Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Pivoting — the ladder

One essay so far against this idea. A ladder is the distinct arguments that stand against one idea, and this one has room to grow.
  1. [ ε 1 ; 1 1 ] x = [ 1 ; 2 ], exact answer (1.000000, 1.000000)with partial pivoting1101U after elimination1.0000001.000000computed xbackward error 0forward error 0without10⁻¹⁷10-1·10¹⁷U after elimination0.0000001.000000computed xbackward error 0.25forward error 0.71no error is raisedgrowth 10¹⁷

    The swap that is not optional

    Run elimination without a row interchange on a matrix that needs one and nothing announces a failure. There is no division by zero, no warning, and an answer of the right shape. It is simply wrong, and how wrong depends on a number you did not look at.

    rung 2 · elimination

All ladders