Orthogonality, measured
Orthogonal is a number
"Q is orthogonal" is a claim about a measurable quantity, ‖QᵀQ − I‖, and on the eight-by-eight Hilbert matrix two standard algorithms return 10⁻¹⁵ and 1 for it. The one that returns 1 still reconstructs the matrix perfectly, which is why nothing warns you.
Two Gram–Schmidts
One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.
A reflection cannot stop being one
Householder QR holds orthogonality at 10⁻¹⁵ whatever the condition number of the matrix, and Gram–Schmidt does not. The reason is not that it is more careful. It is that its Q is built from unit vectors, and rounding a unit vector gives a different reflection rather than a broken one.
The nearest orthogonal matrix
Every field that has to clean up a drifted rotation reaches for QR, and QR does not answer the question. The nearest orthogonal matrix is the orthogonal factor of the polar decomposition — nearer by about a tenth, and, more to the point, the same matrix whatever order the columns were written in. QR's answer changes completely.
An iteration that only multiplies
Newton's iteration for the polar factor needs an inverse every step. Newton–Schulz needs only matrix products — nothing that reads an entry, nothing that pivots — and it converges if and only if every singular value is below √3. At 1.73205 it converges and at 1.73206 it returns an orthogonal matrix that is not the answer, with a residual of 5·10⁻¹⁶ and nothing to say so.
The basis nobody chose on purpose
A method that eliminates a constraint has to pick a basis for its null space, and every basis is correct. Their condition numbers are eight orders apart, the reduced problem inherits the square, and the choice is usually made by a one-line rule nobody thought of as a numerical decision.
A test with no answer in it
A caller with no reference answer can still ask whether a routine answered the right question: reverse the columns, run it again, compare. The polar factor's two answers agree to 10⁻¹⁵ at every conditioning drawn; a QR's differ by 2.353 on matrices whose own norm is 2.449. The test has a floor, and the floor is measurable too.
A rotation that comes back mirrored
Align twenty noisy points and the nearest orthogonal matrix to the answer is a reflection in 7.7 per cent of trials at noise three times the set's thickness and a third of them at ten — at thicknesses of 10⁻², 10⁻³ and 10⁻⁴ alike. The determinant fix is never a small correction. It moves the answer by exactly 2, it costs exactly 4σ₃ of residual, and it leaves the rotation's error at half the noise however thin the set becomes.
A stable block is not a stable basis
Block Gram–Schmidt orthogonalises twice over — between blocks, and inside each one. Householder inside the blocks does not stop the classical between-block step losing orthogonality like κ², 4.2·10⁻³ at κ = 4.3·10⁷, and a second pass does not stop Cholesky QR inside the blocks breaking down at κ = 10⁸. Each level fails only on ill-conditioning placed at its own level, and one variant holds 3·10⁻¹⁵ on every placement.
The right-hand side as one more column
Modified Gram–Schmidt's Q is 4.3·10⁻⁹ from orthogonal at κ = 10⁸, and a least-squares solve that multiplies b by it is wrong by 0.13. Hand the same routine b as an extra column instead and the answer is right to 2.7·10⁻¹⁰ — closer than Householder's 4.0·10⁻⁹. Classical Gram–Schmidt gains nothing from the same trick, to the last bit.
Spread resistances make the loops easy
Scaled to a unit diagonal, the loop equations on the least-resistance tree get easier as a network's resistances spread — from 120 to 5.44 over six decades — and stop depending on the grid's size, while the node equations of the same flow get harder, from 538 to 4.6·10⁴. The spread that ruins the range-space formulation rescues the null-space one, though the loops' density means the work saved is a factor of two, not the factor of nine the iteration counts suggest.
The tree the resistances choose
On a network every basic set is a spanning tree and every null-space basis is a set of loops with entries 0 and ±1, so no tree can make Z badly conditioned. The tree with the best-conditioned Z still gives loop equations 4.8 times worse than the tree of least resistance: the basis has to be chosen against the Hessian, and pivoting finds it only when it pivots on the resistances too.
One number that has to be right
Householder's orthogonality was called structural: a reflection is built from a unit vector, so rounding the vector names a different reflection rather than a broken one. Tested by breaking it, the claim is narrower and sharper. Perturb every component of the reflector by a relative 10⁻², and ‖QᵀQ − I‖ stays at 1.5·10⁻¹⁵ while the factorisation moves to 5·10⁻³. Perturb the one stored scalar by the same amount and ‖QᵀQ − I‖ is 6.5·10⁻². The structure is one degree of freedom, and the departure is four times its relative error.
A triangle where the scalar was
Every level-3 QR assembles a block of reflectors into Q = I − Y T Yᵀ, and T is computed by a recurrence whose inputs are its own previous columns. A block of sixteen carries 136 computed numbers where sixteen separate reflections carry sixteen. The orthogonality it produces is 3.9·10⁻¹⁵ against the single reflector's 7.8·10⁻¹⁶ — a factor of five for a hundred and thirty-six times as many things that have to be right.
Five precise points are five points
Weighting each sighting by its reliability is the standard form of an attitude or registration fit, and it changes how often the nearest orthogonal matrix comes back as a mirror. Measured, the rate is a function of two numbers: the weighted noise over thickness, and the effective count (Σw)²/Σw². Five points with a tenth of the noise, weighted by 1/σ², carry the information of 515 equal points and mirror like five — 7.9 per cent at a noise ratio where twenty points mirror 1.8 and five mirror 9.5. The √m the earlier measurement left unchecked is right, and it counts what carries the thin direction.
A mirror decided in the thin directions
In n dimensions the nearest orthogonal matrix to a noisy alignment is still sometimes a reflection, and the rate at which it is does not depend on n. Three, five and ten dimensions with one thin direction mirror alike; two thin directions mirror like each other in five dimensions and in ten. The rate is the chance that a k × k matrix built from the k thin directions has a negative determinant — 21.7 per cent at a noise ratio of 0.7 for k = 2, measured at 23.0 — and it is well above k independent coin flips. With two or more thin directions the determinant correction still fires, and it no longer rescues the rotation: the answer is eleven noise-widths off whether or not it was mirrored.
What the appended block inherits
Modified Gram–Schmidt on [A b] solves least squares as well as Householder, although its Q is not orthogonal. A block code appends b as one more block. Block modified Gram–Schmidt inherits the rescue at every placement of the ill-conditioning: at κ = 10⁸ the appended block gives 6.9·10⁻¹⁰ where the same Q through Qᵀb gives 8.9·10⁻³. Block classical Gram–Schmidt gets the same wrong answer both ways, to the last bit. And the variant whose Q is orthogonal to 10⁻¹⁵ — two passes with Cholesky QR inside — is a hundred thousand times worse than Householder when the ill-conditioning is inside the blocks, because its R is wrong.
Eight blocks and sixty-four reflections
One block of sixteen reflectors, assembled as I − Y T Yᵀ, departed from orthogonality five times as far as a single reflection, and the question was whether a factorisation of many blocks multiplies that factor. It does not. On 96 × 64 matrices eight blocks of eight end at 1.42·10⁻¹⁴ — within a fifth of the root-sum-square of their own departures, and less than half their sum — while the same factorisation taken one reflector at a time ends at 2.91·10⁻¹⁴. A block departs more than a reflector, and there are an eighth as many of them. And a nearly dependent column that swells the triangle's entries to 10²⁶ costs the product nothing.
The residual the appended block cannot remove
Appending b as one more block made block modified Gram–Schmidt solve least squares as well as Householder, ten million times better than the same Q through Qᵀb at κ = 10⁸ — on problems with no residual. Give b a component outside the range and every stable route's error rises with it, while Qᵀb's, already at κ²u, does not move. The appended block's advantage then falls as one over the residual: 5,400 at a relative residual of 10⁻⁴, 54 at a per cent, none at one. It never falls behind Householder by more than a factor of four. What the residual decides is whether the extra block is worth its synchronisations, and the answer is yes up to a residual of about a per cent.
The factor nobody forms
A blocked Householder factorisation's orthogonal factor, multiplied out, departs from orthogonality half as far in blocks of sixteen as one reflector at a time, and that was read as blocking buying a factor of two. Libraries do not multiply it out. Applied to vectors through its stored blocks — which is how every caller uses it — the same factor departs by 3.1 to 4.0·10⁻¹⁵ at every block size from one to sixty-four, and stops growing after about twenty reflectors instead of adding them up. The factor of two was the price of forming the product, and a factor that is never formed never pays it.
The worst residual belongs to the route
Every stable least-squares route's error rises with the residual at about a fiftieth of the bound κ²uρ, and the question left open was whether a residual aimed at the weak directions closes the gap. It can be aimed exactly: the map from residual to error is, on these problems, one direction and rounding. Aimed, Householder's constant rises from a median of 0.014 to 0.11 — √(m − n) = 6.9 times a typical direction, at 32, 64 and 128 rows to three figures — and stops a factor of nine short of the bound. The direction is each route's own: aimed at one, another route draws a fifth of its worst. And aimed at the appended block, the one-per-cent rule that made it worth its extra block falls to half a per cent, with the block twelve times behind Householder on the same data.
A tree leaks along its leaves
A stable least-squares route's error, for a fixed factorisation, is a linear map from the residual with one dominant direction, and each route leaks along its own. A tree of Householder factorisations — the shape a distributed code factorises in — was expected to have a direction of its own too, with b appended at every leaf inheriting Householder's accuracy there. Measured at κ = 10⁸ on two placements of the ill-conditioning, a tree's map has one direction like every route's, and its aimed constant is of Householder's order: within a fifth of it when the ill-conditioning is spread, two to three times it when it sits between column blocks. Its direction is its own — at a cosine of 0.40 to 0.58 from the sequential route's, 0.13 to 0.18 between blocks — and the split of the rows decides it. Appending b at the leaves changes nothing: the same constant to five figures, the same direction to ten.
Long loops pay before the factor starts
The least-resistance spanning tree makes a network's loop equations well conditioned and its loops long, and the question left was what a direct solver pays for the length, and whether a tree reading the resistances only coarsely buys the conditioning back for less. Counted symbolically under minimum-degree ordering, the least tree's factor costs a median 1.2 to 2.0 times the breadth-first tree's on grids of 4 to 12 points a side, and the loop formulation goes from cheaper than the node equations at 4 × 4 to 3.8 times dearer at 12 × 12, for a conditioning two to three orders better. The price is paid in the loop matrix itself: on every network measured its factor fills in fewer entries than the breadth-first tree's. And no tree between them is a bargain. Kruskal on resistances rounded into bands pays nearly all the extra work until one band holds three quarters of the spread, and by then the conditioning has gone too.
The budget cannot finish the tree
The least-resistance spanning tree makes a network's loop equations well conditioned and its loops long, and the long loops cost a direct solver up to twice the breadth-first tree's factor work. A tree that refused any arc whose loop would run past a budget — 24 arcs, twice the breadth-first tree's longest — was predicted to keep the conditioning within twice the least tree's and save a third of the extra work. It keeps the conditioning and saves nothing. Every budget that binds leaves parts of the network it cannot join, the arcs it refused are needed to finish the tree, and they close loops as long as the least tree's. Half the least tree's loops are already as short as the property behind its conditioning allows; the slack is in the other half, and a greedy budget cannot reach it.
The plateau is set by the length
A Householder factor applied through its reflectors, never formed, loses orthogonality over its first twenty or so reflectors and then stops. The saturation was predicted to be a property of the vector length: on 384-long vectors the growth should continue longer before it stops. It does not. On 96-, 192- and 384-long vectors the departure reaches nine-tenths of its plateau within twenty reflectors, and on the longest it gets there soonest. What the length sets is the plateau itself — 3.6, 5.0 and 6.7·10⁻¹⁵ — which is the unit roundoff times the square root of the length, times a constant between 3.1 and 3.4. Twice the reflectors on the same length move it by nine per cent.
Every trip comes back shorter
An applied Householder factor stops losing orthogonality after about twenty reflectors, at a height that grows like the square root of the vectors' length, because rounding to nearest makes errors of random sign. Rounding every operation toward zero makes errors of one sign, and the plateau was predicted to grow like the length itself — doubling for each doubling. In single precision it does nothing of the kind: from 96 rows to 768, eight times the length, the departure after 64 reflectors grows by a factor of 1.6. An inner product does pile up its errors toward zero, a third of a unit a term, but only when its terms share a sign; Householder's inner products do not, and theirs grow like the square root as before. What rounding toward zero adds is something nearest never does: every vector sent through the factor and back comes home about a unit shorter per reflector, at every length. That shrink is a floor under the departure and has no plateau; rescaling every reflection to its input's length removes it and leaves the departure still climbing.