# CS 3200: Introduction to Scientific Computing Assignment 3 solved

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1. Complete the LU decomposition of the matrix A where the * entries are the unknowns.
.A =
41 2 100 41 2
44 3 *10 0* *
84 0 **1 00 *
     − −     
− =
    
2. Using this decomposition determine the solution of
0
3
16
Ax
    =      
3. The two matrices B and C are superficially “similar” to matrix A above.
41 2 21 2
4 4 3 and 4 4 3
84 2 84 4
B C
  − −  
=− =−
  are their LU decompositions
similar too?
2
4. This is practical example of a small but real-life-type ill-conditioned problem The flow
of water through two very different materials gives this system of linear equations :

−𝐻𝐻1
0
0

0

0
0
−𝑎𝑎𝑎𝑎𝑟𝑟⎦

= 1
∆𝑥𝑥2

−2 1
1 −2 1
1 −2 1
⋱ ⋱ ⋱
1 −(1 + 𝑎𝑎) 𝑎𝑎
⋱ ⋱ ⋱
𝑎𝑎 −2𝑎𝑎 𝑎𝑎
𝑎𝑎 −2𝑎𝑎 𝑎𝑎
𝑎𝑎 −2𝑎𝑎⎦

⎡ ℎ1
ℎ2
ℎ3

ℎ𝑖𝑖

ℎ𝑛𝑛−2
ℎ𝑛𝑛−1
ℎ𝑛𝑛 ⎦

The coefficient a can be very small indeed a = 1.0e-7 giving an ill-conditioned matrix.
Use ∆𝑥𝑥 = 1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛 = 21, 41, 81, 161 .
For values of a = 1.0, 1.0e-1,1.0e-3, 1.0e-5, 1.0e-7 and 1.0e-9, 1.0e-11,
1.0e-13, 1.0e-15 compute the estimated condition number using the matlab
condition number estimator. How does the condition number vary with the
value of a . Explain by using graphs.
If 𝐻𝐻1 = 8 𝑎𝑎𝑎𝑎𝑎𝑎 𝐻𝐻𝑟𝑟 = 4 Solve the system of equations for n= 161, where a =
1.0, a = 1.0e-5 and a = 1.0e-15. Use iterative refinement to check and