Temat: | Rozwiązywanie równania nieliniowego metodami: bisekcji (połowienia), siecznych oraz Newtona (stycznych) | |||||||||||||||||||
Zadanie: | Znaleźć przybliżenie pierwiastka równania: | f(x)= zobacz na tablicy | ||||||||||||||||||
metoda bisekcji | metoda siecznych | metoda stycznych | ||||||||||||||||||
Lokalizacja pierwiastka | Lokalizacja pierwiastka | Lokalizacja pierwiastka | ||||||||||||||||||
xd | xg | dx | xd | xg | dx | xd | xg | dx | ||||||||||||
1,000 | 5,000 | 0,400 | 1,000 | 5,000 | 0,400 | 1,000 | 5,000 | 0,400 | ||||||||||||
x | f(x) | x | f(x) | x | f(x) | |||||||||||||||
1,000 | 1,000 | 0,341 | 1,000 | 1,000 | 0,341 | 1,000 | 1,000 | 0,341 | ||||||||||||
2,000 | 1,400 | 0,285 | 2,000 | 1,400 | 0,285 | 2,000 | 1,400 | 0,285 | ||||||||||||
3,000 | 1,800 | 0,074 | 3,000 | 1,800 | 0,074 | 3,000 | 1,800 | 0,074 | ||||||||||||
4,000 | 2,200 | -0,292 | 4,000 | 2,200 | -0,292 | 4,000 | 2,200 | -0,292 | ||||||||||||
5,000 | 2,600 | -0,784 | 5,000 | 2,600 | -0,784 | 5,000 | 2,600 | -0,784 | ||||||||||||
6,000 | 3,000 | -1,359 | 6,000 | 3,000 | -1,359 | 6,000 | 3,000 | -1,359 | ||||||||||||
7,000 | 3,400 | -1,956 | 7,000 | 3,400 | -1,956 | 7,000 | 3,400 | -1,956 | ||||||||||||
8,000 | 3,800 | -2,512 | 8,000 | 3,800 | -2,512 | 8,000 | 3,800 | -2,512 | ||||||||||||
9,000 | 4,200 | -2,972 | 9,000 | 4,200 | -2,972 | 9,000 | 4,200 | -2,972 | ||||||||||||
10,000 | 4,600 | -3,294 | 10,000 | 4,600 | -3,294 | 10,000 | 4,600 | -3,294 | ||||||||||||
11,000 | 5,000 | -3,459 | 11,000 | 5,000 | -3,459 | 11,000 | 5,000 | -3,459 | ||||||||||||
Kolejne iteracje | Kolejne iteracje | Kolejne iteracje | ||||||||||||||||||
xd | xs | xg | f(xd) | f(xs) | f(xg) | xd | xs | xg | f(xd) | f(xs) | f(xg) | x | f(x) | f'(x) | ||||||
1,000 | 1,000 | 3,000 | 5,000 | 0,341 | -1,359 | -3,459 | 1,000 | 1,000 | 1,359 | 5,000 | 0,341 | 0,298 | -3,459 | 1,000 | 1,000 | 0,341 | 0,040 | |||
2,000 | 1,000 | 2,000 | 3,000 | 0,341 | -0,091 | -1,359 | 2,000 | 1,359 | 1,648 | 5,000 | 0,298 | 0,173 | -3,459 | 2,000 | -7,473 | 2,808 | -0,128 | |||
3,000 | 1,000 | 1,500 | 2,000 | 0,341 | 0,247 | -0,091 | 3,000 | 1,648 | 1,808 | 5,000 | 0,173 | 0,068 | -3,459 | 3,000 | 14,479 | -6,297 | -0,835 | |||
4,000 | 1,500 | 1,750 | 2,000 | 0,247 | 0,109 | -0,091 | 4,000 | 1,808 | 1,869 | 5,000 | 0,068 | 0,021 | -3,459 | 4,000 | 6,935 | -2,861 | 0,295 | |||
5,000 | 1,750 | 1,875 | 2,000 | 0,109 | 0,017 | -0,091 | 5,000 | 1,869 | 1,888 | 5,000 | 0,021 | 0,006 | -3,459 | 5,000 | 16,636 | -9,118 | -1,100 | |||
6,000 | 1,875 | 1,938 | 2,000 | 0,017 | -0,035 | -0,091 | 6,000 | 1,888 | 1,894 | 5,000 | 0,006 | 0,002 | -3,459 | 6,000 | 8,344 | -3,290 | -0,971 | |||
7,000 | 1,875 | 1,906 | 1,938 | 0,017 | -0,009 | -0,035 | 7,000 | 1,894 | 1,895 | 5,000 | 0,002 | 0,000 | -3,459 | 7,000 | 4,955 | -3,448 | -0,260 | |||
8,000 | 1,875 | 1,891 | 1,906 | 0,017 | 0,004 | -0,009 | 8,000 | 1,895 | 1,895 | 5,000 | 0,000 | 0,000 | -3,459 | 8,000 | -8,301 | 3,249 | -0,933 | |||
9,000 | 1,891 | 1,898 | 1,906 | 0,004 | -0,002 | -0,009 | 9,000 | 1,895 | 1,895 | 5,000 | 0,000 | 0,000 | -3,459 | 9,000 | -4,817 | 3,403 | -0,395 | |||
10,000 | 1,891 | 1,895 | 1,898 | 0,004 | 0,001 | -0,002 | 10,000 | 1,895 | 1,895 | 5,000 | 0,000 | 0,000 | -3,459 | 10,000 | 3,793 | -2,502 | -1,295 | |||
1,861 | 0,028 | -0,786 | ||||||||||||||||||
Xs=xg-f(x)*(Xg-Xd)/(F(Xg)-F(Xd)) | 1,896 | -0,001 | -0,820 | |||||||||||||||||
Xs=(Xd+Xg)/2 | 1,895 | 0,000 | -0,819 | |||||||||||||||||
Xn+1=Xn-f(x)/f'(x) |