Методы уточнения корня
Метод половинного деления e=0,001
a |
x |
b |
f(a) |
f(x) |
f(a)*f(x) |
|f(x)|<e |
1,7 |
1,85 |
2 |
-0,13737 |
-0,01828 |
0,0025116 |
|
1,85 |
1,925 |
2 |
-0,01828 |
0,033738 |
-0,0006169 |
|
1,85 |
1,8875 |
1,925 |
-0,01828 |
0,008296 |
-0,0001517 |
|
1,85 |
1,86875 |
1,8875 |
-0,01828 |
-0,00485 |
8,8633E-05 |
|
1,86875 |
1,878125 |
1,8875 |
-0,00485 |
0,00176 |
-8,534E-06 |
|
1,86875 |
1,873438 |
1,878125 |
-0,00485 |
-0,00153 |
7,4387E-06 |
|
1,873438 |
1,875781 |
1,878125 |
-0,00153 |
0,000115 |
-1,768E-07 |
корень |
1,873438 |
1,874609 |
1,875781 |
-0,00153 |
-0,00071 |
1,0881E-06 |
корень |
Метод Ньютона
|
e= |
0,00001 |
|
x |
f(x) |
f'(x) |
|f(x)|<e |
1,7 |
-0,13737 |
2,922119 |
|
1,747009 |
-0,09769 |
2,930059 |
|
1,78035 |
-0,0709 |
2,934159 |
|
1,804514 |
-0,05215 |
2,936425 |
|
1,822273 |
-0,03871 |
2,937742 |
|
1,835449 |
-0,02892 |
2,938541 |
|
1,84529 |
-0,02171 |
2,939042 |
|
1,852675 |
-0,01635 |
2,939367 |
|
1,858237 |
-0,01234 |
2,939583 |
|
1,862437 |
-0,00934 |
2,93973 |
|
1,865613 |
-0,00707 |
2,939833 |
|
1,86802 |
-0,00537 |
2,939905 |
|
1,869845 |
-0,00407 |
2,939958 |
|
1,87123 |
-0,00309 |
2,939995 |
|
1,872282 |
-0,00235 |
2,940023 |
|
1,873081 |
-0,00179 |
2,940044 |
|
1,873688 |
-0,00136 |
2,940059 |
|
1,87415 |
-0,00103 |
2,940071 |
|
1,874501 |
-0,00079 |
2,940079 |
|
1,874768 |
-0,0006 |
2,940086 |
|
1,874972 |
-0,00045 |
2,940091 |
|
1,875126 |
-0,00035 |
2,940094 |
|
1,875244 |
-0,00026 |
2,940097 |
|
1,875333 |
-0,0002 |
2,940099 |
|
1,875401 |
-0,00015 |
2,940101 |
|
1,875453 |
-0,00012 |
2,940102 |
|
1,875492 |
-8,8E-05 |
2,940103 |
|
1,875522 |
-6,7E-05 |
2,940104 |
|
1,875545 |
-5,1E-05 |
2,940104 |
|
1,875562 |
-3,9E-05 |
2,940105 |
|
1,875575 |
-3E-05 |
2,940105 |
|
1,875585 |
-2,2E-05 |
2,940105 |
|
1,875593 |
-1,7E-05 |
2,940106 |
|
1,875599 |
-1,3E-05 |
2,940106 |
|
1,875603 |
-9,9E-06 |
2,940106 |
корень |
Метод простой итерации
|
e= |
0,001 |
|
x |
f(x) |
s(x) |
|f(x0|<e |
1,7 |
-0,1373665 |
1,956703907 |
|
1,956704 |
0,0544055 |
1,843767838 |
|
1,843768 |
-0,0228162 |
1,889021708 |
|
1,889022 |
0,0093503 |
1,870132172 |
|
1,870132 |
-0,0038688 |
1,877888324 |
|
1,877888 |
0,00159444 |
1,874681704 |
|
1,874682 |
-0,0006582 |
1,876003687 |
корень |
1,876004 |
0,00027152 |
1,875458043 |
корень |
1,875458 |
-0,000112 |
1,875683148 |
корень |
Программирование
Шаговый метод отделения корней.
#include<iostream.h>
#include<math.h>
#include<conio.h>
#include<stdio.h>
void main()
{float xn,xk,hx,z,x;
clrscr();
xn=1;
xk=2;
hx=0.1;
for(x=xn;x<=xk;x=x+hx){
z=cos(2/x)+2*sin(1/x)+1/x;
printf(" %f\n ",x);
printf(" %f\n ",z);
printf("\n");
}}
Метод половинного деления
#include<iostream.h>
#include<math.h>
#include<conio.h>
#include<stdio.h>
void main()
{float a,b,x,fa,fx,e;
clrscr();
a=1.8;
b=1.9;
e=0.001;
while(abs(fx)>e)
{x=(a+b)/2;
fa=cos(2/a)-2*sin(1/a)+1/a;
fx=cos(2/x)-2*sin(1/x)+1/x;
if (fa*fx<0)
b=x;
else
a=x;
printf("%f",x);
printf("%f",fx); }}
Уточнение корня методом Ньютона
#include<iostream.h>
#include<math.h>
#include<conio.h>
#include<stdio.h>
void main()
{float x,e,f1x,fx;
clrscr();
x=1.7;
e=0.001;
while(abs(fx)>e)
{fx=cos(2/x)-2*sin(1/x)+1/x;
f1x=2/(x*x)*sin(2/x)+2/(x*x)*cos(1/x)-1/x*x;
x=x-fx/f1x;
printf("%f",x);
printf("%f",fx); }}
Метод простой итерации
#include<iostream.h>
#include<math.h>
#include<conio.h>
#include<stdio.h>
void main()
{float x,e,sx,fx;
clrscr();
x=1.7;
e=0.001;
while(abs(fx)>e)
{x=2/(acos(2*sin(1/x)-1/x));
fx=cos(2/x)-2*sin(1/x)+1/x;
printf("%f",x);
printf("%f",fx); }}
Ручной счёт
Шаговый метод:
F(x)= cos(2/x) - 2*sin(1/x) + 1/x [1;2]
x |
F(x) |
1 |
-1,09909 |
1,1 |
-0,91367 |
1,2 |
-0,74274 |
1,3 |
-0,58961 |
1,4 |
-0,45412 |
1,5 |
-0,33484 |
1,6 |
-0,22987 |
1,7 |
-0,13737 |
1,8 |
-0,05561 |
1,9 |
0,0169 |
2 |
0,081451 |
Метод половинного деления e=0.001
a |
x |
b |
f(a) |
f(x) |
f(a)*f(x) |
|f(x)|<e |
1,7 |
1,85 |
2 |
-0,137 |
-0,018 |
0,0025 |
|
1,85 |
1,925 |
2 |
-0,018 |
0,033 |
-0,00061 |
|
1,85 |
1,8875 |
1,925 |
-0,018 |
0,008 |
-0,00015 |
|
1,85 |
1,86875 |
1,8875 |
-0,0183 |
-0,0049 |
8,86E-05 |
|
1,86875 |
1,8781 |
1,8875 |
-0,0048 |
0,0017 |
-8,53E-06 |
|
1,86875 |
1,87344 |
1,878 |
-0,0049 |
-0,00153 |
7,439E-06 |
|
1,87344 |
1,87578 |
1,878 |
-0,0015 |
0,00012 |
-1,77E-07 |
корень |
Метод Ньютона e=0.00004
x |
f(x) |
f'(x) |
|f(x)|<e |
1,7 |
-0,137 |
2,922 |
|
1,747 |
-0,0977 |
2,93 |
|
1,78 |
-0,0709 |
2,935 |
|
1,804 |
-0,052 |
2,937 |
|
1,822 |
-0,0387 |
2,9378 |
|
1,8355 |
-0,023 |
2,939 |
|
1,85 |
-0,022 |
2,934 |
|
1,853 |
-0,017 |
2,939 |
|
1,858 |
-0,01234 |
2,94 |
|
1,862 |
-0,009 |
2,9398 |
|
1,8657 |
-0,007 |
2,93983 |
|
1,868 |
-0,0054 |
2,93990 |
|
1,8699 |
-0,00407 |
2,939958 |
|
1,87123 |
-0,00309 |
2,939995 |
|
1,87228 |
-0,00235 |
2,940023 |
|
1,87308 |
-0,0018 |
2,940044 |
|
1,8734 |
-0,0014 |
2,94006 |
|
1,874 |
-0,001 |
2,94007 |
|
1,87451 |
-0,0008 |
2,94008 |
|
1,87476 |
-0,0006 |
2,94009 |
|
1,87497 |
-0,00045 |
2,940091 |
|
1,87512 |
-0,00035 |
2,940093 |
|
1,87524 |
-0,00026 |
2,940097 |
|
1,87533 |
-0,0002 |
2,940099 |
|
1,8754 |
-0,00015 |
2,940101 |
|
1,87545 |
-0,00012 |
2,940102 |
|
1,87549 |
-8,8E-05 |
2,940103 |
|
1,87552 |
-6,7E-05 |
2,940104 |
|
1,87555 |
-5,1E-05 |
2,940104 |
|
1,87556 |
-3,9E-05 |
2,940105 |
|
1,87557 |
-3E-05 |
2,940105 |
|
1,87558 |
-2,2E-05 |
2,940105 |
|
1,87559 |
-1,7E-05 |
2,940106 |
|
1,8756 |
-1,3E-05 |
2,940106 |
|
1,875603 |
-9,9E-06 |
2,940106 |
корень |
Метод простой итерации
x |
f(x) |
s(x) |
|f(x0|<e |
0,9 |
0,21 |
0,9525 |
|
0,9525 |
0,0972 |
0,9768 |
|
0,9768 |
0,0469 |
0,9885 |
|
0,9885 |
0,02305 |
0,9943 |
|
0,9943 |
0,011425 |
0,99716 |
|
0,99716 |
0,00569 |
0,998582 |
|
0,99858 |
0,00284 |
0,9992915 |
|
0,99929 |
0,0014176 |
0,99964587 |
|
0,999646 |
0,000701 |
0,999822964 |
корень |