Koeffizienten der Newtonpolynome

Mirko Belickin und Joachim Wagner
Aufgabenblatt 08 (Rechneraufgabe 04)

Koeffizienten in Matrixform (m_{i j}): s_i(x) = m_{i 1} * (x-x_i)³ + m_{i 2} * (x-x_i)² + m_{i 3} * x-x_i + m_{i 4}

Stützstellenart äquidistant, natü:rliche Splines

Funktion 1

n = 2:
-0.000000  0.000000  0.000000 -0.000000
 0.000000 -0.000000  0.000000  0.000000
n = 4:
 0.129006  0.000000 -0.954930 -0.000000
-0.129006  0.607927 -2.94e-17 -1.000000
-0.129006  5.57e-18  0.954930  0.000000
 0.129006 -0.607927  2.66e-17  1.000000
n = 7:
 0.155154  0.000000 -0.996031 -0.000000
 0.038321  0.417799 -0.621016 -0.781831
-0.107370  0.520988  0.221637 -0.974928
-0.172209  0.231862  0.897394 -0.433884
-0.107370 -0.231862  0.897394  0.433884
 0.038321 -0.520988  0.221637  0.974928
 0.155154 -0.417799 -0.621016  0.781831
n = 10:
 0.161108  0.000000 -0.999092 -0.000000
 0.099573  0.303681 -0.808283 -0.587785
-2.06e-16  0.491371 -0.308738 -0.951057
-0.099573  0.491371  0.308738 -0.951057
-0.161108  0.303681  0.808283 -0.587785
-0.161108 -2.17e-17  0.999092  0.000000
-0.099573 -0.303681  0.808283  0.587785
-1.47e-16 -0.491371  0.308738  0.951057
 0.099573 -0.491371 -0.308738  0.951057
 0.161108 -0.303681 -0.808283  0.587785

Funktion 2

n = 2:
-0.000000  0.000000  0.045791 -0.412118
 0.000000 -0.000000  0.045791  0.000000
n = 4:
-0.006494  0.000000  0.440321 -0.412118
 0.006494 -0.087673  0.045791  0.977530
 0.006494  8.22e-19 -0.348739  0.000000
-0.006494  0.087673  0.045791 -0.977530
n = 7:
 0.023891  0.000000 -0.054046 -0.412118
-0.088069  0.184303  0.419876 -0.144874
 0.154872 -0.495087 -0.379283  0.656033
-0.181387  0.699636  0.146698 -0.959639
 0.154872 -0.699636  0.146698  0.959639
-0.088069  0.495087 -0.379283 -0.656033
 0.023891 -0.184303  0.419876  0.144874
n = 10:
 0.107313  0.000000 -0.559666 -0.412118
-0.202548  0.579490  0.483415 -0.793668
 0.043646 -0.514272  0.600808  0.772764
 0.166948 -0.278584 -0.826333  0.442520
-0.115359  0.622937 -0.206497 -0.973848
-0.115359  2.06e-17  0.914789  0.000000
 0.166948 -0.622937 -0.206497  0.973848
 0.043646  0.278584 -0.826333 -0.442520
-0.202548  0.514272  0.600808 -0.772764
 0.107313 -0.579490  0.483415  0.793668

Funktion 3

n = 2:
 5.555556  0.000000 -1.500000  0.300000
-5.555556  5.000000   3.7e-17  0.000000
n = 4:
-6.349206  0.000000 -0.857143  0.300000
   31.746 -2.857143 -1.285714  0.150000
  -31.746    11.429   3.7e-17  0.000000
 6.349206 -2.857143  1.285714  0.150000
n = 7:
 1.916515  0.000000 -1.014077  0.300000
-9.584162  0.492818 -0.971836  0.214286
   36.423 -1.971681 -1.098595  0.128571
 0.000000  7.394313 -0.633798  0.042857
  -36.423  7.394313  0.633798  0.042857
 9.584162 -1.971681  1.098595  0.128571
-1.916515  0.492818  0.971836  0.214286
n = 10:
 0.767342  0.000000 -1.002762  0.300000
-3.836710  0.138122 -0.994475  0.240000
   14.579 -0.552486 -1.019337  0.180000
  -54.481  2.071823 -0.928177  0.120000
  203.346 -7.734807 -1.267956  0.060000
 -203.346    28.867  7.26e-17  0.000000
   54.481 -7.734807  1.267956  0.060000
  -14.579  2.071823  0.928177  0.120000
 3.836710 -0.552486  1.019337  0.180000
-0.767342  0.138122  0.994475  0.240000

Funktion 4

n = 2:
-0.000678  0.000000  0.164634  0.012195
 0.000678 -0.018293  5.59e-18  1.000000
n = 4:
 0.004372  0.000000 -0.080795  0.012195
-0.011787  0.059028  0.184833  0.047059
 0.011787 -0.100102 -1.08e-17  1.000000
-0.004372  0.059028 -0.184833  0.047059
n = 7:
-0.000958  0.000000  0.010779  0.012195
 0.006431 -0.007389 -0.008221  0.023626
-0.010261  0.042225  0.081358  0.062982
 0.000000 -0.036932  0.094969  0.376923
 0.010261 -0.036932 -0.094969  0.376923
-0.006431  0.042225 -0.081358  0.062982
 0.000958 -0.007389  0.008221  0.023626
n = 10:
-0.000467  0.000000  0.005253  0.012195
 0.003623 -0.002524  0.000711  0.018925
-0.011153  0.017040  0.026840  0.033156
 0.058392 -0.043186 -0.020223  0.071633
-0.141105  0.272130  0.391875  0.235849
 0.141105 -0.489839  1.91e-17  1.000000
-0.058392  0.272130 -0.391875  0.235849
 0.011153 -0.043186  0.020223  0.071633
-0.003623  0.017040 -0.026840  0.033156
 0.000467 -0.002524 -0.000711  0.018925

Stützstellenart nach Tschebyscheff-Polynom, natü:rliche Splines

Funktion 1

n = 2:
-0.000000  0.000000  0.000000 -0.000000
 0.000000 -0.000000  0.000000 -0.000000
n = 4:
 0.211525  0.000000 -1.043835 -0.000000
-0.087617  0.583906 -0.506554 -0.795693
-0.087617 -6.05e-17  0.790558 -0.000000
 0.211525 -0.583906 -0.506554  0.795693
n = 7:
 0.166888  0.000000 -1.000100 -0.000000
 0.138337  0.155764 -0.951639 -0.306121
-0.039844  0.517540 -0.364702 -0.925685
-0.174978  0.366966  0.749494 -0.643506
-0.039844 -0.366966  0.749494  0.643506
 0.138337 -0.517540 -0.364702  0.925685
 0.166888 -0.155764 -0.951639  0.306121
n = 10:
 0.166501  0.000000 -0.999998 -0.000000
 0.159662  0.076804 -0.988189 -0.153155
 0.104126  0.290542 -0.824268 -0.564635
-0.024508  0.507652 -0.269507 -0.962212
-0.152197  0.443261  0.563280 -0.825341
-0.152197 -2.06e-16  0.993601 -0.000000
-0.024508 -0.443261  0.563280  0.825341
 0.104126 -0.507652 -0.269507  0.962212
 0.159662 -0.290542 -0.824268  0.564635
 0.166501 -0.076804 -0.988189  0.153155

Funktion 2

n = 2:
-0.000000  0.000000  0.045791 -0.412118
 0.000000 -0.000000  0.045791 -0.000000
n = 4:
-0.002383  0.000000  0.142286 -0.412118
 0.000987 -0.018842  0.092618 -0.080688
 0.000987  1.64e-18 -0.027291 -0.000000
-0.002383  0.018842  0.092618  0.080688
n = 7:
 0.282370  0.000000 -0.847694 -0.412118
-0.172211  0.755012 -0.174767 -0.967729
 0.086999 -0.535181  0.374218  0.622378
-0.067690  0.406683 -0.089495 -0.908175
 0.086999 -0.406683 -0.089495  0.908175
-0.172211  0.535181  0.374218 -0.622378
 0.282370 -0.755012 -0.174767  0.967729
n = 10:
 0.334818  0.000000 -0.857608 -0.412118
 0.021552  0.442454 -0.662711 -0.761270
-0.195968  0.525107  0.574177 -0.840371
 0.128744 -0.645463  0.334539  0.837733
-0.038780  0.323557 -0.473095 -0.352686
-0.038780  7.96e-18  0.426767 -0.000000
 0.128744 -0.323557 -0.473095  0.352686
-0.195968  0.645463  0.334539 -0.837733
 0.021552 -0.525107  0.574177  0.840371
 0.334818 -0.442454 -0.662711  0.761270

Funktion 3

n = 2:
 5.555556  0.000000 -1.500000  0.300000
-5.555556  5.000000   3.7e-17  0.000000
n = 4:
  -11.520  0.000000 -0.911053  0.300000
   18.269 -3.036839 -1.177894  0.212132
  -18.269  8.589486  -2.2e-16  0.000000
   11.520 -3.036839  1.177894  0.212132
n = 7:
 6.574720  0.000000 -1.005792  0.300000
-8.716512  0.585992 -0.988382  0.270291
   18.202 -1.590793 -1.072026  0.187047
-1.55e-14  4.977806 -0.664600  0.066756
  -18.202  4.977806  0.664600  0.066756
 8.716512 -1.590793  1.072026  0.187047
-6.574720  0.585992  0.988382  0.270291
n = 10:
 4.439281  0.000000 -1.000954  0.300000
-5.646739  0.195546 -0.998083  0.285317
   10.696 -0.526308 -1.012177  0.242705
  -27.653  1.603395 -0.940691  0.176336
   86.950 -5.334541 -1.252729  0.092705
  -86.950    18.848  -2.9e-17  0.000000
   27.653 -5.334541  1.252729  0.092705
  -10.696  1.603395  0.940691  0.176336
 5.646739 -0.526308  1.012177  0.242705
-4.439281  0.195546  0.998083  0.285317

Funktion 4

n = 2:
-0.000678  0.000000  0.164634  0.012195
 0.000678 -0.018293  5.59e-18  1.000000
n = 4:
 0.005773  0.000000 -0.035602  0.012195
-0.005480  0.045656  0.084748  0.024096
 0.005480 -0.058972  3.89e-17  1.000000
-0.005773  0.045656 -0.084748  0.024096
n = 7:
-0.001341  0.000000  0.004191  0.012195
 0.002291 -0.003586  0.000995  0.014981
-0.002163  0.013578  0.025947  0.030781
 0.000000 -0.009843  0.039425  0.199570
 0.002163 -0.009843 -0.039425  0.199570
-0.002291  0.013578 -0.025947  0.030781
 0.001341 -0.003586 -0.000995  0.014981
n = 10:
-0.003393  0.000000  0.003542  0.012195
 0.004966 -0.004484  0.001566  0.013465
-0.008931  0.014559  0.014446  0.018513
 0.025892 -0.038786 -0.033792  0.034501
-0.048646  0.156098  0.260533  0.114484
 0.048646 -0.249776 -1.06e-16  1.000000
-0.025892  0.156098 -0.260533  0.114484
 0.008931 -0.038786  0.033792  0.034501
-0.004966  0.014559 -0.014446  0.018513
 0.003393 -0.004484 -0.001566  0.013465

Stützstellenart äquidistant, periodische Splines

Funktion 1

n = 2:
 0.000000 -0.000000  0.000000  0.000000
 0.000000 -0.000000  0.000000  0.000000
n = 4:
 0.129006 -2.23e-17 -0.954930 -0.000000
-0.129006  0.607927  -2.1e-17 -1.000000
-0.129006 -1.05e-17  0.954930  0.000000
 0.129006 -0.607927 -7.81e-17  1.000000
n = 7:
 0.155154  1.11e-17 -0.996031 -0.000000
 0.038321  0.417799 -0.621016 -0.781831
-0.107370  0.520988  0.221637 -0.974928
-0.172209  0.231862  0.897394 -0.433884
-0.107370 -0.231862  0.897394  0.433884
 0.038321 -0.520988  0.221637  0.974928
 0.155154 -0.417799 -0.621016  0.781831
n = 10:
 0.161108 -1.54e-17 -0.999092 -0.000000
 0.099573  0.303681 -0.808283 -0.587785
-1.77e-16  0.491371 -0.308738 -0.951057
-0.099573  0.491371  0.308738 -0.951057
-0.161108  0.303681  0.808283 -0.587785
-0.161108 -1.44e-17  0.999092  0.000000
-0.099573 -0.303681  0.808283  0.587785
-2.65e-16 -0.491371  0.308738  0.951057
 0.099573 -0.491371 -0.308738  0.951057
 0.161108 -0.303681 -0.808283  0.587785

Funktion 2

n = 2:
 0.000000 -0.000000  0.000000  0.000000
 0.000000 -0.000000  0.000000  0.000000
n = 4:
-0.005364  1.15e-17  0.325843  0.000000
 0.005364 -0.072410  2.01e-17  0.977530
 0.005364  1.23e-17 -0.325843  0.000000
-0.005364  0.072410  6.91e-17 -0.977530
n = 7:
 0.030394 -1.57e-17 -0.257313  0.000000
-0.096346  0.234468  0.345606 -0.144874
 0.157236 -0.508769 -0.359739  0.656033
-0.182569  0.704196  0.142789 -0.959639
 0.157236 -0.704196  0.142789  0.959639
-0.096346  0.508769 -0.359739 -0.656033
 0.030394 -0.234468  0.345606  0.144874
n = 10:
 0.126247 -3.24e-17 -0.849967  0.000000
-0.226554  0.681734  0.377154 -0.793668
 0.050070 -0.541659  0.629290  0.772764
 0.165258 -0.271281 -0.834001  0.442520
-0.115021  0.621111 -0.204306 -0.973848
-0.115021  -3.2e-17  0.913693  0.000000
 0.165258 -0.621111 -0.204306  0.973848
 0.050070  0.271281 -0.834001 -0.442520
-0.226554  0.541659  0.629290 -0.772764
 0.126247 -0.681734  0.377154  0.793668

Funktion 3

n = 2:
   22.222   -10.000  4.96e-16  0.300000
  -22.222    10.000  2.29e-16  0.000000
n = 4:
   22.222 -10.000000 -2.07e-16  0.300000
   22.222 -1.48e-16 -1.500000  0.150000
  -22.222    10.000  9.22e-17  0.000000
  -22.222 -1.11e-15  1.500000  0.150000
n = 7:
  102.912   -20.488   8.7e-17  0.300000
  -36.516  5.975430 -1.243905  0.214286
   43.156 -3.414473 -1.024395  0.128571
 0.000000  7.682871 -0.658532  0.042857
  -43.156  7.682871  0.658532  0.042857
   36.516 -3.414473  1.024395  0.128571
 -102.912  5.975430  1.243905  0.214286
n = 10:
  204.678   -28.947  1.58e-16  0.300000
  -58.480  7.894737 -1.263158  0.240000
   29.240 -2.631579 -0.947368  0.180000
  -58.480  2.631579 -0.947368  0.120000
  204.678 -7.894737 -1.263158  0.060000
 -204.678    28.947   2.5e-16  0.000000
   58.480 -7.894737  1.263158  0.060000
  -29.240  2.631579  0.947368  0.120000
   58.480 -2.631579  0.947368  0.180000
 -204.678  7.894737  1.263158  0.240000

Funktion 4

n = 2:
-0.002710  0.036585   3.6e-17  0.012195
 0.002710 -0.036585  5.59e-18  1.000000
n = 4:
 0.007365 -0.031420 -1.16e-18  0.012195
-0.012785  0.068006  0.164634  0.047059
 0.012785 -0.104591 -7.32e-17  1.000000
-0.007365  0.068006 -0.164634  0.047059
n = 7:
-0.002151  0.007259 -5.68e-19  0.012195
 0.006750 -0.009331 -0.005329  0.023626
-0.010341  0.042737  0.080569  0.062982
 -1.8e-18 -0.037035  0.095232  0.376923
 0.010341 -0.037035 -0.095232  0.376923
-0.006750  0.042737 -0.080569  0.062982
 0.002151 -0.009331  0.005329  0.023626
n = 10:
-0.001654  0.005055  1.62e-18  0.012195
 0.003941 -0.003878  0.002118  0.018925
-0.011238  0.017403  0.026463  0.033156
 0.058415 -0.043284 -0.020123  0.071633
-0.141113  0.272158  0.391850  0.235849
 0.141113 -0.489853  1.52e-16  1.000000
-0.058415  0.272158 -0.391850  0.235849
 0.011238 -0.043284  0.020123  0.071633
-0.003941  0.017403 -0.026463  0.033156
 0.001654 -0.003878 -0.002118  0.018925

Stützstellenart nach Tschebyscheff-Polynom, periodische Splines

Funktion 1

n = 2:
 0.000000 -0.000000  0.000000  0.000000
 0.000000 -0.000000  0.000000 -0.000000
n = 4:
 0.211525 -3.51e-16 -1.043835 -0.000000
-0.087617  0.583906 -0.506554 -0.795693
-0.087617 -9.99e-17  0.790558 -0.000000
 0.211525 -0.583906 -0.506554  0.795693
n = 7:
 0.166888  8.16e-17 -1.000100 -0.000000
 0.138337  0.155764 -0.951639 -0.306121
-0.039844  0.517540 -0.364702 -0.925685
-0.174978  0.366966  0.749494 -0.643506
-0.039844 -0.366966  0.749494  0.643506
 0.138337 -0.517540 -0.364702  0.925685
 0.166888 -0.155764 -0.951639  0.306121
n = 10:
 0.166501  7.99e-16 -0.999998 -0.000000
 0.159662  0.076804 -0.988189 -0.153155
 0.104126  0.290542 -0.824268 -0.564635
-0.024508  0.507652 -0.269507 -0.962212
-0.152197  0.443261  0.563280 -0.825341
-0.152197 -2.27e-16  0.993601 -0.000000
-0.024508 -0.443261  0.563280  0.825341
 0.104126 -0.507652 -0.269507  0.962212
 0.159662 -0.290542 -0.824268  0.564635
 0.166501 -0.076804 -0.988189  0.153155

Funktion 2

n = 2:
 0.000000 -0.000000 -0.000000  0.000000
-0.000000  0.000000  0.000000 -0.000000
n = 4:
 0.000912 -2.57e-18 -0.036949  0.000000
-0.000378  0.007215 -0.017931 -0.080688
-0.000378 -3.94e-19  0.027984 -0.000000
 0.000912 -0.007215 -0.017931  0.080688
n = 7:
 0.365872 -1.76e-15 -1.376416  0.000000
-0.208747  0.978285 -0.504490 -0.967729
 0.093158 -0.585634  0.476082  0.622378
-0.070390  0.422906 -0.111155 -0.908175
 0.093158 -0.422906 -0.111155  0.908175
-0.208747  0.585634  0.476082 -0.622378
 0.365872 -0.978285 -0.504490  0.967729
n = 10:
 1.005188 -1.33e-14 -1.923268  0.000000
-0.258112  1.328330 -1.338151 -0.761270
-0.157318  0.338454  0.792592 -0.840371
 0.121476 -0.601246  0.269351  0.837733
-0.037523  0.313072 -0.453654 -0.352686
-0.037523  4.04e-17  0.417046 -0.000000
 0.121476 -0.313072 -0.453654  0.352686
-0.157318  0.601246  0.269351 -0.837733
-0.258112 -0.338454  0.792592  0.840371
 1.005188 -1.328330 -1.338151  0.761270

Funktion 3

n = 2:
   22.222   -10.000  4.96e-16  0.300000
  -22.222    10.000  2.29e-16  0.000000
n = 4:
   64.760   -17.071 -4.89e-16  0.300000
   11.111  5.53e-06 -1.500000  0.212132
  -11.111  7.071064 -1.67e-16  0.000000
  -64.760  5.53e-06  1.500000  0.212132
n = 7:
  707.873   -54.690  -1.9e-15  0.300000
  -46.789  8.401841 -1.375176  0.270291
   23.771 -3.282917 -0.949058  0.187047
-8.87e-15  5.295293 -0.706988  0.066756
  -23.771  5.295293  0.706988  0.066756
   46.789 -3.282917  0.949058  0.187047
 -707.873  8.401841  1.375176  0.270291
n = 10:
 2846.171  -109.896 -4.04e-15  0.300000
 -150.379    15.475 -1.386383  0.285317
   30.754 -3.748461 -0.886678  0.242705
  -31.568  2.374961 -0.977837  0.176336
   88.086 -5.545192 -1.242965  0.092705
  -88.086    18.953  2.18e-16  0.000000
   31.568 -5.545192  1.242965  0.092705
  -30.754  2.374961  0.977837  0.176336
  150.379 -3.748461  0.886678  0.242705
-2846.171    15.475  1.386383  0.285317

Funktion 4

n = 2:
-0.002710  0.036585   3.6e-17  0.012195
 0.002710 -0.036585  5.59e-18  1.000000
n = 4:
 0.009085 -0.022237  1.77e-18  0.012195
-0.005791  0.049611  0.072161  0.024096
 0.005791 -0.060950 -5.24e-18  1.000000
-0.009085  0.049611 -0.072161  0.024096
n = 7:
-0.004588  0.007597  7.09e-18  0.012195
 0.002467 -0.004672  0.002607  0.014981
-0.002189  0.013813  0.025434  0.030781
 1.44e-19 -0.009887  0.039602  0.199570
 0.002189 -0.009887 -0.039602  0.199570
-0.002467  0.013813 -0.025434  0.030781
 0.004588 -0.004672 -0.002607  0.014981
n = 10:
-0.014565  0.012961  1.14e-17  0.012195
 0.005535 -0.006286  0.002940  0.013465
-0.009010  0.014939  0.014002  0.018513
 0.025908 -0.038877 -0.033660  0.034501
-0.048650  0.156122  0.260498  0.114484
 0.048650 -0.249788 -1.06e-16  1.000000
-0.025908  0.156122 -0.260498  0.114484
 0.009010 -0.038877  0.033660  0.034501
-0.005535  0.014939 -0.014002  0.018513
 0.014565 -0.006286 -0.002940  0.013465

Letzte Änderung: 08. Dezember 1999
mail: jwagner@informatik.uni-osnabrueck.de