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Файл:Алгоритмизация в инженерных задачах. Учебное пособие
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ȽɅȺȼȺ 6
ɂɇɌȿɊɉɈɅəɐɂə
ɂ ɗɄɋɌɊȺɉɈɅəɐɂə ɎɍɇɄɐɂɃ
6.1. Ⱥɩɩɪɨɤɫɢɦɚɰɢɹ, ɢɧɬɟɪɩɨɥɹɰɢɹ
ɢ ɷɤɫɬɪɚɩɨɥɹɰɢɹ ɮɭɧɤɰɢɣ
Ⱥɩɩɪɨɤɫɢɦɚɰɢɟɣ (ɩɪɢɛɥɢɠɟɧɢɟɦ) ɮɭɧɤɰɢɢ ɧɚɡɵɜɚɟɬɫɹ ɧɚɯɨɠɞɟɧɢɟ ɬɚ-
ɤɨɣ ɮɭɧɤɰɢɢ (ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɟɣ ɮɭɧɤɰɢɢ), ɤɨɬɨɪɚɹ ɛɵɥɚ ɛɵ ɛɥɢɡɤɚ ɡɚɞɚɧɧɨɣ.
Ɍɨɱɧɨɫɬɶ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɡɚɞɚɟɬɫɹ ɢɡɧɚɱɚɥɶɧɨ ɢ ɡɚɜɢɫɢɬ ɨɬ ɫɤɨɪɨɫɬɢ ɪɨɫɬɚ
ɮɭɧɤɰɢɢ ɢ ɞɢɚɩɚɡɨɧɚ ɟɟ ɡɧɚɱɟɧɢɣ.
ȼ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɩɪɢɛɥɢɠɟɧɢɟ ɫɬɪɨɢɬɫɹ ɧɚ ɞɢɫɤɪɟɬɧɨɦ ɧɚɛɨɪɟ ɬɨɱɟɤ,
ɚɩɩɪɨɤɫɢɦɚɰɢɸ ɧɚɡɵɜɚɸɬ ɬɨɱɟɱɧɨɣ ɢɥɢ ɞɢɫɤɪɟɬɧɨɣ. ȿɫɥɢ ɚɩɩɪɨɤɫɢɦɚɰɢɹ ɩɪɨɜɨɞɢɬɫɹ ɧɚ ɧɟɩɪɟɪɵɜɧɨɦ ɦɧɨɠɟɫɬɜɟ ɬɨɱɟɤ (ɨɬɪɟɡɤɟ), ɬɨ ɚɩɩɪɨɤɫɢɦɚɰɢɹ ɧɚɡɵɜɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɢɥɢ ɢɧɬɟɝɪɚɥɶɧɨɣ. ɉɪɢɦɟɪɨɦ ɧɟɩɪɟɪɵɜɧɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ
ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɪɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ, ɬ. ɟ. ɡɚɦɟɧɚ ɧɟɤɨɬɨɪɨɣ
ɮɭɧɤɰɢɢ ɫɬɟɩɟɧɧɵɦ ɦɧɨɝɨɱɥɟɧɨɦ (ɧɚɩɪɢɦɟɪ, ɞɥɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɜɵɱɢɫɥɟɧɢɹ
ɢɧɬɟɝɪɚɥɚ ɦɟɬɨɞɨɦ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ ɢɥɢ ɬɪɚɩɟɰɢɣ).
Ⱥɩɩɪɨɤɫɢɦɚɰɢɹ ɩɪɢɦɟɧɹɟɬɫɹ ɞɥɹ ɪɟɲɟɧɢɹ ɩɪɢɤɥɚɞɧɵɯ ɡɚɞɚɱ ɦɟɬɨɞɨɦ ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ (ɆɄɗ). ɆɄɗ ɜɨɡɧɢɤ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɜ
ɦɟɯɚɧɢɤɟ ɫɩɥɨɲɧɵɯ ɫɪɟɞ.
ɆɄɗ – ɷɬɨ ɨɞɢɧ ɢɡ ɩɪɢɛɥɢɠɟɧɧɵɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ
ɭɪɚɜɧɟɧɢɣ. Ɉɧ ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɬɨɦ, ɱɬɨ ɥɸɛɨɟ ɧɟɩɪɟɪɵɜɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ
ɮɢɡɢɱɟɫɤɨɣ ɩɟɪɟɦɟɧɧɨɣ u (ɯ, ɭ, z, t) ɜ ɪɚɫɱɟɬɧɨɣ ɨɛɥɚɫɬɢ, ɧɚɩɪɢɦɟɪ ɞɟɮɨɪɦɚɰɢɸ ɢɥɢ ɬɟɦɩɟɪɚɬɭɪɧɨɟ ɩɨɥɟ, ɦɨɠɧɨ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɬɶ ɧɚɛɨɪɨɦ ɤɭɫɨɱɧɨɧɟɩɪɟɪɵɜɧɵɯ ɮɭɧɤɰɢɣ, ɨɩɪɟɞɟɥɟɧɧɵɯ ɧɚ ɤɨɧɟɱɧɨɦ ɱɢɫɥɟ ɩɨɞɨɛɥɚɫɬɟɣ (ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ). Ⱦɚɧɧɵɟ ɷɥɟɦɟɧɬɵ ɢɦɟɸɬ ɨɛɳɢɟ ɭɡɥɨɜɵɟ ɬɨɱɤɢ ɢ ɜ ɫɨɜɨɤɭɩɧɨɫɬɢ ɚɩɩɪɨɤɫɢɦɢɪɭɸɬ ɮɨɪɦɭ ɨɛɥɚɫɬɢ. Ⱦɥɹ ɪɚɡɛɢɟɧɢɹ ɪɚɫɱɟɬɧɨɣ ɨɛɥɚɫɬɢ ɧɚ ɷɥɟɦɟɧɬɵ ɢɫɩɨɥɶɡɭɟɬɫɹ ɫɩɟɰɢɚɥɶɧɵɣ ɚɥɝɨɪɢɬɦ ɩɨɤɪɵɬɢɹ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɣ ɚɜɬɨɦɚɬɢɱɟɫɤɭɸ ɝɟɧɟɪɚɰɢɸ ɫɟɬɤɢ (ɪɢɫ. 30).
Ɋɢɫ. 30. ɉɪɢɦɟɪ ɫɟɬɤɢ
91
Ⱦɚɥɟɟ ɩɨɞɛɢɪɚɸɬ ɪɟɲɟɧɢɟ, ɤɨɬɨɪɨɟ ɨɱɟɧɶ ɛɥɢɡɤɨ ɤ ɬɨɱɧɨɦɭ. Ɏɭɧɤɰɢɹ,
ɤɨɬɨɪɚɹ ɩɪɢɧɢɦɚɟɬɫɹ ɡɚ ɪɟɲɟɧɢɟ Ⱦɍ, ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɛɧɨɣ. Ɇɟɠɞɭ ɩɪɨɛɧɨɣ
ɮɭɧɤɰɢɟɣ ɢ ɬɨɱɧɨɣ ɜɫɟɝɞɚ ɫɭɳɟɫɬɜɭɟɬ ɨɲɢɛɤɚ – ɧɟɜɹɡɤɚ. ɐɟɥɶ ɚɥɝɨɪɢɬɦɢɡɚɰɢɢ
ɆɄɗ – ɦɢɧɢɦɢɡɢɪɨɜɚɬɶ ɧɟɜɹɡɤɭ.
ɂɧɬɟɪɩɨɥɹɰɢɟɣ ɧɚɡɵɜɚɸɬ ɬɚɤɭɸ ɪɚɡɧɨɜɢɞɧɨɫɬɶ ɚɩɩɪɨɤɫɢɦɚɰɢɢ, ɩɪɢ ɤɨ-
ɬɨɪɨɣ ɤɪɢɜɚɹ ɩɨɫɬɪɨɟɧɧɨɣ ɮɭɧɤɰɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɞɢɫɤɪɟɬɧɨɦɭ ɧɚɛɨɪɭ ɬɨɱɟɤ
ɢ ɩɪɨɯɨɞɢɬ ɬɨɱɧɨ ɱɟɪɟɡ ɧɢɯ.
ȼ ɧɚɭɱɧɵɯ ɢ ɢɧɠɟɧɟɪɧɵɯ ɪɚɫɱɟɬɚɯ ɱɚɫɬɨ ɩɪɢɯɨɞɢɬɫɹ ɨɩɟɪɢɪɨɜɚɬɶ ɧɚɛɨɪɚɦɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ, ɩɨ ɤɨɬɨɪɵɦ ɬɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɮɭɧɤɰɢɸ,
ɤɨɬɨɪɚɹ ɦɨɝɥɚ ɛɵ ɫ ɜɵɫɨɤɨɣ ɬɨɱɧɨɫɬɶɸ ɞɚɜɚɬɶ ɩɪɨɦɟɠɭɬɨɱɧɵɟ ɡɧɚɱɟɧɢɹ.
ɋɭɳɟɫɬɜɭɟɬ ɬɚɤɠɟ ɛɥɢɡɤɚɹ ɤ ɢɧɬɟɪɩɨɥɹɰɢɢ ɡɚɞɚɱɚ, ɤɨɬɨɪɚɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ
ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɤɚɤɨɣ-ɥɢɛɨ ɫɥɨɠɧɨɣ ɮɭɧɤɰɢɢ ɞɪɭɝɨɣ, ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɮɭɧɤɰɢɟɣ.
ȿɫɥɢ ɧɟɤɨɬɨɪɚɹ ɮɭɧɤɰɢɹ ɫɥɢɲɤɨɦ ɫɥɨɠɧɚ ɞɥɹ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɵɯ ɜɵɱɢɫɥɟɧɢɣ,
ɦɨɠɧɨ ɩɨɩɵɬɚɬɶɫɹ ɜɵɱɢɫɥɢɬɶ ɟɟ ɡɧɚɱɟɧɢɟ ɜ ɧɟɫɤɨɥɶɤɢɯ ɬɨɱɤɚɯ, ɚ ɩɨ ɧɢɦ ɩɨɫɬɪɨɢɬɶ, ɬ. ɟ. ɢɧɬɟɪɩɨɥɢɪɨɜɚɬɶ, ɛɨɥɟɟ ɩɪɨɫɬɭɸ ɮɭɧɤɰɢɸ. ɍɩɪɨɳɟɧɢɟ ɮɭɧɤɰɢɢ
ɜɧɨɫɢɬ ɩɨɝɪɟɲɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɣ, ɧɨ ɜ ɧɟɤɨɬɨɪɵɯ ɤɥɚɫɫɚɯ ɡɚɞɚɱ ɷɬɨ ɞɨɩɭɫɬɢɦɨ
ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɪɚɫɱɟɬɚ ɢ ɫɧɢɠɟɧɢɹ ɜɪɟɦɟɧɧɨɣ ɫɥɨɠɧɨɫɬɢ ɚɥɝɨɪɢɬɦɚ.
ɉɭɫɬɶ ɡɚɞɚɧ ɞɢɫɤɪɟɬɧɵɣ ɧɚɛɨɪ ɬɨɱɟɤ xi (i = 0,1…n), ɧɚɡɵɜɚɟɦɵɯ ɭɡɥɚɦɢ
ɢɧɬɟɪɩɨɥɹɰɢɢ, ɩɪɢɱɟɦ ɫɪɟɞɢ ɷɬɢɯ ɬɨɱɟɤ ɧɟɬ ɫɨɜɩɚɞɚɸɳɢɯ, ɚ ɬɚɤɠɟ ɡɧɚɱɟɧɢɹ
ɮɭɧɤɰɢɢ yi ɜ ɷɬɢɯ ɬɨɱɤɚɯ. Ɍɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɮɭɧɤɰɢɸ g(x), ɩɪɨɯɨɞɹɳɭɸ ɱɟɪɟɡ ɜɫɟ ɡɚɞɚɧɧɵɟ ɭɡɥɵ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɪɢɬɟɪɢɟɦ ɛɥɢɡɨɫɬɢ ɮɭɧɤɰɢɢ ɹɜɥɹɟɬɫɹ
g(xi) = yi. ȼ ɤɚɱɟɫɬɜɟ ɮɭɧɤɰɢɢ g(x) ɨɛɵɱɧɨ ɜɵɛɢɪɚɟɬɫɹ ɩɨɥɢɧɨɦ, ɤɨɬɨɪɵɣ ɧɚɡɵɜɚɸɬ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɦ ɩɨɥɢɧɨɦɨɦ.
ȼ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɩɨɥɢɧɨɦ ɟɞɢɧ ɞɥɹ ɜɫɟɣ ɨɛɥɚɫɬɢ ɢɧɬɟɪɩɨɥɹɰɢɢ, ɝɨɜɨɪɹɬ, ɱɬɨ ɢɧɬɟɪɩɨɥɹɰɢɹ ɝɥɨɛɚɥɶɧɚɹ.
ȼ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɦɟɠɞɭ ɪɚɡɥɢɱɧɵɦɢ ɭɡɥɚɦɢ ɩɨɥɢɧɨɦɵ ɪɚɡɥɢɱɧɵ, ɝɨɜɨɪɹɬ ɨ ɤɭɫɨɱɧɨɣ ɢɥɢ ɥɨɤɚɥɶɧɨɣ ɢɧɬɟɪɩɨɥɹɰɢɢ. ɇɚɣɞɹ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ, ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ f(x) ɦɟɠɞɭ ɭɡɥɚɦɢ (ɩɪɨɜɟɫɬɢ ɢɧɬɟɪɩɨɥɹɰɢɸ ɜ ɭɡɤɨɦ ɫɦɵɫɥɟ ɫɥɨɜɚ), ɚ ɬɚɤɠɟ ɨɩɪɟɞɟɥɢɬɶ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ f(x)
ɞɚɠɟ ɡɚ ɩɪɟɞɟɥɚɦɢ ɡɚɞɚɧɧɨɝɨ ɢɧɬɟɪɜɚɥɚ (ɩɪɨɜɟɫɬɢ ɷɤɫɬɪɚɩɨɥɹɰɢɸ).
ɉɭɫɬɶ ɢɦɟɟɬɫɹ n ɡɧɚɱɟɧɢɣ xi, ɤɚɠɞɨɦɭ ɢɡ ɤɨɬɨɪɵɯ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɜɨɟ ɡɧɚɱɟɧɢɟ yi. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɚɤɭɸ ɮɭɧɤɰɢɸ F, ɱɬɨ:
F(xi) = y
(i = 0,1…n),
i
ɝɞɟ x
– ɭɡɥɵ ɢɧɬɟɪɩɨɥɹɰɢɢ; ɩɚɪɵ (xi, ɭi) – ɬɨɱɤɢ ɞɚɧɧɵɯ; ɪɚɡɧɢɰɚ ɦɟɠɞɭ ɫɨɫɟɞ-
i
ɧɢɦɢ ɡɧɚɱɟɧɢɹɦɢ (xi – x
) – ɲɚɝ; ɮɭɧɤɰɢɹ F(x) – ɢɧɬɟɪɩɨɥɢɪɭɸɳɟɣ ɮɭɧɤɰɢɟɣ
i-1
ɢɥɢ ɢɧɬɟɪɩɨɥɹɧɬɨɦ.
Ɏɭɧɤɰɢɸ F ɨɩɪɟɞɟɥɹɸɬ ɜ ɨɛɳɟɦ ɜɢɞɟ ɦɧɨɝɨɱɥɟɧɚ ɫɬɟɩɟɧɢ n:
n
o
1n
−
1
−
.axa...xaxa)x(F
++++=
n1n
Ⱥɧɚɥɢɬɢɱɟɫɤɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɨɥɢɧɨɦɚ ai ɦɨɠɧɨ ɧɚɣɬɢ ɢɡ ɫɢɫɬɟɦɵ ɢɡ
(n+1) ɭɪɚɜɧɟɧɢɹ ɫ (n+1) ɧɟɢɡɜɟɫɬɧɵɦɢ [11]:
92
n
°
°
°
®
°
°
°
¯
0o
n
1o
.....
n
no
1n
−
01
−
1n
11
−
1n
n1
−
−
−
=++++
;yaxa...xaxa
0n01n
=++++
;yaxa...xaxa
1n11n
=++++
.yaxa...xaxa
nnn1n
ɇɨ ɞɥɹ ɚɥɝɨɪɢɬɦɢɡɚɰɢɢ ɞɚɧɧɵɣ ɦɟɬɨɞ ɫɥɢɲɤɨɦ ɬɪɭɞɨɟɦɤɢɣ, ɢɦɟɸɬɫɹ ɛɨɥɟɟ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɵɟ ɦɟɬɨɞɢɤɢ. [11]
6.2. Ɇɟɬɨɞɵ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ ɩɨɥɢɧɨɦɨɦ
ɉɨɥɢɧɨɦ ɇɶɸɬɨɧɚ
ȿɫɥɢ ɮɭɧɤɰɢɹ y = f(x) ɨɩɪɟɞɟɥɟɧɚ ɧɚ ɨɬɪɟɡɤɟ [a; b] ɢ ɡɚɞɚɧɚ ɫɜɨɢɦɢ ɡɧɚɱɟ-
ɧɢɹɦɢ y
ɜ ɪɚɜɧɨɨɬɫɬɨɹɳɢɯ ɭɡɥɚɯ x
i
ɧɚɯɨɠɞɟɧɢɹ ɢɧɬɟɪɩɨɥɢɪɭɸɳɟɣ ɮɭɧɤɰɢɢ ɩɪɢɦɟɧɹɸɬ ɩɨɥɢɧɨɦ ɇɶɸɬɨɧɚ.
Ɂɚɞɚɱɚ ɢɧɬɟɪɩɨɥɹɰɢɢ ɩɨɥɢɧɨɦɨɦ ɇɶɸɬɨɧɚ ɪɟɲɚɟɬɫɹ ɜ ɞɜɭɯ ɫɥɭɱɚɹɯ:
ɚ) ɤɨɝɞɚ ɬɨɱɤɚ x
א[a; b] ɧɚɯɨɞɢɬɫɹ ɜ ɧɚɱɚɥɟ ɬɚɛɥɢɰɵ ɡɧɚɱɟɧɢɣ, ɢɫɩɨɥɶɡɭɟɬ-
i
ɫɹ ɩɟɪɜɚɹ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɚɹ ɮɨɪɦɭɥɚ ɇɶɸɬɨɧɚ [13]
yqy)x(P
()( )
+
+−−
!n
xx
−
i
q
=
,yyy
,
h
+
Ɍɨɝɞɚ
≈= ,
n
1n
+
h
=Δ
y
)x(
+
)!1n(
ɛ) ɤɨɝɞɚ ɬɨɱɤɚ x
א[a; b] ɧɚɯɨɞɢɬɫɹ ɜ ɤɨɧɰɟ ɬɚɛɥɢɰɵ ɡɧɚɱɟɧɢɣ, ɢɫɩɨɥɶɡɭɟɬ-
i
ɫɹ ɜɬɨɪɚɹ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɚɹ ɮɨɪɦɭɥɚ ɇɶɸɬɨɧɚ [13]
yqy)x(P
+Δ+=
nnn
()( )
+
1nq...1qq
−++
!n
א[a; b], ɬ. ɟ. x
i
() ()( )
−
1qq
!2
,y
0
+
,yyy
2
y
1k
i
+Δ+=
ooin
1nq...1qq
n
Δ
−=Δ
i1ii
= xi + h, h = const>0, ɬɨ ɞɥɹ
i+1
−−
+Δ
0
k
2q1qq
!3
k
Δ−Δ=Δ
1i
+
)x(P)x(fy
n
)1n(
+
+
() ()( )
!2
n
Δ
0
∏
0k
=
1qq
2
y
1n
,y
,
−⋅
)kq()x(f
+Δ
[]
2q1qq
++
!3
3
+Δ
y
0
0
i
.
b;ax ∈
3
y
+Δ
2n
−−
yy =Δ .
ii
93
−
xx
n
=
q
1n
+
h
=Δ
y
)x(
+
)!1n(
.
h
n
)1n(
+
∏
0k
=
.
+⋅
)kq()x(f
ɉɨɥɢɧɨɦ Ʌɚɝɪɚɧɠɚ
ɉɨɥɢɧɨɦ Ʌɚɝɪɚɧɠɚ ɹɜɥɹɟɬɫɹ ɭɧɢɜɟɪɫɚɥɶɧɵɦ ɦɟɬɨɞɨɦ ɢɧɬɟɪɩɨɥɹɰɢɢ ɢ
ɩɪɢɦɟɧɹɟɬɫɹ, ɟɫɥɢ ɮɭɧɤɰɢɹ y = f(x) ɨɩɪɟɞɟɥɟɧɚ ɧɚ ɨɬɪɟɡɤɟ [a; b] ɢ ɡɚɞɚɧɚ ɫɜɨɢɦɢ ɡɧɚɱɟɧɢɹɦɢ yi ɜ ɧɟ ɪɚɜɧɨɨɬɫɬɨɹɳɢɯ ɭɡɥɚɯ,
§
n
¨
n
¨
⋅=
¦
¨
=
0i
¨
©
(y)x(L
∏
in
≠
ik
=
0k
≈=
y
n
−
∏
=
0k
=Δ
)x(
)xx(
k
+
)!1n(
+
−
xx
k
xx
−
ki
,
)x(L)x(f)x(y
n
)1n(
+
[13] .
)x(f
:
hxx
+≠
i1i
·
¸
,
¸
)
¸
¸
¹
6.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 6.1. Ɋɚɫɫɱɢɬɚɬɶ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɦɧɨɝɨɱɥɟɧ Ʌɚɝɪɚɧɠɚ ɞɥɹ
ɮɭɧɤɰɢɢ, ɡɚɞɚɧɧɨɣ ɞɢɫɤɪɟɬɧɨ ɜ ɬɚɛɥɢɱɧɨɣ ɮɨɪɦɟ (ɬɚɛɥɢɰɚ 11).
Ɍɚɛɥɢɰɚ 11
ɂɫɯɨɞɧɵɟ ɞɚɧɧɵɟ
x
i
ɭ
i
1 2 3 5
1 5 14 81
Ɋɟɲɟɧɢɟ. Ⱦɥɹ ɨɩɢɫɚɧɢɹ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɩɨɥɢɧɨɦ Ʌɚ-
ɝɪɚɧɠɚ 3-ɣ ɫɬɟɩɟɧɢ, ɬ. ɤ. ɡɚɞɚɧɨ ɜɫɟɝɨ 4 ɡɧɚɱɟɧɢɹ, ɩɟɪɜɨɟ ɢɡ ɤɨɬɨɪɵɯ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɯ
.
0
Ɋɟɲɟɧɢɟ ɢɳɟɦ ɜ ɜɢɞɟ
·
¸
−
xx
k
.
¸
)
−
xx
¸
ki
¸
¹
¦
§
n
¨
n
¨
⋅=
(y)x(L
∏
in
¨
0i
=
ik
≠
¨
0k
=
©
94
ɭ
n
14
)5x)(3x)(2x(
−−−
1)x(L
⋅=
)5x)(2x)(1x(
⋅+
23
−−−
)53)(23)(13(
−−−
.1x3x2x
−+−=
5
⋅+
)51)(31)(21(
−−−
81
⋅+
−−−
)5x)(3x)(1x(
−−−
+
)52)(32)(12(
−−−
)3x)(2x)(1x(
−−−
=
)145)(25)(15(
ȼ ɬɚɛɥɢɰɟ 12. ɩɪɢɜɟɞɟɧɵ ɬɨɱɧɵɟ ɢ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɧɵɟ ɞɚɧɧɵɟ ɞɥɹ ɫɪɚɜ-
ɧɟɧɢɹ.
ɋɪɚɜɧɟɧɢɟ ɞɚɧɧɵɯ
Ɍɚɛɥɢɰɚ 12
x
i
i
1 2 3 5
1 5 14 81
L(xi) 1 5 17 89
Ɋɢɫ. 31. ɋɪɚɜɧɟɧɢɟ ɬɨɱɧɨɣ (y) ɢ ɢɧɬɟɪɩɨɥɢɪɭɸɳɟɣ ɮɭɧɤɰɢɢ (L)
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɨɬɥɢɱɢɟ ɢɧɬɟɪɩɨɥɹɰɢɢ ɮɭɧɤɰɢɢ ɨɬ ɚɩɩɪɨɤɫɢɦɚɰɢɢ?
1.
ɑɬɨ ɬɚɤɨɟ ɷɤɫɬɪɚɩɨɥɹɰɢɹ ɮɭɧɤɰɢɢ?
2.
ɉɟɪɟɱɢɫɥɢɬɟ ɦɟɬɨɞɵ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ ɩɨɥɢɧɨɦɨɦ.
3.
ɇɚɡɨɜɢɬɟ ɨɛɥɚɫɬɢ ɩɪɢɦɟɧɟɧɢɹ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ ɩɨɥɢɧɨɦɚɦɢ
4.
ɇɶɸɬɨɧɚ ɢ Ʌɚɝɪɚɧɠɚ.
95
ȽɅȺȼȺ 7
ɑɂɋɅȿɇɇɈȿ Ɋȿɒȿɇɂȿ ɈȻɕɄɇɈȼȿɇɇɕɏ
ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ
ɉȿɊȼɈȽɈ ɉɈɊəȾɄȺ
Ɍɟɨɪɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ – ɪɚɡɞɟɥ ɦɚɬɟɦɚɬɢɤɢ, ɜ ɤɨɬɨɪɨɦ
ɢɡɭɱɚɸɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢ ɫɜɹɡɚɧɧɵɟ ɫ ɧɢɦɢ ɡɚɞɚɱɢ. ɇɚ ɩɪɚɤɬɢɤɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɸɬɫɹ ɞɥɹ ɨɩɢɫɚɧɢɹ ɞɢɧɚɦɢɤɢ ɮɢɡɢɱɟɫɤɢɯ ɫɢɫɬɟɦ.
ɍɩɪɨɳɟɧɧɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ – ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɜ ɤɨɬɨɪɨɦ ɧɟɢɡɜɟɫɬɧɨɣ ɜɟɥɢɱɢɧɨɣ ɹɜɥɹɟɬɫɹ ɧɟɤɨɬɨɪɚɹ ɮɭɧɤɰɢɹ. ɉɪɢ ɷɬɨɦ ɜ ɫɚɦɨɦ ɭɪɚɜɧɟɧɢɢ
ɮɢɝɭɪɢɪɭɸɬ ɩɪɨɢɡɜɨɞɧɵɟ ɪɚɡɧɨɝɨ ɩɨɪɹɞɤɚ
Ɋɚɡɥɢɱɚɸɬ ɨɛɵɤɧɨɜɟɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ (ɈȾɍ) ɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ (Ⱦɍɑɉ).
Ɉɛɵɤɧɨɜɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (ɈȾɍ) – ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɫɜɹɡɵɜɚɸɳɟɟ ɨɞɧɭ ɧɟɡɚɜɢɫɢɦɭɸ ɩɟɪɟɦɟɧɧɭɸ ɯ, ɢɫɤɨɦɭɸ ɮɭɧɤɰɢɸ ɭ(ɯ) ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɟ ɞɨ n-ɝɨ ɩɨɪɹɞɤɚ [11, 13, 27]:
ɇɚɢɜɵɫɲɢɣ ɩɨɪɹɞɨɤ ɩɪɨɢɡɜɨɞɧɨɣ, ɜɯɨɞɹɳɟɣ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɧɚɡɵɜɚɟɬɫɹ ɩɨɪɹɞɤɨɦ ɭɪɚɜɧɟɧɢɹ.
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ – ɷɬɨ ɭɪɚɜɧɟɧɢɟ,
ɫɨɞɟɪɠɚɳɟɟ ɧɟɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ ɨɬ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɢ ɢɯ ɱɚɫɬɧɵɟ
ɩɪɨɢɡɜɨɞɧɵɟ.
Ɋɟɲɟɧɢɹɦɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (7.1) ɹɜɥɹɸɬɫɹ ɤɪɢɜɵɟ ɭ(ɯ) ɧɚ
ɩɥɨɫɤɨɫɬɢ ɭ, x, ɧɚɡɵɜɚɟɦɵɟ ɢɧɬɟɝɪɚɥɶɧɵɦɢ ɤɪɢɜɵɦɢ.
Ⱦɥɹ ɪɟɲɟɧɢɹ ɈȾɍ ɚɧɚɥɢɬɢɱɟɫɤɢɦ ɫɩɨɫɨɛɨɦ ɧɟɨɛɯɨɞɢɦɨ ɩɪɢɦɟɧɹɬɶ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ
ɧɭɸ, ɩɪɨɯɨɞɹɳɭɸ ɱɟɪɟɡ ɡɚɞɚɧɧɭɸ ɬɨɱɤɭ, ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
ɜ ɜɢɞɟ ɡɧɚɱɟɧɢɣ ɩɪɨɢɡɜɨɞɧɵɯ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ x
ɜɤɥɸɱɢɬɟɥɶɧɨ:
. ɑɬɨɛɵ ɜɵɞɟɥɢɬɶ ɢɡ ɦɧɨɠɟɫɬɜɚ ɢɧɬɟɝɪɚɥɶɧɵɯ ɤɪɢɜɵɯ ɟɞɢɧɫɬɜɟɧ-
′
°
′
ȼ ɦɚɬɟɦɚɬɢɤɟ ɧɚɯɨɠɞɟɧɢɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (1), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ
ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ (2), ɧɚɡɵɜɚɟɬɫɹ ɡɚɞɚɱɟɣ Ʉɨɲɢ ɞɥɹ ɈȾɍ [11, 13, 27]:
°
®
....
°
−
°
¯
ɨɬ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ.
,ɭ)ɯ(ɭ
=
00
=
,ɭ)ɯ(ɭ
10
)1n(
=
)n(
−
1n0
(1)
.0))x(ɭ),...,ɯ(ɭ),ɯ(ɭ,x(F
=
ɞɨ ɩɨɪɹɞɤɚ (n-1)
0
(2)
.ɭ)ɯ(ɭ
96
′
=
®
¯
),y,x(fy
(3)
=
.y)x(y
00
Ʌɢɲɶ ɞɥɹ ɨɝɪɚɧɢɱɟɧɧɵɯ ɤɥɚɫɫɨɜ ɈȾɍ ɪɚɡɪɚɛɨɬɚɧɵ ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ. Ʉ ɧɢɦ ɨɬɧɨɫɹɬɫɹ:
1. ɍɪɚɜɧɟɧɢɹ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ, ɤɨɬɨɪɵɟ ɪɟɲɚɸɬɫɹ ɜ ɢɧ-
ɬɟɝɪɚɥɚɯ.
2. Ʌɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ.
3. ɇɟɤɨɬɨɪɵɟ ɫɩɟɰɢɚɥɶɧɵɟ ɜɢɞɵ ɭɪɚɜɧɟɧɢɣ [27].
7.1. Ɇɟɬɨɞ ɗɣɥɟɪɚ
Ɂɚɦɟɧɹɹ ɜ (3) ɩɪɨɢɡɜɨɞɧɭɸ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɤɚɠɞɨɝɨ i-ɝo ɭɡɥɚ ɫɟɬɤɢ ɪɚɡɧɨɫɬɧɵɦ ɨɬɧɨɲɟɧɢɟɦ, ɩɪɢɯɨɞɢɦ ɤ ɦɟɬɨɞɭ ɗɣɥɟɪɚ [11, 13, 27]:
°
®
°
¯
i1i
+
=
h
).x(fy
=
00
),y,x(f
ii
yy
−
ɢɥɢ
+
®
=
¯
⋅+=
).x(fy
00
),y,x(fhyy
iii1i
Ɋɢɫ. 32. Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɦɟɬɨɞɚ ɗɣɥɟɪɚ
97
Ɇɟɬɨɞ ɗɣɥɟɪɚ ɢɦɟɟɬ ɩɨɝɪɟɲɧɨɫɬɶ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɭɸ ɲɚɝɭ h ɜ ɩɟɪɜɨɣ
ɫɬɟɩɟɧɢ, ɢ ɹɜɥɹɟɬɫɹ ɦɟɬɨɞɨɦ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɬɨɱɧɨɫɬɢ.
Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɦɟɬɨɞɚ ɗɣɥɟɪɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɟɫɥɢ ɢɡɜɟɫɬɧɨ
ɡɧɚɱɟɧɢɟ xi, yi, ɬɨ ɦɨɠɧɨ ɧɚɣɬɢ ɩɪɨɢɡɜɨɞɧɭɸ y′ ɨɬ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ x
(ɬɚɧɝɟɧɫ ɭɝɥɚ ɧɚɤɥɨɧɚ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɢɧɬɟɝɪɚɥɶɧɨɣ ɤɪɢɜɨɣ). ɉɪɨɞɨɥɠɚɹ ɤɚɫɚɬɟɥɶɧɭɸ ɞɨ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɥɢɧɢɟɣ x
= xi + h, ɩɨɥɭɱɢɦ y
i+1
i+1
.
7.2. Ɇɟɬɨɞ ɗɣɥɟɪɚ-Ʉɨɲɢ
(Ɇɟɬɨɞ Ɋɭɧɝɟ-Ʉɭɬɬɚ ɜɬɨɪɨrɨ ɩɨɪɹɞɤɚ)
Ɇɟɬɨɞ ɗɣɥɟɪɚ-Ʉɨɲɢ ɹɜɥɹɟɬɫɹ ɦɟɬɨɞɨɦ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ (ɛɨɥɟɟ ɬɨɱɧɵɦ
ɦɟɬɨɞɨɦ ɪɟɲɟɧɢɹ ɈȾɍ). Ɋɚɫɱɟɬ ɜɟɞɭɬ ɩɨ ɫɥɟɞɭɸɳɢɦ ɡɚɜɢɫɢɦɨɫɬɹɦ:
Δ+=
+
°
°
°
®
°
°
°
¯
hyy
+=
i1i
+
h
=Δ
y
2
h
y
2
ɇɚ ɜɫɟɦ ɢɧɬɟɪɜɚɥɟ ɪɚɫɱɟɬɚ ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ ɲɚɝɚ.
Ƚɟɨɦɟɬɪɢɱɟɫɤɢ ɦɟɬɨɞ ɨɡɧɚɱɚɟɬ (ɪɢɫ. 33), ɱɬɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɚɩɪɚɜɥɟɧɢɟ
ɢɧɬɟɝɪɚɥɶɧɨɣ ɤɪɢɜɨɣ ɜ ɢɫɯɨɞɧɨɣ ɬɨɱɤɟ (ɯ
), ɚ ɜ ɤɚɱɟɫɬɜɟ ɨɤɨɧɱɚɬɟɥɶɧɨɝɨ ɜɵɛɢɪɚɟɬɫɹ ɫɪɟɞɧɟɟ ɢɡ ɷɬɢɯ ɧɚɩɪɚɜɥɟɧɢɣ.
ɭ
i+1
,yyy
ii1i
Δ+Δ=Δ
,yyy
2i1ii
),y,x(f
ii1i
⋅++=Δ
)).y,x(fhy,hx(f
iiii2i
+
⋅++
))y,x(fhy,x(f)yx(f
iii1iii
. [11, 13, 27]
2
, yi) ɢ ɜɨ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɬɨɱɤɟ (x
i
i+1
i
,
Ɋɢɫ. 33. Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɦɟɬɨɞɚ ɗɣɥɟɪɚ-Ʉɨɲɢ
98
7.3. Ɇɟɬɨɞ Ɋɭɧɝɟ-Ʉɭɬɬɚ ɱɟɬɜɟɪɬɨɝɨ ɩɨɪɹɞɤɚ
ȼ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɩɪɚɤɬɢɤɟ ɧɚɢɛɨɥɟɟ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɦɟɬɨɞ ɊɭɧɝɟɄɭɬɬɚ ɱɟɬɜɟɪɬɨɝɨ ɩɨɪɹɞɤɚ.
ȼɟɥɢɱɢɧɵ ɭ
ɜɵɱɢɫɥɹɸɬɫɹ ɩɨ ɫɥɟɞɭɸɳɢɦ ɮɨɪɦɭɥɚɦ:
i+1
,yyy
+
°
°
y
°
°
1
i
°
°
®
2
i
°
°
3
°
i
°
°
4
°
¯
i
Δ+=
ii1i
1
6
⋅=
2
1
i
ii
),y,x(fhk
ii
h
x(fhk
y,
2
h
++⋅=
x(fhk
y,
2
++⋅=
4
3
),kk2k2k(
+++=Δ
i
i
1
1
++⋅=
),k
iii
2
1
2
),k
iii
2
3
).ky,hx(fhk
iii
[11, 13, 27].
ȼ ɞɚɧɧɨɦ ɦɟɬɨɞɟ ɭɫɪɟɞɧɟɧɢɟ ɩɪɨɜɨɞɢɬɫɹ ɩɨ ɬɪɺɦ ɬɨɱɤɚɦ, ɮɨɪɦɭɥɚ ɗɣɥɟɪɚ ɧɚ ɤɚɠɞɨɦ ɨɬɪɟɡɤɟ ɢɫɩɨɥɶɡɭɟɬɫɹ 4 ɪɚɡɚ: ɜ ɧɚɱɚɥɟ ɨɬɪɟɡɤɚ, ɞɜɚɠɞɵ ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ ɢ ɜ ɤɨɧɰɟ ɨɬɪɟɡɤɚ (ɪɢɫ. 34).
Ɋɢɫ. 34. Ɋɚɡɛɢɟɧɢɟ ɲɚɝɚ h ɜ ɦɟɬɨɞɟ Ɋɭɧɝɟ-Ʉɭɬɬɚ
4
1
Ɇɟɬɨɞ Ɋɭɧɝɟ-Ʉɭɬɬɚ ɹɜɥɹɟɬɫɹ ɱɟɬɵɪɟɯɷɬɚɩɧɵɦ, ɜ ɧɟɦ
k...k ɹɜɥɹɸɬɫɹ
i
i
ɱɚɫɬɧɵɦɢ ɩɪɢɪɚɳɟɧɢɹɦɢ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɥɟɜɨɦ ɭɡɥɟ x
ɲɚɝɚ, ɜ ɭɡɥɟ x
+ h/2 ɢ ɜ ɩɪɚɜɨɦ ɭɡɥɟ xi. ɉɨ ɫɪɚɜɧɟɧɢɸ ɫ ɨɩɢɫɚɧɧɵɦɢ ɜɵɲɟ ɦɟ-
i
ɬɨɞɚɦɢ ɨɧ ɢɦɟɟɬ ɛɨɥɟɟ ɜɵɫɨɤɭɸ ɬɨɱɧɨɫɬɶ, ɧɨ ɧɟɜɵɫɨɤɭɸ ɫɤɨɪɨɫɬɶ ɩɨɢɫɤɚ ɪɟɲɟɧɢɹ.
99
i
7.4. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 7.1.
′
xyy −=
, ɫ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɯ0 = 0; ɭ0 = 1,5; ɧɚ ɨɬɪɟɡɤɟ [0; 1,5 ] ɫ ɲɚ-
Ɋɟɲɢɬɶ ɦɟɬɨɞɨɦ ɗɣɥɟɪɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
ɝɨɦ h = 0,25.
Ɋɟɲɟɧɢɟ. ɉɨ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ ɡɚɩɨɥɧɹɟɦ 0-ɸ ɫɬɪɨɤɭ ɞɚɧɧɵɯ (ɬɚɛɥɢ-
ɰɚ 13)
Ɍɚɛɥɢɰɚ ɞɚɧɧɵɯ
Ɍɚɛɥɢɰɚ 13
i xi yi
′
xyy −=
iii
yhy
⋅=Δ
′
iii
0 0 1,5 1,5 0,375
1 0,25 1,875 1,625 0,406
2 0,5 2,281 1,781 0,445
3 0,75 2,726 1,976 0,494
4 1,0 322 2,221 0,555
5 1,25 3,775 2,525 0,631
6 1,5 4,407
Ⱦɥɹ 1-ɣ ɫɬɪɨɤɢ ɩɪɢɦɟɧɹɟɦ ɮɨɪɦɭɥɭ ɗɣɥɟɪɚ )y,x(fhyy
ɇɚɩɪɢɦɟɪ.
0001
+
=⋅+=⋅+=
⋅+=
.875,15,125,05,1)y,x(fhyy
.
iii1i
′
111
=−=−=
.
625,125,0875,1xyy
′
⋅=Δ
11
=⋅=
406,0625,125,0yhy
ɢ ɬ. ɞ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɤɚɱɟɫɬɜɟ ɨɬɜɟɬɚ ɩɨɥɭɱɚɟɦ ɮɭɧɤɰɢɸ y(x), ɡɚɞɚɧɧɭɸ ɜ
ɬɚɛɥɢɱɧɨɣ ɮɨɪɦɟ. Ⱦɥɹ ɩɪɢɜɟɞɟɧɢɹ ɮɭɧɤɰɢɢ ɤ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɟɟ
ɧɭɠɧɨ ɢɧɬɟɪɩɨɥɢɪɨɜɚɬɶ.
Ɂɚɞɚɱɚ 7.2. Ɋɟɲɢɬɶ ɦɟɬɨɞɨɦ Ɋɭɧɝɟ-Ʉɭɬɬɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
′
xyy −=
, ɫ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɯ0 = 0; ɭ0 = 1,5; ɧɚ ɨɬɪɟɡɤɟ [0; 1,5 ] ɫ ɲɚ-
ɝɨɦ h = 0,25.
Ɋɟɲɟɧɢɟ.
,yyy
Δ+=
+
1
y
1
0
1
ii
6
00
ii1i
4
3
2
i
i
),kk2k2k(
+++=Δ
i
=−⋅=⋅=
;375,0)05,1(25,0)y,x(fhk
100
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