Exercise 1: Why y Was Fine but x Was Catastrophically Wrong — Possible Solution ==================================================================== WHY y CAME OUT NEARLY PERFECT ------------------------------ y is solved directly from the eliminated second row, which after elimination reads approximately (1 - 10^16) y = (2 - 10^16). Both sides of this equation are dominated by the huge 10^16 term contributed by the tiny original pivot's enormous multiplier, and that huge term appears consistently on both sides, so it largely cancels out of the RATIO b2_new/a22_new even though each individual number involved is huge. The huge multiplier does introduce some rounding error into this step, but because y is obtained as a straightforward division with no separate cancellation-prone subtraction feeding into it afterward, that error stays extremely small in relative terms - verified at about 1.2 * 10^-16. WHY x CAME OUT CATASTROPHICALLY WRONG ------------------------------ x is recovered afterward via back-substitution: x = (1 - y) / 1e-16. Even though y itself carries only a tiny error, the subtraction (1 - y) is exactly Chapter 4's catastrophic cancellation pattern, since y is extremely close to 1 (the true value of y is approximately 0.9999999999999999, so 1 - y is a very small number built almost entirely from whatever small error y carries). Then DIVIDING that already-degraded subtraction result by the tiny pivot, 1e-16, multiplies whatever relative error survived the subtraction by a factor of 10^16 - turning y's already-small absolute error into a massive relative error in x. WHY THE SAME PROCESS PRODUCES SUCH DIFFERENT OUTCOMES ------------------------------ y is computed by a single division with no dangerous subtraction immediately before it. x is computed by a dangerous subtraction (two nearly-equal numbers, 1 and y) followed immediately by division by an extremely small number - stacking exactly the two error-amplifying patterns this course has spent multiple chapters establishing (Chapter 4's cancellation and Chapter 6's near-zero-denominator division) directly on top of each other. It isn't that "the elimination process is unreliable" in some general sense - it's that x's specific computational path happens to route through both known danger patterns at once, while y's path routes through neither. WHY THIS WORKS AS AN ANSWER ------------------------------ The explanation traces each variable's own distinct computational path separately rather than treating "the algorithm" as a single undifferentiated source of error, and identifies the specific combination (cancellation immediately followed by division by a tiny number) that explains why x's path was so much more dangerous than y's, tying the explanation directly to mechanisms already established in Chapters 4 and 6.