In this section, we report
our experiments on the accuracy and response time of the condition
number prediction methods. We use SVMlight
for SVM regression. There are 277 matrices from Matrix Market tested
in the experiments. We use altogether 60 matrix features. First,
We test how accurate the predicted condition numbers are, compared
with the directly computed condition numbers. The figure below
illustrates the percentage of all matrices for which the relative
differences between the computed values and the predicted values
of the condition number are within 10, 10^2, 10^3, 10^4, respectively. 

Given
a matrix, the time used to obtain the condition number is referred
to as the response
time. The response time for the LAPACK method is the time to compute
the condition number using LAPACK routines.The response time for the
prediction method includes the time to compute matrix features and
the time for prediction. Here
we also compare the response time for prediction using the
whole matrix features and using half of the features. Table
1 shows the average response time for the 277 matrices used
in our tests. The prediction methods are 15 times faster
than using LAPACK on average. 
6
seconds is also an acceptable time for an online query systems.
Prediction with feature selection is only slightly faster
than without any feature selection. Using half of the features
does not mean reducing the time cost by half.
TABLE 1

Average
response time (in seconds).

LAPACK

Prediction
(all)

Prediction
(FS)

99.23

6.56

6.32


The
prediction method is especially advantageous in response
time for large size matrices. For example, in Table 2 the
average LAPACK response time for the 78 matrices with size
larger than 2000 is around 6 minutes, while using the prediction
methods, the response time is only about 20 seconds.
TABLE 2

Average
response time for larger size matrices (in seconds).

Size 
Num of Mat. 
LAPACK 
Prediction(all) 
Prediction(FS) 
>=
1000 
119 
227.22 
15.17 
14.62 
>=
2000 
78 
340.74 
22.81 
21.99 
Table 3 gives some examples of how the prediction methods
exceed the LAPACK method in response time. For instance, LAPACK
uses about two and a half hours to compute the condition number
of the matrix ADD20, the prediction methods only need less
than one second.
TABLE 3 
Matrix name 
Size 
nnz 
LAPACK 
Prediction(all) 
Prediction(FS) 
ADD20 
2395 
13151 
8206.7 
0.94 
0.81 
CRY10000 
10000 
49699 
2262.4 
23.41 
24.01 
LNS_3937 
3937 
25407 
2977.0 
1.38 
1.33 
PSMIGR_1 
3140 
543160 
2129.8 
15.82 
3.31 

