by unimodular elimination and once from the gcds of every minor, which share no algorithm at all.
A basis that describes its lattice badly
The same set of points has infinitely many bases, they are all correct, and they are not equally useful. One measurement separates them — the product of the vectors' lengths over the lattice determinant — and the determinant is the invariant the reduction may not change, which is what makes the reduction checkable.
An exact answer to a measured problem
The residual is the zero vector, nothing was rounded at any step, and the answer is wrong in its first digit. Data accurate to fourteen places, an exact solve of the system it defines, and an error of 10⁻⁵ — because conditioning was never a statement about arithmetic and removing the arithmetic error removes none of it.
A relation among digits that were not there
A lattice finds the exact integer combination of several numbers that vanishes, and it needs enough digits of them to do it. Give it more digits than the numbers have and it finds a relation anyway — among the rounding, with the same confidence and no warning. Recovery runs 8 of 8 at twelve digits and 1 of 8 at forty.
Rounding a coordinate in the wrong basis
The nearest lattice point is found by writing the target in the basis and rounding each coordinate, which is three lines of arithmetic and is wrong. On a basis skewed by forty it lands a mean of 26 times too far and a worst of 40.02 — the basis's own orthogonality defect, to three figures — and the identical three lines on the reduced basis are exact at every target.
The knob and the rounding
The Lovász parameter is the number a lattice reduction is specified by, and moving it from 0.50 to 0.99 strengthens the proved bound from 16.00 to 1.83, costs 91 per cent more steps, and returns a basis with the same orthogonality defect. The one rounding nobody writes down decides everything: above 2⁵³ the reduction returns a basis 12,345 times worse than it should, with the determinant invariant equal to one throughout.
A bound on every intermediate at once
Fraction-free elimination's intermediates are minors of the original, which is a theorem about exactness. It is also a bound: Hadamard's inequality applies to every minor, so one inequality bounds the whole run before it starts. The bound on the k-th step is the one on (k+1)×(k+1) minors, not the one on the whole matrix — and on a 10×10 with entries in ±6 the difference is ten bits, with the run reaching 2.7 bits a step against the bound's 3.4.
Three orders and one last entry
Over the integers there is no stability to pivot for, so a fraction-free elimination swaps rows only when the pivot is zero. Choosing a pivot for length instead does change the sequence of minors — the smallest-nonzero rule makes seven exchanges where the natural order makes none and keeps the profile two bits lower through the middle. It cannot change the peak. The last entry of the elimination is the determinant, and the determinant does not know what order it was computed in.
The room a relation has to stand out
A lattice search for an integer relation returns its shortest vector, and the proposal was to return the gap to the next one as well, so a caller could tell a relation from an accident. Measured, the gap is a certificate with a budget: the digits the numbers really have, shared among all but one of them, less the size of the relation. A found relation's gap sits half a digit under that budget, accidents stay near zero, and a one-digit gap vouches for 81 of 96 relations among three numbers and for 1 of 35 among six. The test numbers the proposal came from turned out to have relations of their own.
Two precisions guard the other edge
Run a lattice relation search at N digits and again at N/10, and accept its answer only if both runs return the same vector. Among six measured numbers, where the gap between the shortest and next vector vouches for one found relation in 35, the two runs agree on 34. Past a double's sixteen digits they never once agree on an accident, 0 of 561. They do agree on 62 accidents at fifteen digits or fewer — approximate relations that really are the shortest vector there — and the gap, which cannot see a relation among six numbers, can see those.
The pivot is in every product
A fraction-free elimination's intermediates are minors, and Hadamard bounds a minor by the lengths of the rows it is made of — so the rule that picks the smallest pivot entry looked like a proxy for a rule that picks the shortest row. Measured, the two keep the bit-length profile equally low: 0.969 and 0.966 of the natural order's area. They part on what the arithmetic costs. Counting every multiplication and division at the product of its operands' lengths, the smallest-pivot rule costs 0.70 of the natural order and the shortest-row rule 0.84, because the pivot multiplies every entry of the step and divides every entry of the next. Three per cent of area is thirty per cent of arithmetic.
The digits between the two searches
A lattice relation search run at N digits and again at N/10 never agrees with itself on a relation among the rounding, but it does agree on 62 approximate relations at fifteen digits or fewer — accidents that really are the shortest vector there, and that dropping a digit was said to be unable to dislodge, since a combination that cancels to D digits cancels to D − 1. It cancels, but it stops being the shortest. Two digits apart the accepted accidents fall to 17 and three digits apart to 5, while the relations kept fall from 711 of 755 to 652 and 602 — every one of the losses at fifteen digits or fewer. Past a double's sixteen digits the separation costs nothing and buys nothing.
What reading the next pivot buys
In a fraction-free elimination every pivot is paid for twice — it multiplies every entry of its own step and divides every entry of the next — and the rule that picks the smallest pivot entry left a median 20 per cent above the least arithmetic any row order reaches. A rule that charges two steps ahead brings the median matrix to within 2.5 per cent, and on one matrix of twenty-four costs 1.87 times the least, worse than the smallest pivot ever does. Charging three steps ahead finds the least of all 40,320 orders on thirteen matrices and is never more than 27 per cent above it. And none of it pays: choosing that way costs forty to two hundred and sixty times the elimination it chooses.
The field decides it, usually
A matrix whose rank depends on the field it is read over was built, the first time, from its invariant factors outward, because random integer matrices never seemed to show the effect. Random 0/1 matrices show it at almost every size that is not tiny. At twenty rows, 99.8% of them are invertible over the rationals, 29% modulo two, 56% modulo three — and 71% of the ones the rationals call invertible are singular modulo two. Modulo two they obey, corank by corank, the law for uniformly random matrices over that field; modulo three and five, which their entries cannot fill, they converge to that field's law anyway.
An order found once knows what the rows know
Searching for a fraction-free elimination's least-cost row order costs forty to two hundred and sixty times the elimination, so it can only pay if one order serves a whole family. The prediction was that it cannot: the best order depends on which rows happen to have short entries, so a reused order should do no better than the natural one. On independent entries and on a shared zero pattern that holds — another member's best order is an ordinary order, at the middle of the 40,320. On a family that shares its row scales it fails completely: another member's order costs a median 1.13 of the least where the natural order costs 1.88. But on every family, including that one, the rule that just takes the smallest pivot entry is cheaper than any reused or trained order — 1.24, 1.08 and 1.01. What an order carries from one matrix to the next is what the family shares — and only when that is a particular matrix, not a visible structure, does an order trained on the family beat the rule.
Every entry under the determinant
The Hermite form's swell was put down to its transform — the record of how the answer was reached — while the form's own entries stayed bounded. Instrumented separately, the form's intermediates swell exactly as much: 3,371 bits against the transform's 3,366 for a random 48 × 48 matrix whose determinant has 219. Carried out modulo the determinant instead, every entry stays within the determinant's width, the answer is identical on every matrix to n = 80, and it is almost always the identity with the whole determinant in its last pivot and last column. What the bound does not buy is speed: counted in bit operations the modular route does more work until just past n = 72, because its steps multiply numbers the determinant's size where plain elimination multiplies wide numbers by small quotients.
The swell was one-sided
The Smith form can be computed modulo the determinant, and it was predicted to return the right invariant factors, to stay within twice the determinant's width, and to cost about what the modular Hermite form costs instead of the far larger count of the plain Smith routine. The first three hold. The comparison does not: the plain Smith routine never needed a far larger count. With column operations as well as row operations, its smallest-entry pivot is a unit after a step or two, every entry it holds is a minor of the matrix, and nothing it writes is wider than the determinant — at every size to sixty-four, at a fiftieth of the modular route's work. The swell the Hermite form suffered belonged to doing elimination with one hand.