University of Wyoming Student Name:
Department of Mathematics
Spring 2008
Instructor: Man-Chung Yeung
Numerical Analysis 4340, Test 1 (2/19/2008)
1. (50 points) Consider the linear system
2x1+x2−3x3+4x4= 2
−3x1+2x2+5x3−3x4= 1
8x1+x2−3x3+2x4= 4
−4x1+2x2+3x3−x4= 5.
a) Solve the linear system by Gaussian elimination with pivoting.
b) Solve the linear system by
(a) finding A−1with Gauss-Jordan method with pivoting,
(b) computing A−1b,
where Ais the coefficient matrix and bthe right hand side of the linear system.
c) Which method requires more operation? Why?
d) Find the condition number K(A).
Solution.
a)
2 1 −3 4 |2
−3 2 5 −3|1
8 1 −3 2 |4
−4 2 3 −1|5
⇒(r1↔r3)
8 1 −3 2 |4
−3 2 5 −3|1
2 1 −3 4 |2
−4 2 3 −1|5
⇒(r2←r2+3
8r1, r3←r3+−2
8r1, r4←r4+4
8r1)
8 1 −3 2 |4
0 2.375 3.875 −2.25 |2.5
0 0.75 −2.25 3.5|1
0 2.5 1.5 0 |7
⇒(r2↔r4)
8 1 −3 2 |4
0 2.5 1.5 0 |7
0 0.75 −2.25 3.5|1
0 2.375 3.875 −2.25 |2.5
⇒(r3←r3+−0.75
2.5r2, r4←r4+−2.375
2.5r2)
8 1 −3 2 |4
0 2.5 1.5 0 |7
0 0 −2.7 3.5| −1.1
0 0 2.45 −2.25 | −4.15
