Themes
Where the algebra stops being the arithmetic
Every identity in linear algebra is exact, and every implementation of it is not. The gap is not a defect to be apologised for — it is the subject. QᵀQ is the identity in the algebra and is measurably not the identity in the arithmetic, and the difference between those two sentences is what this site is about.
Whose fault is it
A good algorithm returns the exact answer to a nearby problem. So when the answer is wrong there are two possible authors, and they are separately measurable: the backward error is what the algorithm did, the condition number is what the problem did to it. Almost every essay here reports both.
Identical algebra, different arithmetic
The subject supplies pairs of algorithms that a textbook derivation cannot tell apart and a computer can: classical against modified Gram–Schmidt, the normal equations against QR, elimination with and without a row swap. Each pair is one derivation and two behaviours, and the second is only visible if you run it.
Measured, not asserted
No decomposition is drawn on this site without its residual printed beside it. ‖A − QR‖/‖A‖, ‖QᵀQ − I‖, ‖PA − LU‖/‖A‖ — the numbers are on the figures, and a build in which a factorisation figure fails to carry one does not complete.
Two routes to a number
A determinant from the pivots and from cofactor expansion. A singular value from one-sided Jacobi and from the eigenvalues of AᵀA. A Hilbert solution in floating point and in exact rational arithmetic. Neither route can confirm itself, and where they disagree the disagreement is the finding.
Precision as the knob
The decompositions here are written against an arithmetic context rather than against plus and times, so the same code runs at any significand width from sixteen bits to fifty-three. Where a figure has a slider, the slider is usually the mantissa — and every frame it can reach was generated, and asserted, at build time.
Exact ground truth
Most numerical demonstrations estimate their own error by computing the same thing more carefully, which is circular. The Hilbert matrix has a closed-form rational inverse, so for that whole class of problems the answer is known in BigInt integers and the error is measured rather than approximated.
Assertions that reject
An assertion is evidence only if it can fail. Every check on this site is fed a counterexample somewhere in the build — the orthogonality test is shown a shear, the reconstruction test is shown the wrong matrix, the growth measurement is shown Wilkinson's matrix — and the build fails if any of them accepts.