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