10.15

(a)Determine the condition number for the following system using
 the row-sum norm. Do not normalize the system.

A = [ 1 4 9 16 25;4 9 16 25 36; 9 16 25 36 49; 16 25 36 49 64;25 36 49 64 81]

How many digits of precision will be lost due to ill-conditioning?
(b)Repeat (a), but scale the matrix by making the maximum element in each
row equal to one.


Matlab Code:

A = [ 1 4 9 16 25;4 9 16 25 36; 9 16 25 36 49; 16 25 36 49 64;25 36 49 64 81]

%(a)

conditionA = cond(A)

significant = 10^(log10(cond(A)) - 7.2)

%(b)

b = [ 1/25 0 0 0 0;0 1/36 0 0 0;0 0 1/49 0 0;0 0 0 1/64 0;0 0 0 0 1/81];

A_scaled = b * A

conditionA_scaled = cond(A_scaled)

significant_scaled = 10^(log10(cond(A_scaled)) - 7.2)

RESULTS

A =

     1     4     9    16    25

     4     9    16    25    36

     9    16    25    36    49

    16    25    36    49    64

    25    36    49    64    81

conditionA =

     3.145210075463109e+17

significant =

     1.984493397046558e+10

A_scaled =

   0.040000000000000   0.160000000000000   0.360000000000000   0.640000000000000   1.000000000000000

   0.111111111111111   0.250000000000000   0.444444444444444   0.694444444444444   1.000000000000000

   0.183673469387755   0.326530612244898   0.510204081632653   0.734693877551020   1.000000000000000

   0.250000000000000   0.390625000000000   0.562500000000000   0.765625000000000   1.000000000000000

   0.308641975308642   0.444444444444444   0.604938271604938   0.790123456790123   1.000000000000000

conditionA_scaled =

     9.237149093911445e+16

significant_scaled =

     5.828247062861999e+09

These results show that both matrices are substantially ill-formed and there is a large loss of significance