Solve the following (given as the augmented matrix): [3 1-4 7 - 2 3 1 -5 2 0 5 1

Question

Solve the following (given as the augmented matrix): [3 1-4 7 - 2 3 1 -5 2 0 5 10] Here is a system of equations that is called "ill- conditioned, " meaning that the solution is not easy to get accurately, Section 2.4 discusses this; here, we give a "taste" of the problem. This is the system as an aug- [3 1 4 9 8 -6 -8 -6 -1 2 3 4] You can see that x = [1, 1, 1]^T is the solution. Confirm the solution by doing naive Gaussian elimination using exact arithmetic (use fractions throughout). Now get the solution using only three significant figures in your computations. Observe that the solution is different. Compute the solution when the system is changed only slightly: Change the coefficient in the first column of the first row to 3.1. Use more precise computations, perhaps single or even double precision. Observe that this makes a large change in the solution. Use Gaussian elimination with partial pivoting to solve the equations of Exercise 13. Are any row interchanges needed?

Explanation / Answer

15-

Interchange Rows R2 & R3 (partial pivoting) we get

3 1 -4 | 7

2 0 5 | 10

-2 3 1 | -5

Apply R2 = R2 - 2/3R1 and R3 = R3 + 2/3R1

3 1 -4 | 7

0 -2/3 23/3 | 16/3

0 11/3 -5/3 | -1/3

R2 = 3R2 and R3 = 3R3

3 1 -4 | 7

0 -2 23 | 16

0 11 -5 | -1

R3 = 2R3 + 11R2

3 1 -4 | 7

0 -2 23 | 16

0 0 243 | 174

So x3 = 0.716

and -2x2 + 23 x3 = 16

x2 = 0.234

and 3x1 +0.234 -4(0.716) = 7

x1 = 3.209

So the solution is ( 3.209, 0.234, 0.716 )

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