THE FOLD / GLITCH / DIVIDE BY ZERO / THE CONDITION NUMBER
THE CONDITION NUMBER
the residual is small and every digit is wrong
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A small residual is the thing everyone checks and it proves almost nothing. Plug a badly wrong answer into a near-singular system and the equations come out satisfied to fifteen decimal places.
LIT verified live. for
LIT verified live. for
A = [[1,1],[1,1.0001]] the determinant is 1.0e-4 and the condition number is κ = 40,004. Perturbing the right-hand side by a relative 10-10 over 2,000 random directions, the worst relative change in the solution is 40,002 times larger — within 0.005% of κ, which is exactly the bound doing its job. And a deliberately wrong answer, off by 1.414 in norm, leaves a residual of 1.0e-4: the error is 14,142 times the residual, so a check on ‖Ax−b‖ reports success while every digit of x is wrong.2 HOW IT WAS WEAVED · AI + HUMAN
The condition number, the perturbation bound and the residual/error distinction are the first chapter of numerical linear algebra — Wilkinson, and Higham’s Accuracy and Stability.
AVAN (AI) measured the amplification rather than quoting the bound, because a bound is an upper limit and the question is whether it is attained. It is: 40,002 against a κ of 40,004. The second number is the one worth carrying — the residual is what a program can compute without knowing the answer, and it is precisely the quantity that stays small when the answer is wrong, because a nearly-singular matrix maps a large error onto a small one by construction.
AVAN (AI) measured the amplification rather than quoting the bound, because a bound is an upper limit and the question is whether it is attained. It is: 40,002 against a κ of 40,004. The second number is the one worth carrying — the residual is what a program can compute without knowing the answer, and it is precisely the quantity that stays small when the answer is wrong, because a nearly-singular matrix maps a large error onto a small one by construction.
3 ONE DIMENSION
The residual is small. Every digit is wrong.
4 TWO DIMENSIONS · INTERACTIVE
Nudge the matrix toward singular and watch both numbers move.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object.
AVAN’s addition (the inverse-companion): the forward reading is that ill-conditioning amplifies error. The inverse is that the amplification and the reassurance are the same map. A squashes a large change in x into a small change in b — that is what near-singular means — and the residual is computed by applying A. So the very property that makes the answer untrustworthy is the property that makes the check come back clean, and it is not a coincidence or a weakness in the check: no function of
Ax−b can do better, because A has already thrown the information away.LIT for A = [[1,1],[1,1.0001]] the determinant is 1.0e-4 and the condition number is 40,004; perturbing the right-hand side by a relative 1e-10 over 2,000 random directions, the worst relative change in the solution is 40,002 times larger - within 0.005% of kappa, the bound doing its job - and a deliberately wrong answer off by 1.414 in norm leaves a residual of 1.0e-4, so the error is 14,142 times the residual and a check on the residual reports success while every digit of x is wrong
FIG The condition number, the perturbation bound and the residual/error distinction are the first chapter of numerical linear algebra - Wilkinson, and Higham's Accuracy and Stability. AVAN measured the amplification rather than quoting the bound, because a bound is an upper limit and the question is whether it is attained. It is: 40,002 against a kappa of 40,004. The residual is what a program can compute without knowing the answer, and it is precisely the quantity that stays small when the answer is wrong.
FIG The condition number, the perturbation bound and the residual/error distinction are the first chapter of numerical linear algebra - Wilkinson, and Higham's Accuracy and Stability. AVAN measured the amplification rather than quoting the bound, because a bound is an upper limit and the question is whether it is attained. It is: 40,002 against a kappa of 40,004. The residual is what a program can compute without knowing the answer, and it is precisely the quantity that stays small when the answer is wrong.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN