Practice question · Sort into groups
Sort each 2 by 2 system by its conditioning, judging from how nearly parallel its two equations are.
Groups: Well-conditioned · Ill-conditioned
- x + y = 2 and x + 1.0001y = 2.0001
- x = 1 and y = 1
- x + y = 2 and x - y = 0
- 2x + 3y = 5 and 4x + 6.0001y = 10
Hints
- Compare the ratios of the coefficients in the two equations of each system.
- If one equation is nearly a multiple of the other, the lines are nearly parallel.
Show the answer
Well-conditioned: x + y = 2 and x - y = 0, x = 1 and y = 1
Ill-conditioned: x + y = 2 and x + 1.0001y = 2.0001, 2x + 3y = 5 and 4x + 6.0001y = 10
Why
In s1 and s3 the second equation is nearly a multiple of the first, so the lines are nearly parallel and their crossing point slides enormously under tiny data changes. s2 and s4 are perpendicular, so their intersection is pinned down firmly.
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