분야
pdfConsider the following linear equation Ax = b, where
A =
10 −2 1 1
2 5 1 −1
3 −2 9 1
4 −1 2 11
, b =
10
7
11
16
(i) 10 point • Find all solutions of the linear equation
• How many solutions does the linear equation abovehave? Justify your
answer.
• Moreover, suppose that you construct the Jacobi algorithm to solve
the above linear equation. Does the Jacobi algorithm converge to the
solution of Ax = b? Justify your answer.
(ii) 10 point • Construct the MATLAB code of the Jacobi algorithm to solve the
above linear equation.
• Given the initial condition x0 =
0 0 0 0>
, iterate your Jacobi
algorithm for two times.
• Compute kx
∗ − x2k
2 = (x
∗ − x2)
>(x
∗ − x2), where x
∗
is the solution
from (i) and x2 is the solution obtained from two iterations of your
Jacobi algorithm.
(iii) 10 point • Construct the MATLAB code of the Gauss-Seidel algorithm to solve
the above linear equation.
• Given the initial condition x0 =
0 0 0 0>
, iterate your GaussSeidel algorithm for two times.
• Compute kx
∗ − x2k
2 = (x
∗ − x2)
>(x
∗ − x2), where x
∗
is the solution
from (i) and x2 is the solution obtained from two iterations of yo
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