Studies of Singular Value Decomposition in Terms of Integrable Systems
Masashi Iwasaki · Kyoto University Research Information Repository (Kyoto University) · 2004
iii 4.6 4.7 4.8 4.9 4.10 pqd and the dLV algorithns (u-axis).The dashed, dotted and solid lines are given by the DK, the pqd and the dl"Y algorithms, respectively.A graph of rearranged relative errors in computed singular values ak ofBi by the DK, the pqd and the dLY algorithms from small to large.The dashed, dotted and solid lines are given by the DK, the pqd and the dLV algoritlms, respectively.A graph ofthe suMx k for ordering singular values ok according to magnitude (x-axis) and relative errors in computed singular values ofB2 by the DK, the pqd and the dLV algoritlms (y-axis).The dashed, dotted and solid lines are given by the DK, the pqd and the dlJV algorithms, respectively.A graph ofthe suMx k for ordering singular values ak according to magnitude (x-axis) and relative errors in computed singular values ofB3 by the DK, the pqd and the dLV algoritlms (y-axis).The dashed, dotted and solid lines are in given by the DK, the pqd and the dLV algorithms, respectively.A graph of the suMx k fbr ordering singular values ak according to magnitude (x-axis) and relative errors in computed singular values of B4 by the DK, the pqd and the dlLJV algorithms (y-axis).The dashed, dotted and solid lines are given by the DK, the pqd and the dLV algorithms, respectively.A graph ofiteration number in the dl.V algorithn (x-axis) and estimated error bounds of singular values Iak -0kl (y-axis).The solid, dotted and dashed lines correspond to the cases where k = 1, 2 and 3, respectively.43 43 44 44 45 46 5.1 5.2 5.3 Evolution Pv(n) --> pv(n+i) A graph ofthe suMx k for ordering singular values ak according to magnitude (x-axis) and relative errors in computed singular values ofB4 by the sl and dLV routines (u-axis).The red soild and green dashed Iines are given by the sl and dl"]V routines, respectively A graph ofthe suMx k for ordering singular values ak according to magnimde (x-axis) and relative errors in computed singular values of B4 by the sl, DBDSQR and DLASQ routines (y-axis).The red soild, green dashed and blue dotted lines are given by the sl, DBDSQR and DLASQ routines, respectively.50 59 60 iv List of Tables 3.1 3.2 3.3 4.1 4.2 4.3 4.4 5.1 5.2 Choice of the step-size 6(") Timing of deflation in Cases 2-5 Operation number in Cases 2-5 Complexity ofpqd, dqd and dLY algorithms Four cases of upper bidiagonal matrices Singular values in fbur cases of 100 × 100 matrices Computational time of the DK, the pqd and the dLY algorithms (sec.)Computational time ofthe sl and the dlJV routines (sec.)Computational time of the sl, the DBDSQR and the DLASQ routines (sec.) 26