Condition number of a

The condition number

κ(A) ≡ cond(A)=

indicates the sensitivity of the solution to perturbations in A and b (where function cond(A) is MatLab’s function).

The condition number is always in the range 1 ≤ κ(A)≤∞, where κ(A) is a mathematical property of A. Any algorithm will produce a solution that is sensitive to perturbations in A and b if κ(A) is large. In exact math a matrix is either singular or non-singular. κ(A) = ∞ for a singular matrix. κ(A) indicates how close A is to being numerically singular. A matrix with large κ is said to be ill-conditioned

Summary of Limits to Numerical Solution of Ax = b:

1. κ(A) indicates how close A is to being numerically singular

2. If κ(A) is “large”, A is ill-conditioned and even the best numerical algorithms will produce a solution, ˆx that cannot be guaranteed to be close to the true solution, x

3. In practice, Gaussian elimination with partial pivoting and back substitution produces a solution with a small residual

r = b − Aˆx

even if κ(A) is large.

The condition number

3.4 Individual tasks

1. Compute the condition number of matrices A and C. Solve the system by means MatLab. Make the error in one element of the matrix and one element of the vector right. Again, choose the system and compare the solutions. Write down the error estimates.

2. Solve the system Cx = D using the formula of Cramer.

3. Check the program for solving the linear methods a) Jacobi, b) Seidel for a given system of linear algebraic equations.

4. Perform calculations with varying accuracy, calculating the required number of iterations. Construct the graphs of the number of iterations on the accuracy for different methods.

Table 3.1 – Variants of the tasks

№1

A				B
1	0.47	-0.11	0.55	.1.33
0.42	1	0.35	0.17	1.29
-0.25	0.67	1	0.36	2.11
0.54	-0.32	-0.74	1	0.10

C			D
1	2	3	13
2	3	5	4
3	5	9	17

№2

0.63	1	0.11	0.34	2.08
0.17	1.18	-0.45	0.11	0.17
0.31	-0.15	1.17	-2.35	1.28
0.58	0.21	-3.45	-1.18	0.05

1	2	3	0.55
1	4	9	1.35
1	8	27	3.55

№3

0.77	0.04	-0.21	0.18	1.24
-0.45	1.23	-0.06	0	-0.88
-0.26	-0.34	1.11	0	0.62
-0.05	0.26	-0.34	1.12	-1.17

0.42	1.43	0.27	1
1.43	-0.84	0.93	2
0.27	0.93	-0.48	3

№4

0.79	-0.12	0.34	0.16	-0.64
-0.34	1.18	-0.17	0.18	1.42
-0.16	-0.34	0.85	0.31	-0.42
-0.12	0.26	0.08	0.75	0.83

0.64	0.54	-0.33	3
0.54	-0.92	0.24	2
-0.33	0.24	0.78	1

№5

-0.68	-0.18	0.02	0.21	-1.83
0.16	-0.88	-0.14	0.27	0.65
0.37	0.27	-1.02	-0.24	-2.23
0.12	0.21	-0.18	-0.75	1.13

0.5	1.77	0.39	1.5
0.84	1.79	0.95	2.5
0.24	1.03	-0.41	3

№6

-0.58	-0.32	0.03	0	-0.44
0.11	-1.26	-0.36	0	-1.42
0.12	0.08	-1.14	-0.24	0.83
0.15	-0.35	-0.18	0	1.42

0.19	0.51	0.86	0.35
0.51	0.32	0.95	0.42
0.86	0.95	-0.12	0.45

№7

-0.83	0.31	-0.18	0.22	1.71
-0.21	-0.67	0	0.22	-0.62
0.32	-0.18	-0.95	-0.19	0.89
0.12	0.28	-0.14	-1	-0.94

0.64	1.54	-0.33	0.3
1.54	-0.92	0.24	0.2
-0.33	0.24	0.78	0.1

№8

-0.87	0.27	-0.22	-0.18	-1.21
-0.21	-1	-0.45	0.18	0.33
0.12	0.13	-0.33	0.18	0.48
0.33	-0.41	0	-1	1.21

0.55	1.77	0.39	1.5
0.84	1.79	0.95	2.5
0.24	1.03	-0.41	3

№9

-0.81	-0.07	0.38	-0.21	0.81
-0.22	-0.92	0.11	0.33	0.64
0.51	-0.07	-0.81	-0.11	1.71
0.33	-0.41	0	-1	1.21

0.59	1.77	1.39	1.5
0.84	1.79	0.95	2.5
1.24	1.03	-0.41	3

№10

-1	0.22	-0.11	0.31	-2.7
0.38	-1	-0.12	0.22	1.5
0.11	0.23	1	-0.51	1.2
0.17	-0.21	0.31	-1	0.17

1.42	1.43	0.27	0.1
1.43	-0.84	0.93	0.2
0.27	0.93	-0.48	0.3

№11

-0.93	-0.08	0.11	-1.18	0.51
0.18	-0.48	0	0.21	-1.17
0.13	0.31	-1	-0.21	1.02
0.08	0	-0.33	-0.72	0.28

-0.93	-0.08	0.11	1.18
0.18	-0.48	0	0.21
0.13	0.31	-1	0.210

№12

-0.95	-0.06	-0.12	0.14	2.17
0.04	-1.12	0.08	0.11	1.4
0.11	0.12	0	1.03	0.8
0.34	0.08	-1.06	0.14	2.1

-1	-0.07	0.21	0.92
1	0.03	-0.42	0.92
-0.03	1	-0.04	1.2

№13

0	-0.19	0.27	-0.88	1.2
-0.33	-1	-0.07	0.21	0.92
0.11	0	1.03	-0.42	0.92
-0.92	-0.03	0	-0.04	1.2

-1	-0.07	0.21	0.92
0.07	0.03	-0.42	0.92
0.21	0.42	-0.04	1.2

№14

-0.88	-0.23	0.25	-0.16	1.24
0.33	0.03	-0.84	-0.32	-1.15
0.14	-0.66	-0.18	0.24	0.89
0.12	-0.05	0	-0.85	0.57

0.08	-0.12	-0.77	0.32
0.25	0.22	0.14	-1
-0.77	-0.14	0.06	-0.12

№15

0.12	-1	0.32	-0.18	0.72
0.08	-0.12	-0.77	0.32	0.58
0.25	0.22	0.14	-1	-1.56
-0.77	-0.14	0.06	-0.12	-1.21

0.08	0.25	-0.77	0.32
0.25	0.22	0.14	-1
-0.77	0.14	0.06	-0.12

№16

-0.86	0.23	0.18	0.17	1.42
0.12	-1.14	0.08	0.09	0.83
0.16	0.24	-1	-0.35	-1.21
0.23	-0.08	0.05	-0.75	-0.65

1	2	3	0.55
2	4	9	0.35
3	9	5	0.55

№17

76	21	6	-34	-142
12	-114	8	9	83
16	24	-100	-35	-121
23	-8	5	-75	85

1	2	4	0.55
1	4	16	1.35
1	8	64	3.55

№18

-83	27	-13	-11	142
5	-68	13	24	26
9	54	127	36	23
13	27	34	156	49

0.64	0.53	-0.33	3
0.53	-0.92	0.23	2
-0.33	0.23	0.78	1

№19

1	2	3	9	1.11
2	1	9	4	1.16
3	9	1	4	1.24
9	1	3	4	1.55

5	3	1	11
3	5	3	17
1	3	5	19

№20

-1	0.28	-0.17	0.06	-21
0.52	-1	0.12	0.17	117
0.17	-0.18	-0.79	0	0.81
0.11	0.22	0.03	-0.95	-0.72

11	12	13	13
12	13	15	4
13	15	19	17