Use Gaussian elimination and three-digit chopping arithmetic to solve the following linear systems, and compare the approximations to the actual solution.
a. 0.03x1 + 58.9x2 = 59.2,
5.31x1 − 6.10x2 = 47.0
Actual solution [10, 1]
b. 3.03x1 − 12.1x2 + 14x3 = −119,
−3.03x1 + 12.1x2 − 7x3 = 120,
6.11x1 − 14.2x2 + 21x3 = −139.
Actual solution [0, 10, 1/7]
c. 1.19x1 + 2.11x2 − 100x3 + x4 = 1.12,
14.2x1 − 0.122x2 + 12.2x3 − x4 = 3.44,
100x2 − 99.9x3 + x4 = 2.15,
15.3x1 + 0.110x2 − 13.1x3 − x4 = 4.16
Actual solution [0.176, 0.0126,−0.0206,−1.18].
d. πx1 − ex2 +√2x3 −√3x4 =√11,
π2x1 + ex2 − e2x3 + 3/7 x4 = 0,
√5x1 −√6x2 + x3 −√2x4 = π,
π3x1 + e2x2 −√7x3 + 1/9 x4 =√2.
Actual solution [0.788,−3.12, 0.167, 4.55].