n -
, (7.21)
(7.22)
(7.23)
(7.23) (7.21)
(7.24)
(7.22) zl
. (7.25)
-
. (7.26)
Kl ,1, Kl ,2, Kl ,3 Kl ,4 :
7.5.
,
Excel. . :
.
h = 0,1. Excel , 7.4.
7.4 . 7.10.
1) f_1(), f_2() f_3() :
Function f_1(ByVal x, ByVal y_1, ByVal y_2, ByVal y_3)
f_1 = 2 * x * y_1 / y_2
End Function
Function f_2(ByVal x, ByVal y_1, ByVal y_2, ByVal y_3)
f_2 = 8 * y_3 / (3 * y_2)
End Function
Function f_3(ByVal x, ByVal y_1, ByVal y_2, ByVal y_3)
f_3 = 3 * y_3 / y_1
End Function
Kl , I .
2) B 3, C 3, D 3 0, 1, 0.
xi 0; 0,1; 0,2; ; 1.
E 3, F 3, G 3 =SIN(A3), =COS(A3) =-SIN(A3).
7.10
A | B | C | D | E | F | G | |
1 | h= | 0,1 |
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2 | xi | y1i | y2i | y3i | y1=sinx | y2=cosx | y3=-sinx |
3 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
4 | 0,1 | 0,099833 | 0,995004 | -0,09983 | 0,09983 | 0,995 | -0,099833 |
5 | 0,2 | 0,198669 | 0,980067 | -0,19867 | 0,19867 | 0,98007 | -0,198669 |
6 | 0,3 | 0,29552 | 0,955337 | -0,29552 | 0,29552 | 0,95534 | -0,29552 |
7 | 0,4 | 0,389418 | 0,921061 | -0,38942 | 0,38942 | 0,92106 | -0,389418 |
8 | 0,5 | 0,479425 | 0,877583 | -0,47943 | 0,47943 | 0,87758 | -0,479426 |
9 | 0,6 | 0,564642 | 0,825336 | -0,56464 | 0,56464 | 0,82534 | -0,564642 |
10 | 0,7 | 0,644217 | 0,764843 | -0,64422 | 0,64422 | 0,76484 | -0,644218 |
11 | 0,8 | 0,717356 | 0,696707 | -0,71736 | 0,71736 | 0,69671 | -0,717356 |
12 | 0,9 | 0,783326 | 0,621611 | -0,78333 | 0,78333 | 0,62161 | -0,783327 |
13 | 1 | 0,84147 | 0,540303 | -0,84147 | 0,84147 | 0,5403 | -0,841471 |
B E, , - y1=sinx.
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:
1) u (x), [ a, b ]
, (7.27)
u (a) = u (b) = 0. (7.28)
(7.27), (7.28) .
x = a x = b , . f (x) , u (x) .
2)
(7.29)
(7.30)
(7.29), (7.30)
(7.31)
(7.32)
: . ui u (x) . (7.31) (5.11):
. (7.33)
(7.31), (7.32) ui
(7.34)
(7.34) p (x) ≥ 0 . [1].
(7.34)
(7.35)
, .
1. . (7.35)
(7.36)
(7.36) (7.35) , , ui
(7.37)
(7.35), (7.37) ,
(7.38)
2. . . (7.38) :
(7.39)
(7.40)
7. 6.
.
Excel. N = 10, .. h = π/10 ≈ 0,31415926.
x, p (x) f (x), (7.36) (7.40). 7.11 Excel .
1) A 1: A 15, B 1: B 15 A 4: H 4 7.11. x 0 B 5, 3,1415926 B 6, B 5: B 6 B 15.
2) a, b h B 1, B 2, B 3 .
, .
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3) p (x), .. C 5: C 16 , p (x) = 1.
4) D 5: D 16 =-2*SIN(x),
f (x) = 2sin x. D 5 D 5 D 15.
5) E 5 0, E 6
=1/(2+C6*h^2-E5) E 6 E 13. E 14 0.
6) F 5 0, F 6
=(a-D6*h^2)/(2+C6*h^2-E5), F 7 =(F6-D7*h^2)/(2+C7*h^2-E6) F7 F 14.
7) . G 14 =F14, G 13 =E13*G14+F13 G 13 G 6.
8) H 5: H 15 . H 5 =sin(x) H 5 H 15.
, 0,01.
7.11
A | B | C | D | E | F | G | H | |
1 | a= | 0 |
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2 | b= | 0 |
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3 | h= | 0,31415926 |
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4 | i | x | p(x) | f(x) | alfa | beta | ui | sin(x) |
5 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 |
6 | 1 | 0,31415926 | 1 | -0,61803 | 0,476486 | 0,029064 | 0,310289 | 0,309017 |
7 | 2 | 0,62831852 | 1 | -1,17557 | 0,616443 | 0,089439 | 0,590204 | 0,587785 |
8 | 3 | 0,94247778 | 1 | -1,61803 | 0,674649 | 0,168077 | 0,812347 | 0,809017 |
9 | 4 | 1,25663704 | 1 | -1,90211 | 0,702224 | 0,249857 | 0,954971 | 0,951057 |
10 | 5 | 1,5707963 | 1 | -2 | 0,71609 | 0,320271 | 1,004116 | 1 |
11 | 6 | 1,88495556 | 1 | -1,90211 | 0,723272 | 0,367423 | 0,954971 | 0,951057 |
12 | 7 | 2,19911482 | 1 | -1,61803 | 0,727048 | 0,383239 | 0,812347 | 0,809017 |
13 | 8 | 2,51327408 | 1 | -1,17557 | 0,72905 | 0,363988 | 0,590204 | 0,587785 |
14 | 9 | 2,82743334 | 1 | -0,61803 | 0 | 0,310289 | 0,310289 | 0,309017 |
15 | 10 | 3,1415926 | 1 | -1,1E-07 |
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| 5,36E-08 |
, Excel.
7. 7.
Excel. h = 0,1, .. [1, 2] N = 10 . 7.4 . E, F G, (7.27) (7.31) .
a = 2, b = 12 h = 0,1, B 1: B 3 2, 12 0,1.
, 7.12.
p (x), .. C 5: C 16 , p (x) = 0.
D 5 =6*x+2, f (x) = 6 x + 2, D 5 D 15.
H 5 =x^3+x^2 H 5 H 15.
7.12.
7.12
A | B | C | D | E | F | G | H | |||
1 | a= | 2 |
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2 | b= | 12 |
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3 | h= | 0,1 |
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4 | i | x | p(x) | f(x) | alfa | beta | ui | x^3+x^2 | ||
5 | 0 | 1 | 0 | 8 | 0 | 0 |
| 2 | ||
6 | 1 | 1,1 | 0 | 8,6 | 0,5 | 0,957 | 2,54100000 | 2,54100000 | ||
7 | 2 | 1,2 | 0 | 9,2 | 0,666667 | 0,576667 | 3,16800000 | 3,16800000 | ||
8 | 3 | 1,3 | 0 | 9,8 | 0,75 | 0,359 | 3,88700000 | 3,88700000 | ||
9 | 4 | 1,4 | 0 | 10,4 | 0,8 | 0,204 | 4,70400000
| 4,70400000 | ||
10 | 5 | 1,5 | 0 | 11 | 0,833333 | 0,078333 | 5,62500000 | 5,62500000 | ||
11 | 6 | 1,6 | 0 | 11,6 | 0,857143 | -0,03229 | 6,65600000 | 6,65600000 | ||
12 | 7 | 1,7 | 0 | 12,2 | 0,875 | -0,135 | 7,80300000 | 7,80300000 | ||
13 | 8 | 1,8 | 0 | 12,8 | 0,888889 | -0,23378 | 9,07200000 | 9,07200000 | ||
14 | 9 | 1,9 | 0 | 13,4 | 0 | 10,469 | 10,46900000 | 10,46900000 | ||
15 | 10 | 2 | 0 | 14 |
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| 12 |
(7.34) (7.31), (7.32) . (7.33) (!) .