(a) Find α and b such that αx₁+ x₂= b, x₁+ x₂= 3 has

(i) A unique solution

(ii) Infinitely many solutions

(iii) No solutions.

(b) Apply the Gauss elimination to the following two systems and compare the calculations step by step. Explain why the elimination fails if no solution exists.

(c) Why may a computer program give you the result that a homogeneous linear system has only the trivial solution although you know its coefficient determinant to be zero?

(d) Pivoting. Solve System (A) (below) by the Gauss elimination first without pivoting. Show that for any fixed machine word length and sufficiently small ϵ > 0 the computer gives x₂ = 1 and then x₁ = 0. What is the exact solution? Its limit as ϵ †’ 0?. Then solve the system by the Gauss elimination with pivoting. Compare and comment.

(e) Solve System (B) by the Gauss elimination and three-digit rounding arithmetic, choosing 

(i) the first equation, 

(ii) the second equation as pivot equation. (Remember to round to 3S after each operation before doing the next, just as would be done on a computer!) Then use four-digit rounding arithmetic in those two calculations. Compare and comment.

Transcribed Image Text:

X1 + х2 + x3 3D 3 4x1 + 2x2 — хз — 5 9х1 + 5х2 — хз — 13 Х1+ X2 + хз — 3 4x1 + 2x2 — хз — 5 9х1 + 5х2 — хз — 12. (A) EX1 + x2 = 1 x1 + x2 = 2 (В) 4.03х1 + 2.16х2 — — 4.61 6.21x1 + 3.35х2 3 —7.19