Structural Health Monitoring Using Smart Sensors - ideals ...

Singular value
Singular value
error ratio
10 0 Imote2 Sv
Matlab Sv0
10 -20
0 20 40 60 80 100
|Sv - Sv0|/|Sv0|
10 -20
0 20 40 60 80 100
Order of singular values
(a)
Singular value
Singular value
error ratio
10 0 Imote2 Sv
Matlab Sv0
10 -20
0 20 40 60 80 100
10 0 |Sv - Sv0|/|Sv0|
10 -20
0 20 40 60 80 100
Order of singular values
Figure 7.2. SVD accuracy check on 100 x 100 matrix (Numerical recipes functions): (a)
smaller singular values; and (b) larger singular values.
(b)
Singular value
10 0 Imote2 Sv
Matlab Sv0
10 -20
0 20 40 60 80 100
10 0 |Sv - Sv0|/|Sv0|
10 -20
0 20 40 60 80 100
Order of singular values
Singular value
error ratio
Singular value
Singular value
error ratio
10 0 Imote2 Sv
Matlab Sv0
10 -20 0 20 40 60 80 100
10 0 |Sv - Sv0|/|Sv0|
10 -20
0 20 40 60 80 100
Order of singular values
(a)
(b)
Figure 7.3. SVD accuracy check on 100 x 100 matrix (CLAPACK functions): (a) smaller
singular values; and (b) larger singular values.
are consequently distributed equally in log scale, allowing a check of the algorithms for
singular values of various magnitudes. A matrix is reconstructed from the calculated
singular values and the two unitary matrices obtained in the SVD. This SVD and
reconstruction are performed on both the Imote2 and a PC. Figures 7.2 and 7.3 show
singular values estimated on the two platforms and the ratio of the numerical difference
between the two to the singular values on a PC. The accuracy of the SVD on the Imote2
and on a PC is numerically the same. As another accuracy indicator, the 2-norm of the
difference between the original matrix and the matrix reconstructed from the SVD results
are calculated as shown in Table 7.1. These values are as small as . The SVD
implementation on the Imote2 is considered to be accurate to within the precision of the
data type.
Limitations on the matrix size are also examined. Because smart sensors have limited
memory, numerical operations on large matrices are not feasible. An double
precision matrix occupies n B of memory. Therefore the two unitary matrices of SVD
105
– n n