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Файл:Алгоритмизация в инженерных задачах. Учебное пособие
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ȼ ɦɟɬɨɞɟ ɝɪɚɞɢɟɧɬɚ ɲɚɝ
ɝɞɚ ɩɪɢɯɨɞɢɬɫɹ ɞɟɥɚɬɶ ɛɨɥɶɲɨɟ ɱɢɫɥɨ ɢɬɟɪɚɰɢɣ. ȼɫɥɟɞɫɬɜɢɟ ɷɬɨɝɨ ɛɨɥɟɟ ɷɮɮɟɤɬɢɜɧɨ ɩɪɢɦɟɧɹɬɶ ɦɟɬɨɞ ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɫɩɭɫɤɚ Ʉɨɲɢ, ɩɪɢ ɤɨɬɨɪɨɦ ɲɚɝ
ɛɢɪɚɟɬɫɹ ɢɡ ɭɫɥɨɜɢɹ ɦɢɧɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɢ f(
′
0)(f =λ
(ɩɪɨɢɡɜɨɞɧɚɹ
ɢ ɧɚɯɨɞɹɬɫɹ ɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ λ ).
ɩɨɫɬɨɹɧɧɵɣ, ɨɧ ɦɨɠɟɬ ɨɤɚɡɚɬɶɫɹ ɦɚɥɵɦ, ɢ ɬɨ-
λ
λ ɜɵ-
) ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ ɯ*
λ
Ɋɢɫ. 21. Ȼɥɨɤ-ɫɯɟɦɚ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɝɪɚɞɢɟɧɬɧɨɝɨ ɫɩɭɫɤɚ
ɫ ɮɢɤɫɢɪɨɜɚɧɧɵɦ ɲɚɝɨɦ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ
ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɫɩɭɫɤɚ
1. Ɉɩɪɟɞɟɥɹɸɬ ɧɚɱɚɥɶɧɵɣ ɜɟɤɬɨɪ ɚɪɝɭɦɟɧɬɨɜ ɮɭɧɤɰɢɢ, ɩɪɢɧɚɞɥɟɠɚɳɢɣ
ɞɨɩɭɫɬɢɦɨɣ ɨɛɥɚɫɬɢ ɡɧɚɱɟɧɢɣ f(x).
2.
Ɉɩɪɟɞɟɥɹɸɬ grad f(x0) ɢɥɢ –grad f(x0).
71
λ
k
3.
x
Ɋɚɫɫɱɢɬɵɜɚɸɬ ɲɚɝ
Ɉɩɪɟɞɟɥɹɸɬ ɭɬɨɱɧɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɪɧɹ ɭɪɚɜɧɟɧɢɹ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
4.
(k+1)
= x
± λ grad f(x
(k)
ɢɡ ɭɪɚɜɧɟɧɢɹ 0)(f =
λ
), ɡɧɚɤ «+» ɩɪɢɦɟɧɹɟɬɫɹ, ɟɫɥɢ ɧɟɨɛɯɨɞɢɦɨ ɦɚɤɫɢɦɢ-
(k)
ɡɢɪɨɜɚɬɶ ɮɭɧɤɰɢɸ, «–» – ɞɥɹ ɦɢɧɢɦɢɡɚɰɢɢ.
– ɟɫɥɢ
)1k(
, ɪɟɲɟɧɢɟ ɧɚɣɞɟɧɨ;
ε<+)x(gradf
′
.
– ɟɫɥɢ ɧɟɬ, ɬɨ ɩɟɪɟɯɨɞ ɤ ɩ. 2.
ȿɫɥɢ grad f(x
(k)
) = 0, ɬɨ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɬɨɱɧɵɦ.
ɉɪɟɢɦɭɳɟɫɬɜɨɦ ɦɟɬɨɞɚ ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɫɩɭɫɤɚ ɹɜɥɹɟɬɫɹ ɟɝɨ ɩɪɨɫɬɨɬɚ ɢ
ɭɦɟɧɶɲɟɧɢɟ ɨɛɴɟɦɨɜ ɪɚɫɱɟɬɨɜ, ɬɚɤ ɤɚɤ grad f(x) ɜɵɱɢɫɥɹɟɬɫɹ ɧɟ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ,
ɱɬɨ ɫɭɳɟɫɬɜɟɧɧɨ ɞɥɹ ɡɚɞɚɱ ɛɨɥɶɲɨɣ ɪɚɡɦɟɪɧɨɫɬɢ.
ɇɟɞɨɫɬɚɬɤɨɦ ɹɜɥɹɟɬɫɹ ɬɨ, ɱɬɨ ɲɚɝɢ ɞɨɥɠɧɵ ɛɵɬɶ ɦɚɥɵɦɢ, ɱɬɨɛɵ ɧɟ ɩɪɨɩɭɫɬɢɬɶ ɬɨɱɤɭ ɨɩɬɢɦɭɦɚ.
ɉɪɢ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɢ ɦɟɬɨɞɨɜ ɝɪɚɞɢɟɧɬɧɨɝɨ ɫɩɭɫɤɚ ɧɟɨɛɯɨɞɢɦɨ ɩɪɚɜɢɥɶɧɨ
ɜɵɛɪɚɬɶ ɲɚɝ ɢɬɟɪɚɰɢɣ:
ɞɥɹ ɦɟɬɨɞɚ ɝɪɚɞɢɟɧɬɧɨɝɨ ɫɩɭɫɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦ ɲɚɝɨɦ ɲɚɝ Ȝ ɧɟɨɛɯɨɞɢ-
−
ɦɨ ɜɵɛɢɪɚɬɶ ɦɟɧɶɲɟ 0,01, ɢɧɚɱɟ ɦɟɬɨɞ ɪɚɫɯɨɞɢɬɫɹ (ɦɟɬɨɞ ɦɨɠɟɬ ɪɚɫɯɨɞɢɬɶɫɹ ɢ
ɩɪɢ ɬɚɤɨɦ ɲɚɝɟ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɢɫɫɥɟɞɭɟɦɨɣ ɮɭɧɤɰɢɢ);
−
ɝɪɚɞɢɟɧɬɧɵɣ ɦɟɬɨɞ ɫ ɞɪɨɛɥɟɧɢɟɦ ɲɚɝɚ ɧɟ ɨɱɟɧɶ ɱɭɜɫɬɜɢɬɟɥɟɧ ɤ ɜɵɛɨɪɭ
ɩɚɪɚɦɟɬɪɨɜ. Ɉɞɢɧ ɢɡ ɜɚɪɢɚɧɬɨɜ ɜɵɛɨɪɚ ɩɚɪɚɦɟɬɪɨɜ:
,
1,0=ε
−
ɞɥɹ ɦɟɬɨɞɚ ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɫɩɭɫɤɚ ɲɚɝ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɫ ɩɨɦɨɳɶɸ
,
10=λ
…
δ⋅λ=λ
01
, ɩɪɢ
δ⋅λ=λ
−1kk
;
95,0=δ
«ɡɨɥɨɬɨɝɨ ɫɟɱɟɧɢɹ» [19, 20].
4.3. ɍɫɥɨɜɧɚɹ ɨɩɬɢɦɢɡɚɰɢɹ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ.
Ɇɟɬɨɞ ɲɬɪɚɮɧɵɯ ɮɭɧɤɰɢɣ
Ȼɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɢɧɠɟɧɟɪɧɵɯ ɡɚɞɚɱ ɫɜɹɡɚɧɨ ɫ ɨɩɬɢɦɢɡɚɰɢɟɣ ɩɪɢ
ɧɚɥɢɱɢɢ ɧɟɤɨɬɨɪɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɨɝɪɚɧɢɱɟɧɢɣ. Ɍɚɤɢɟ ɨɝɪɚɧɢɱɟɧɢɹ ɫɭɳɟɫɬɜɟɧɧɨ
ɭɦɟɧɶɲɚɸɬ ɪɚɡɦɟɪɵ ɨɛɥɚɫɬɢ, ɜ ɤɨɬɨɪɨɣ ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɨɢɫɤ ɨɩɬɢɦɭɦɚ. ɉɪɨɰɟɫɫ ɨɩɬɢɦɢɡɚɰɢɢ ɫɬɚɧɨɜɢɬɫɹ ɛɨɥɟɟ ɫɥɨɠɧɵɦ, ɬɚɤ ɤɚɤ ɨɫɧɨɜɧɨɟ ɭɫɥɨɜɢɟ ɨɩɬɢɦɚɥɶɧɨɫɬɢ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɨɬɨɪɵɦ ɨɩɬɢɦɭɦ ɞɨɥɠɟɧ ɧɚɯɨɞɢɬɶɫɹ ɜ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɨɱɤɟ (ɝɪɚɞɢɟɧɬ ɪɚɜɟɧ ɧɭɥɸ) ɦɨɠɟɬ ɧɟ
ɜɵɩɨɥɧɹɬɶɫɹ (ɟɫɥɢ ɫɬɚɰɢɨɧɚɪɧɚɹ
ɬɨɱɤɚ ɧɟ ɜɯɨɞɢɬ ɜ ɈȾɁ ɮɭɧɤɰɢɢ).
Ɉɝɪɚɧɢɱɟɧɢɹ ɜ ɩɨɞɨɛɧɵɯ ɡɚɞɚɱɚɯ ɦɨɝɭɬ ɛɵɬɶ ɡɚɞɚɧɵ ɜ ɜɢɞɟ ɪɚɜɟɧɫɬɜ ɢ
ɧɟɪɚɜɟɧɫɬɜ.
ɇɚɩɪɢɦɟɪ, ɦɢɧɢɦɢɡɢɪɨɜɚɬɶ
)x...x,x(f
ɩɪɢ ɨɝɪɚɧɢɱɟɧɢɹɯ [19, 20]:
n21
K...2,1k,0)x...x,x(h
n21
== ;
J...2,1j,0)x...x,x(g
n21j
=≥
.
72
m
λλλ=λ
ɂɞɟɹ ɦɟɬɨɞɚ ɲɬɪɚɮɧɵɯ ɮɭɧɤɰɢɣ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɩɨɢɫɤɟ ɨɩɬɢɦɚɥɶɧɨɝɨ
ɡɧɚɱɟɧɢɹ ɧɨɜɨɣ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ F(x) = f(x) + H(x), ɤɨɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɫɭɦɦɨɣ
ɢɫɯɨɞɧɨɣ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɢ ɧɟɤɨɬɨɪɨɣ ɮɭɧɤɰɢɢ H(x), ɨɩɪɟɞɟɥɹɟɦɨɣ ɫɢɫɬɟɦɨɣ
ɨɝɪɚɧɢɱɟɧɢɣ ɢ ɧɚɡɵɜɚɟɦɨɣ ɲɬɪɚɮɧɨɣ ɮɭɧɤɰɢɟɣ. ɒɬɪɚɮɧɵɟ ɮɭɧɤɰɢɢ ɫɬɪɨɹɬ
ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɨɛɟɫɩɟɱɢɬɶ ɥɢɛɨ ɛɵɫɬɪɨɟ ɜɨɡɜɪɚɳɟɧɢɟ ɜ ɞɨɩɭɫɬɢɦɭɸ ɨɛɥɚɫɬɶ, ɥɢɛɨ ɧɟɜɨɡɦɨɠɧɨɫɬɶ ɜɵɯɨɞɵ ɢɡ ɧɟɟ.
Ɉɫɧɨɜɧɵɦ ɦɟɬɨɞɨɦ ɪɟɲɟɧɢɹ
ɬɚɤɢɯ ɡɚɞɚɱ ɹɜɥɹɟɬɫɹ ɦɟɬɨɞ ɦɧɨɠɢɬɟɥɟɣ Ʌɚɝɪɚɧɠɚ, ɤɨɬɨɪɵɣ ɩɨɡɜɨɥɹɟɬ ɩɟɪɟɜɨɞɢɬɶ ɡɚɞɚɱɢ ɭɫɥɨɜɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɜ ɡɚɞɚɱɢ
ɨɩɬɢɦɢɡɚɰɢɢ ɛɟɡ ɨɝɪɚɧɢɱɟɧɢɣ.
ȿɫɥɢ ɭɫɥɨɜɢɹ ɡɚɞɚɧɵ ɫɢɫɬɟɦɨɣ ɪɚɜɟɧɫɬɜ, ɜɜɨɞɢɦ ɧɟɨɩɪɟɞɟɥɟɧɧɵɟ ɦɧɨɠɢɬɟ-
ɥɢ
ɢ ɫɨɫɬɚɜɥɹɟɦ ɮɭɧɤɰɢɸ Ʌɚɝɪɚɧɠɚ
),..,(
21
¦
m
=
1i
)x(h)x(f),x(L
λ+=λ
ii
ɤɨɬɨɪɚɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɭ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɢ ɩɪɨɢɡɜɟɞɟɧɢɣ ɦɧɨɠɢɬɟɥɟɣ
Ʌɚɝɪɚɧɠɚ
λ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɮɭɧɤɰɢɢ ɨɝɪɚɧɢɱɟɧɢɣ.
i
Ɂɚɩɢɫɵɜɚɸɬ ɧɟɨɛɯɨɞɢɦɵɟ ɭɫɥɨɜɢɹ ɛɟɡɭɫɥɨɜɧɨɝɨ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɢ Ʌɚ-
ɝɪɚɧɠɚ
, ɩɪɢɪɚɜɧɢɜɚɹ ɧɭɥɸ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ
),x(L λ
x ɢ ɩɨ
i
.
λ
i
Ⱦɚɥɟɟ ɧɚɯɨɞɹɬ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɚɦɢ ɛɟɡɭɫɥɨɜɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ (ɧɚɩɪɢɦɟɪ, ɦɟɬɨɞɨɦ ɝɪɚɞɢɟɧɬɧɨɝɨ ɫɩɭɫɤɚ). ȿɫɥɢ ɪɟɲɟɧɢɟ ɧɟ ɜɯɨɞɢɬ ɜ
ɈȾɁ, ɬɨ ɦɟɧɹɸɬ ɤɨɷɮɮɢɰɢɟɧɬ
λ . ɂ ɩɪɨɜɨɞɹɬ ɨɩɬɢɦɢɡɚɰɢɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɨɜɨɣ
ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ.
ȿɫɥɢ ɨɝɪɚɧɢɱɟɧɢɹ ɡɚɞɚɧɵ ɜ ɜɢɞɟ ɧɟɪɚɜɟɧɫɬɜ, ɬɨ ɦɨɠɟɬ ɩɪɢɦɟɧɹɬɶɫɹ ɧɟɫɤɨɥɶɤɨ ɜɢɞɨɜ ɲɬɪɚɮɧɵɯ ɮɭɧɤɰɢɣ [19, 20]:
−
ɤɜɚɞɪɚɬɢɱɧɵɣ ɲɬɪɚɮ ɜɢɞɚ
ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɣ ɲɬɪɚɮ ɜɢɞɚ
−
−
ɨɛɪɚɬɧɚɹ ɮɭɧɤɰɢɹ ɜɢɞɚ
¦
m
λ+=λ
¦
=
1i
m
¦
m
=
=
1
()x(f),x(L
λ+=λ
i
1i
i
2
;
))x(g()x(f),x(L
ii
λ−=λ
1i
).
)x(g
));x(glg()x(f),x(L
ii
Ɍɚɤ ɤɚɤ ɢɫɫɥɟɞɭɟɦɚɹ ɮɭɧɤɰɢɹ ɨɛɵɱɧɨ ɢɦɟɟɬ ɷɤɫɬɪɟɦɭɦɵ ɜ ɤɪɚɣɧɢɯ ɬɨɱɤɚɯ, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɮɭɧɤɰɢɸ-ɨɝɪɚɧɢɱɟɧɢɟ, ɬɨ ɧɟɨɛɯɨɞɢɦɨ ɜɜɨɞɢɬɶ ɜ ɪɚɫɱɟɬ
ɭɫɥɨɜɢɹ Ʉɭɧɚ-Ɍɚɤɤɟɪɚ:
>λ
,0
i
®
¯
ii
=λ
.0)x(q
4.4. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 4.1.
.
15,0=ε
ɇɚɣɬɢ ɦɢɧɢɦɭɦ ɮɭɧɤɰɢɢ
2
1
2
ɩɪɢ ɬɨɱɧɨɫɬɢ
x25x)x(f +=
2
Ɋɟɲɢɦ ɡɚɞɚɱɭ ɝɪɚɞɢɟɧɬɧɵɦ ɦɟɬɨɞɨɦ ɫ ɩɨɫɬɨɹɧɧɵɦ ɲɚɝɨɦ, ɩɪɢɦɟɦ
.
035,0=λ
,
73
λ
1.
2.
3.
ª
*x
«
−=5,1
¬
4.
*
5. ɉɪɢɧɢɦɚɟɦ
2
º
ɉɪɢɧɢɦɚɟɦ ɧɚɱɚɥɶɧɭɸ ɬɨɱɤɭ
ɇɚɯɨɞɢɦ ɜɟɤɬɨɪ ɝɪɚɞɢɟɧɬ
ȼɵɱɢɫɥɹɟɦ
−=
®
1
¯
2
ª
.
=
0x
»
«
2
¼
¬
1ɯ2
º
ª
=∇
)x(f
, ɪɚɫɩɢɫɵɜɚɟɦ ɜ ɤɨɨɪɞɢɧɚɬɚɯ.
)0x(f0x1x ∇
1
1
.
»
«
2ɯ50
¼
¬
λ−=
;42x
λ−=
.1002x
1
86,1
º
– ɩɪɟɞɩɨɥɚɝɚɟɦɨɟ ɪɟɲɟɧɢɟ.
»
¼
®
¯
;86,1x
1
1
2
−==.5,1x
Ⱦɟɥɚɟɦ ɩɪɨɜɟɪɤɭ ɧɚ ɨɤɨɧɱɚɧɢɟ ɜɵɱɢɫɥɟɧɢɣ
*
ε<∇ )x(f
⋅
86,12
ª
1
=∇
)x(f
«
−⋅
¬
ª
º
=
«
»
)5,1(50
−
¬
¼
221
=−+=∇
, ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɨɞɨɥɠɚɟɦ ɪɚɫɱɟɬɵ.
ε>∇ )x(f
112
=∇λ−=
2
1
®
2
¯
2
λ−=
;72,386,1x
λ+−=
.755,1x
2
=
;73,1x
1
®
2
¯
2
=
.125,1x
74
72,3
75
º
,
»
¼
,
092,75)75(72,3)x(f
...2,1,0n),x(fxx
6.
Ⱦɟɥɚɟɦ ɩɪɨɜɟɪɤɭ ɧɚ ɨɤɨɧɱɚɧɢɟ ɜɵɱɢɫɥɟɧɢɣ
⋅
73,12
ª
1
=∇
)x(f
«
⋅
125,150
¬
46,3
º
ª
º
=
«
»
¬
¼
,
»
25,56
¼
221
=+=∇
*
, ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɨɞɨɥɠɚɟɦ ɪɚɫɱɟɬɵ.
ε>∇ )x(f
ɉɪɢ
*
1
®
*
==.0x
¯
2
*
;071,0x
, ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɬɨɱɤɚ ɹɜɥɹɟɬɫɹ
15,0142,1)x(f
=ε<=∇
,
356,5625,5646,3)x(f
ɷɤɫɬɪɟɦɭɦɨɦ f(x1, x2).
ɉɪɢɦɟɪ ɤɨɞɚ ɩɪɢɥɨɠɟɧɢɹ:
{
float x1,x2,k,e,df1,df2,x11,x22,df11,df22,mf,f1,f2;
k = 0.035;
e = 0.15;
x1 = StrToFloat(Edit1->Text);
x2 = StrToFloat(Edit2->Text);
do {
//1. ɧɚɯɨɞɢɦ ɩɪɨɢɡɜɨɞɧɭɸ
df11 = 2*x1;
df22 = 50*x2;
//2. ɪɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɨɜ ɮɭɧɤɰɢɢ ɜ ɢɫɯɨɞɧɨɣ ɧɭɥɟɜɨɣ ɬɨɱɤɟ
x11 = x1-k*df11;
x22 = x2-k*df22;
//3. ɪɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɨɬ ɧɚɣɞɟɧɧɵɯ ɚɪɝɭɦɟɧɬɨɜ
f1 = 2*x11;
f2 = 2*x22;
//4. ɪɚɫɫɱɢɬɵɜɚɟɦ ɦɨɞɭɥɶ ɜɟɤɬɨɪɚ ɩɪɨɢɡɜɨɞɧɨɣ
mf = pow((f1*f1+f2*f2),0.5);
75
λ
//5. ɩɪɢɫɜɚɢɜɚɟɦ ɡɧɚɱɟɧɢɟ ɫɥɟɞɭɸɳɟɣ ɧɭɥɟɜɨɣ ɬɨɱɤɢ
x1 = x11-k*f1;
x2 = x22-k*f2;
}
while (mf>e);
Edit3->Text = FloatToStr(x11);
Edit4->Text = FloatToStr(x22);
Ɂɚɞɚɱɚ 4.2. Ɇɢɧɢɦɢɡɢɪɨɜɚɬɶ ɮɭɧɤɰɢɸ
ɨɝɪɚɧɢɱɟɧɢɢ
02x1x5)x(g ≥−−=
.
)42x()41x()x(f −+−=
22
ɩɪɢ
ɉɪɢɦɟɧɹɹ ɤɜɚɞɪɚɬɢɱɧɵɣ ɲɬɪɚɮ, ɫɨɫɬɚɜɢɦ ɮɭɧɤɰɢɸ Ʌɚɝɪɚɧɠɚ
m
¦
2222
.
ii
1i
=
)2x1x5()42x()41x())x(g()x(f),x(L −−λ+−+−=λ+=λ
ɍɪɚɜɧɟɧɢɹ, ɨɩɪɟɞɟɥɹɸɳɢɟ ɫɬɚɰɢɨɧɚɪɧɭɸ ɬɨɱɤɭ, ɢɦɟɸɬ ɜɢɞ
L
∂
1x
∂
L
∂
2x
∂
,
0)2x1x5(2)41x(2
=−−λ−−=
.
0)2x1x5(2)42x(2
=−−λ−−=
Ⱦɚɥɟɟ ɨɩɬɢɦɢɡɢɪɭɟɦ ɮɭɧɤɰɢɸ ɩɪɢ ɪɚɡɥɢɱɧɵɯ λ ɦɟɬɨɞɨɦ ɝɪɚɞɢɟɧɬɧɨɝɨ
ɫɩɭɫɤɚ ɩɪɢ ɲɚɝɟ k = 0.015 ɢ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɢ e = 0.15.
ȼɵɛɢɪɚɟɦ ɧɭɥɟɜɭɸ ɬɨɱɤɭ
, ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɨɝɪɚɧɢɱɟɧɢɹɦ,
)2x,1x(xɨ=
ɢ ɩɪɨɜɨɞɹɬ ɨɩɬɢɦɢɡɚɰɢɸ. Ʉɚɠɞɭɸ ɩɨɥɭɱɟɧɧɭɸ ɬɨɱɤɭ ɨɩɬɢɦɭɦɚ
ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɜɟɪɹɬɶ ɧɚ ɜɵɩɨɥɧɟɧɢɟ ɨɝɪɚɧɢɱɟɧɢɣ ɩɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ. ȿɫɥɢ
ɪɟɲɟɧɢɟ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɝɪɚɧɢɱɟɧɢɹɦ, ɬɨ ɧɟɨɛɯɨɞɢɦɨ ɩɟɪɟɫɱɢɬɵɜɚɬɶ ɡɧɚɱɟ-
*
x ɩɪɢ ɞɪɭɝɨɦ ɡɧɚɱɟɧɢɢ λ .
ɧɢɟ
ɉɨ ɦɟɪɟ ɩɪɨɜɟɞɟɧɢɹ ɪɚɫɱɟɬɨɜ ɡɧɚɱɟɧɢɟ ɨɩɬɢɦɭɦɚ ɛɭɞɟɬ ɫɯɨɞɢɬɶɫɹ ɤ ɨɞɧɨɦɭ ɡɧɚɱɟɧɢɸ
Ɍɚɤ, ɩɪɢ
ɢɫɯɨɞɧɨɦɭ ɧɟɪɚɜɟɧɫɬɜɭ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɧɟɨɛɯɨɞɢɦɨ ɢɡɦɟɧɢɬɶ
ɉɪɢɧɢɦɚɟɦ 2=λ ,
.
)2x,1x(x*=
ɢ 1=λ ɢɦɟɟɦ
)2,2(x0=
)2,2(x0=
, ɱɬɨ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ
)98,2,98,2(x*=
λ .
,
.
)79,2;79,2(x*=
)2x,1x(x*=
ɉɪɢɧɢɦɚɟɦ
ɉɪɢɧɢɦɚɟɦ
,
10=
,
30=λ
,
)2,2(x0=
,
)2,2(x0=
.
)57,2;57,2(x*=
.
)52,2;52,2(x*=
76
Ⱦɚɥɶɧɟɣɲɟɟ ɢɡɦɟɧɟɧɢɟ ɩɚɪɚɦɟɬɪɚ
ɜɵɯɨɞɧɨɝɨ ɡɧɚɱɟɧɢɹ ɨɩɬɢɦɭɦɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɢɧɢɦɚɟɦ ɡɚ ɢɫɤɨɦɨɟ ɡɧɚɱɟɧɢɟ
.
)52,2;52,2(x*=
ɉɪɢɦɟɪ ɤɨɞɚ ɩɪɢɥɨɠɟɧɢɹ:
float x1,x2,e,k,df1,df2,x11,x22,df11,df22,mf,f1,f2,r;
k = 0.015;
e = 0.15;
x1 = StrToFloat(Edit1->Text);
x2 = StrToFloat(Edit2->Text);
r = StrToFloat(Edit6->Text);
do {
//1. ɧɚɯɨɞɢɦ ɩɪɨɢɡɜɨɞɧɭɸ
df11 = 2*(x1-4)-r*2*(5-x1-x2);
df22 = 2*(x2-4)-r*2*(5-x1-x2);
//2. ɪɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɨɜ ɮɭɧɤɰɢɢ ɜ ɢɫɯɨɞɧɨɣ ɧɭɥɟɜɨɣ ɬɨɱɤɟ
x11 = x1-k*df11;
x22 = x2-k*df22;
//3. ɪɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɨɬ ɧɚɣɞɟɧɧɵɯ ɚɪɝɭɦɟɧɬɨɜ
f1 = 2*(x11-4)-r*2*(5-x11-x22);
f2 = 2*(x22-4)-r*2*(5-x11-x22);
//4. ɪɚɫɫɱɢɬɵɜɚɟɦ ɦɨɞɭɥɶ ɜɟɤɬɨɪɚ ɩɪɨɢɡɜɨɞɧɨɣ
mf = pow((f1*f1+f2*f2),0.5);
//5. ɩɪɢɫɜɚɢɜɚɟɦ ɡɧɚɱɟɧɢɟ ɫɥɟɞɭɸɳɟɣ ɧɭɥɟɜɨɣ ɬɨɱɤɢ
x1 = x11-k*f1;
x2 = x22-k*f2;
}
while (mf>e);
Edit3->Text = FloatToStr(x11);
Edit4->Text = FloatToStr(x22);
ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɜɥɢɹɟɬ ɧɚ ɢɡɦɟɧɟɧɢɟ
λ
77
Ɂɚɞɚɱɚ 4.3. Ɍɪɟɛɭɟɬɫɹ ɩɟɪɟɩɪɚɜɢɬɶ 400 ɦ
3
ɫɵɩɭɱɟɝɨ ɦɚɬɟɪɢɚɥɚ ɱɟɪɟɡ ɪɟɤɭ.
Ⱦɥɹ ɩɟɪɟɜɨɡɤɢ ɝɪɭɡɚ ɧɟɨɛɯɨɞɢɦɨ ɫɤɨɧɫɬɪɭɢɪɨɜɚɬɶ ɤɨɧɬɟɣɧɟɪ. ɂɡɜɟɫɬɧɵ ɫɥɟɞɭɸɳɢɟ ɞɚɧɧɵɟ: ɫɬɨɢɦɨɫɬɶ ɤɚɠɞɨɝɨ ɪɟɣɫɚ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɛɟɪɟɝ ɪɟɤɢ ɢ ɨɛɪɚɬɧɨ ɪɚɜɧɚ 4,2 ɪɭɛ.; ɫɬɨɢɦɨɫɬɶ ɦɚɬɟɪɢɚɥɨɜ ɞɥɹ ɢɡɝɨɬɨɜɥɟɧɢɹ ɞɧɚ ɤɨɧɬɟɣɧɟɪɚ
ɫɨɫɬɚɜɥɹɟɬ 20 ɪɭɛ/ɦ
ɤɨɧɬɟɣɧɟɪɚ
– 20 ɪɭɛ/ɦ
2
; ɛɨɤɨɜɵɯ ɫɬɟɧɨɤ ɤɨɧɬɟɣɧɟɪɚ ɫɨɫɬɚɜɥɹɟɬ 5 ɪɭɛ/ɦ2; ɤɪɵɲɤɢ
2
.
ɋɤɨɧɫɬɪɭɢɪɭɣɬɟ ɤɨɧɬɟɣɧɟɪ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɦɢɧɢɦɢɡɢɪɨɜɚɬɶ ɩɨɥɧɵɟ ɡɚɬɪɚɬɵ ɧɚ ɩɟɪɟɜɨɡɤɭ ɝɪɭɡɚ.
ɉɪɢɧɹɬɶ ɞɥɹ ɪɚɫɱɟɬɨɜ ɲɚɝ ɢɡɦɟɧɟɧɢɹ ɜɟɥɢɱɢɧ k = 0,001; ɬɨɱɧɨɫɬɶ ɟ = 0,1.
ɐɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɨɛɳɟɣ ɫɬɨɢɦɨɫɬɢ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɫɬɨɢɦɨɫɬɢ ɢɡɝɨɬɨɜɥɟɧɢɹ ɢ
ɫɬɨɢɦɨɫɬɢ ɩɟɪɟɜɨɡɤɢ
F=S +S
.
12
Ɉɛɨɡɧɚɱɢɦ ɫɬɨɪɨɧɵ ɤɨɧɬɟɣɧɟɪɚ ɞɥɹ ɩɟɪɟɜɨɡɤɢ a, b, c, ɬɨɝɞɚ ɫɬɨɢɦɨɫɬɶ ɩɟɪɟɜɨɡɤɢ ɫɨɫɬɚɜɢɬ
400 4,2
S=
1
⋅
abc
⋅⋅
,
ɚ ɫɬɨɢɦɨɫɬɶ ɢɡɝɨɬɨɜɥɟɧɢɹ ɤɨɧɬɟɣɧɟɪɚ
S =2ab20+2ac5+2bc5=40ab+10ac+10bc.⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅ ⋅⋅ ⋅⋅
2
ɐɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɜɢɞ
400 4,2
F= +40 a b+10 a c+10 b c.
⋅
abc
⋅⋅
⋅⋅ ⋅⋅ ⋅⋅
Ƚɪɚɞɢɟɧɬ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɢɫɬɟɦɨɣ ɭɪɚɜɧɟɧɢɣ ɱɚɫɬɧɵɯ
ɩɪɨɢɡɜɨɞɧɵɯ
F
∂
−=
°
a
∂
c
a
b
°
F
∂
°
−=
®
b
∂
°
F
∂
°
−=
°
c
∂
¯
2.4400
⋅
2
cba
⋅⋅
2.4400
⋅
2
cba
⋅⋅
2.4400
⋅
2
cba
⋅⋅
;c10b40
⋅+⋅+
;c10a40
.
⋅+⋅+
.b10a10
⋅+⋅+
Ⱦɚɥɟɟ ɦɢɧɢɦɢɡɢɪɭɟɦ ɰɟɥɟɜɭɸ ɮɭɧɤɰɢɸ ɦɟɬɨɞɨɦ ɝɪɚɞɢɟɧɬɧɨɝɨ ɫɩɭɫɤɚ ɫ
ɩɨɫɬɨɹɧɧɵɦ ɲɚɝɨɦ ɢ ɩɨɥɭɱɚɟɦ ɨɩɬɢɦɚɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ
78
=
°
®
°
¯
ɉɪɢɦɟɪ ɤɨɞɚ ɩɪɢɥɨɠɟɧɢɹ:
float a,b,c,e,k,df1,df2,df3,a11,b11,c11,df11,df22,df33,mf,f1,f2,f3;
k = 0.001;
e = 0.1;
a = StrToFloat(Edit1->Text);
b = StrToFloat(Edit2->Text);
c = StrToFloat(Edit3->Text);
do {
//1. ɧɚɯɨɞɢɦ ɩɪɨɢɡɜɨɞɧɭɸ
df11 = -400*4.2*b*c/((a*b*c)*(a*b*c))+40*b+10*c;
df22 = -400*4.2*a*c/((a*b*c)*(a*b*c))+40*a+10*c;
df33 = -400*4.2*b*a/((a*b*c)*(a*b*c))+10*a+10*b;
//2. ɪɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɨɜ ɮɭɧɤɰɢɢ ɜ ɢɫɯɨɞɧɨɣ ɧɭɥɟɜɨɣ ɬɨɱɤɟ
a11 = a-k*df11;
b11 = b-k*df22;
c11 = c-k*df33;
//3. ɪɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɨɬ ɧɚɣɞɟɧɧɵɯ ɚɪɝɭɦɟɧɬɨɜ
f1 = -400*4.2*b11*c11/((a11*b11*c11)*(a11*b11*c11))+40*b11+10*c11;
f2 = -400*4.2*a11*c11/((a11*b11*c11)*(a11*b11*c11))+40*a11+10*c11;
f3 = -400*4.2*b11*a11/((a11*b11*c11)*(a11*b11*c11))+10*a11+10*b11;
//4. ɪɚɫɫɱɢɬɵɜɚɟɦ ɦɨɞɭɥɶ ɜɟɤɬɨɪɚ ɩɪɨɢɡɜɨɞɧɨɣ
mf = pow((f1*f1+f2*f2+f3*f3),0.5);
//5. ɩɪɢɫɜɚɢɜɚɟɦ ɡɧɚɱɟɧɢɟ ɫɥɟɞɭɸɳɟɣ ɧɭɥɟɜɨɣ ɬɨɱɤɢ
a = a11-k*f1;
b = b11-k*f2;
c = c11-k*f3;
}
while (mf>e);
;ɦ39,1a
=
=
;ɦ39,1b
.ɦ6,5c
79
Edit4->Text = FloatToStr(a11);
Edit5->Text = FloatToStr(b11);
Edit6->Text = FloatToStr(c11);
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
Ʉɚɤɢɟ ɬɪɟɛɨɜɚɧɢɹ ɩɪɟɞɴɹɜɥɹɸɬɫɹ ɤ ɤɪɢɬɟɪɢɹɦ ɨɩɬɢɦɢɡɚɰɢɢ?
1.
ɉɟɪɟɱɢɫɥɢɬɟ ɜɢɞɵ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɪɟɲɟɧɢɹ
2.
ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɡɚɞɚɱ.
Ʉɚɤ ɤɥɚɫɫɢɮɢɰɢɪɭɸɬɫɹ ɡɚɞɚɱɢ ɨɩɬɢɦɢɡɚɰɢɢ?
3.
ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɦɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɭɫɥɨɜ-
4.
ɧɵɯ ɢ ɛɟɡɭɫɥɨɜɧɵɯ ɡɚɞɚɱ.
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɬɶ ɝɪɚɞɢɟɧɬɧɵɯ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ?
5.
ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɩɪɢɦɟɧɟɧɢɹ ɝɪɚɞɢɟɧɬɧɵɯ ɦɟɬɨɞɨɜ ɨɩɬɢ-
6.
ɦɢɡɚɰɢɢ (ɝɪɚɞɢɟɧɬɧɨɝɨ ɫɩɭɫɤɚ, ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɫɩɭɫɤɚ).
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɨɫɨɛɟɧɧɨɫɬɶ ɦɟɬɨɞɚ ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɫɩɭɫɤɚ?
7.
Ʉɚɤɢɟ ɜɢɞɵ ɨɝɪɚɧɢɱɟɧɢɣ ɩɪɢɦɟɧɹɸɬɫɹ ɜ ɡɚɞɚɱɚɯ ɨɩɬɢɦɢɡɚɰɢɢ?
8.
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɦɟɬɨɞɚ ɦɧɨɠɢɬɟɥɟɣ Ʌɚɝɪɚɧɠɚ ɞɥɹ ɨɩɬɢɦɢɡɚ-
9.
ɰɢɢ ɭɫɥɨɜɧɵɯ ɡɚɞɚɱ?
Ʉɚɤɢɟ ɜɢɞɵ ɲɬɪɚɮɧɵɯ ɮɭɧɤɰɢɣ ɩɪɢɦɟɧɹɸɬɫɹ ɜ ɦɟɬɨɞɟ ɦɧɨɠɢɬɟɥɟɣ
10.
Ʌɚɝɪɚɧɠɚ?
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