Numerical-Methods

Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. The residual is the check everybody runs after a solve, and it is the one check that cannot detect the most common way a solve goes wrong. The condition number can: it converts the precision of the inputs into an error bar on the output, before any statistics are involved....

Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. Forming X’X squares the condition number, so the formula everybody learns spends twice the digits of the two factorisations nobody is taught. Underneath, least squares is a right angle — the fit is the foot of a perpendicular onto the column space — and reading it that way explains why QR needs no inversion at all, and why the one check that defines a least-squares solution is blind to a normal-equations fit that is wrong by 321 percent....