Table 5 | |||||||
Initial x | Initial y | Stepsize h | Appx. Y | Exa. Y | Error | k1 | k2 |
0.00000 | 1.00000 | 0.10000 | 1.00000 | 1.00000 | 0.00000 | 0.1 | 0.121 |
0.1000000 | 1.00000 | 0.10000 | 1.1105 | 1.11111 | 0.00061 | 0.1 | 0.146531 |
0.2000000 | 1.11050 | 0.10000 | 1.233765513 | 1.25000 | 0.01623 | 0.123321 | 0.184168 |
0.3000000 | 1.23377 | 0.10000 | 1.387510219 | 1.42857 | 0.04106 | 0.152218 | 0.237076 |
0.4000000 | 1.38751 | 0.10000 | 1.582157194 | 1.66667 | 0.08451 | 0.192518 | 0.314947 |
0.5000000 | 1.58216 | 0.1000000 | 1.835890108 | 2.00000 | 0.16411 | 0.250322 | 0.006266 |
Table 6 | |||||||
Initial x | Initial y | Stepsize h | Appx. Y | Exa. Y | Error | k1 | k2 |
0 | 1 | 0.05 | 1 | 1 | 0 | 0.05 | 0.055125 |
0.05 | 1 | 0.05 | 1.0525625 | 1.052631579 | -6.9079E-05 | 0.055394 | 0.061378 |
0.1 | 1.0525625 | 0.05 | 1.110948907 | 1.111111111 | -0.0001622 | 0.06171 | 0.068756 |
0.15 | 1.110948907 | 0.05 | 1.176182339 | 1.176470588 | -0.00028825 | 0.06917 | 0.077545 |
0.2 | 1.176182339 | 0.05 | 1.249540038 | 1.25 | -0.00045996 | 0.078068 | 0.088127 |
0.25 | 1.249540038 | 0.05 | 1.332637341 | 1.333333333 | -0.00069599 | 0.088796 | 0.101024 |
0.3 | 1.332637341 | 0.05 | 1.427547224 | 1.428571429 | -0.0010242 | 0.101895 | 0.11696 |
0.35 | 1.427547224 | 0.05 | 1.536974305 | 1.538461538 | -0.00148723 | 0.118115 | 0.136966 |
0.4 | 1.536974305 | 0.05 | 1.664514529 | 1.666666667 | -0.00215214 | 0.13853 | 0.162549 |
0.45 | 1.664514529 | 0.05 | 1.815054023 | 1.818181818 | -0.0031278 | 0.164721 | 0.195975 |
0.5 | 1.815054023 | 0.05 | 1.995402285 | 2 | -0.00459772 | 0.199082 | 0.240788 |
Table 7 | |||||||
Initial x | Initial y | Stepsize h | Appx. Y | Exa. Y | Error | k1 | k2 |
0 | 1 | 0.025 | 1 | 1 | 0 | 0.025 | 0.026266 |
0.025 | 1 | 0.025 | 1.025625 | 1.025641026 | -1.6026E-05 | 0.026298 | 0.027664 |
0.05 | 1.025625 | 0.025 | 1.051923467 | 1.052631579 | -0.00070811 | 0.027664 | 0.029138 |
0.075 | 1.051923467 | 0.025 | 1.078930846 | 1.081081081 | -0.00215023 | 0.029102 | 0.030693 |
0.1 | 1.078930846 | 0.025 | 1.106685431 | 1.111111111 | -0.00442568 | 0.030619 | 0.032337 |
0.125 | 1.106685431 | 0.025 | 1.135228679 | 1.142857143 | -0.00762846 | 0.032219 | 0.034073 |
0.15 | 1.135228679 | 0.025 | 1.164605572 | 1.176470588 | -0.01186502 | 0.033908 | 0.035911 |
0.175 | 1.164605572 | 0.025 | 1.194865028 | 1.212121212 | -0.01725618 | 0.035693 | 0.037857 |
0.2 | 1.194865028 | 0.025 | 1.226060373 | 1.25 | -0.02393963 | 0.037581 | 0.03992 |
0.225 | 1.226060373 | 0.025 | 1.258249888 | 1.290322581 | -0.03207269 | 0.03958 | 0.042109 |
0.25 | 1.258249888 | 0.025 | 1.291497448 | 1.333333333 | -0.04183589 | 0.041699 | 0.044435 |
0.275 | 1.291497448 | 0.025 | 1.32587326 | 1.379310345 | -0.05343709 | 0.043948 | 0.04691 |
0.3 | 1.32587326 | 0.025 | 1.361454731 | 1.428571429 | -0.0671167 | 0.046339 | 0.049547 |
0.325 | 1.361454731 | 0.025 | 1.398327491 | 1.481481481 | -0.08315399 | 0.048883 | 0.05236 |
0.35 | 1.398327491 | 0.025 | 1.436586603 | 1.538461538 | -0.10187494 | 0.051595 | 0.055367 |
0.375 | 1.436586603 | 0.025 | 1.476338004 | 1.6 | -0.123662 | 0.054489 | 0.058586 |
0.4 | 1.476338004 | 0.025 | 1.517700238 | 1.666666667 | -0.14896643 | 0.057585 | 0.062038 |
0.425 | 1.517700238 | 0.025 | 1.560806538 | 1.739130435 | -0.1783239 | 0.060903 | 0.065749 |
0.45 | 1.560806538 | 0.025 | 1.605807372 | 1.818181818 | -0.21237445 | 0.064465 | 0.069745 |
0.475 | 1.605807372 | 0.025 | 1.652873554 | 1.904761905 | -0.25188835 | 0.0683 | 0.074061 |
0.5 | 1.652873554 | 0.025 | 1.702200096 | 2 | -0.2977999 | 0.072437 | 0.078733 |
FLOWCHART
Remark. The local error in the Algorithm 3.3 is . To achieve this error, we are forced to more computation or in other words spend time to compute and . It all depends on the nature of the function to estimate the time consumed for the computation. The cost we pay for higher accuracy is more computation. Also, to reduce the local error, we need smaller values of the step size h, which again results in large number of computation. Each computation leads to move of rounding errors. In other words, reduction in discretization error may lead to increase in rounding off error. The moral is that the indiscriminate reduction of step-size need not mean more accuracy.