Ɂɚɞɚɱɚ 38 Ⱦɨɜɟɫɬɢ. ɳɨ ɦɟɬɨɞ, ɹɤɨɦɭ ɜɿɞɩɨɜɿɞɚɽ ɰɹ ɬɚɛɥɢɰɹ Ȼɚɬɱɟɪɚ, ɦɚɽ ɬɪɟɬɿɣ ɫɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ.
4. Ɇɟɬɨɞɢ ɱɟɬɜɟɪɬɨɝɨ |
ɩɨɪɹɞɤɭ |
m |
4, |
R W |
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O W5 , p 4. ɇɚɣɛɿɥɶɲ |
ɩɨɲɢɪɟɧɿ ɦɟɬɨɞɢ: |
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Ɋɨɡɝɥɹɧɟɦɨ ɹɤ ɩɨɜ’ɹɡɚɧɢɣ ɩɨɪɹɞɨɤ ɦɟɬɨɞɭ p ɿ ɤɿɥɶɤɿɫɬɶ ɫɬɚɞɿɣ m :
ɤɿɥɶɤɿɫɬɶ ɫɬ |
ɞɿɣ M |
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ɬɨɱɧɿɫɬɶ ɦɟɬɨɞɭ P |
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əɫɧɨ |
ɱɨɦɭ |
ɦɟɬɨɞɢ |
Ɋɭɧɝɟ-Ʉɭɬɬɚ |
ɩɪɢ m |
5 |
ɪɿɞɤɨ ɡɚɫɬɨɫɨɜɭɸɬɶɫɹ: |
ɡɛɿɥɶɲɟɧɧɹ ɤɿɥɶɤɨɫɬɿ ɫɬɚɞɿɣ (ɚ ɡɧɚɱɢɬɶ ɡɛɿɥɶɲɟɧɧɹ ɨɛ‘ɽɦɭ ɨɛɱɢɫɥɸɜɚɥɶɧɨʀ ɪɨɛɨɬɢ) ɧɟ ɞɚɽ ɜɢɝɪɚɲɭ ɜ ɬɨɱɧɨɫɬɿ.
Ɍɟɨɪɟɦɚ ɇɟɯɚɣ m- ɫɬɚɞɿɣɧɢɣ ɦɟɬɨɞ Ɋɭɧɝɟ-Ʉɭɬɬɚ ɦɚɽ p -ɣ ɫɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ ɧɚ ɤɪɨɰɿ, ɚ f t,u ɡɚɞɨɜɨɥɶɧɹɽ ɭɦɨɜɭ Ʌɿɩɲɢɰɹ. Ɍɨɞɿ ɦɟɬɨɞ ɦɚɽ p -ɣ
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ɫɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ ɿ ɞɥɹ ɩɨɯɢɛɤɢ ɦɟɬɨɞɿɜ Ɋɭɧɝɟ-Ʉɭɬɬɚ ɦɚɽ ɦɿɫɰɟ ɬɚɤɚ ɨɰɿɧɤɚ: |
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1W Rj (W) - |
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zn |
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yn |
u tn |
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d T eD0T maxj |
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\ j |
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O Wp , |
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ɞɟ \ j |
ɩɨɯɢɛɤɚ ɚɩɪɨɤɫɢɦɚɰɿʀ ɦɟɬɨɞɚ, |
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D0 pLm 1 BLW0 |
m 1, L – |
ɫɬɚɥɚ Ʌɿɩɲɢɰɹ, B |
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max |
fij |
,T max |
tn |
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i, j |
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10.4. Ɇɟɬɨɞɢ ɡ ɤɨɧɬɪɨɥɟɦ ɬɨɱɧɨɫɬɿ ɧɚ ɤɪɨɰɿ [ȻɀɄ, 365 - 367].
ɑɚɫɬɨ ɜ ɯɨɞɿ ɪɨɡɪɚɯɭɧɤɿɜ ɧɟɨɛɯɿɞɧɨ ɡɦɿɧɸɜɚɬɢ ɤɪɨɤ ɿɧɬɟɝɪɭɜɚɧɧɹ, ɤɨɧɬɪɨɥɸɸɱɢ ɜɟɥɢɱɢɧɭ ɩɨɯɢɛɤɢ ɦɟɬɨɞɭ ɧɚ ɤɪɨɰɿ. ɉɪɢ ɩɪɚɤɬɢɱɧɿɣ ɨɰɿɧɰɿ ɰɿɽʀ ɜɟɥɢɱɢɧɢ ɦɨɠɧɚ. ɧɚɩɪɢɤɥɚɞ, ɩɨɫɬɭɩɚɬɢ ɬɚɤ.
ɉɟɪɲɢɣ ɩɿɞɯɿɞ ɜɢɤɨɪɢɫɬɨɜɭɽ ɩɪɢɧɰɢɩ Ɋɭɧɝɟ. Ƚɨɥɨɜɧɢɣ ɱɥɟɧ ɩɨɯɢɛɤɢ ɧɚ
ɨɞɧɨɦɭ ɤɪɨɰɿ (tn |
,tn 1) ɿɧɬɟɝɪɭɜɚɧɧɹ ɦɚɽ ɜɢɝɥɹɞ |
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M p 1 0 W p 1 |
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p 1! . |
ȼ ɪɟɡɭɥɶɬɚɬɿ ɞɜɨɯ ɤɪɨɤɿɜ ɿɧɬɟɝɪɭɜɚɧɧɹ ɨɞɧɨɤɪɨɤɨɜɢɦ ɦɟɬɨɞɨɦ, ɧɚɩɪɢɤɥɚɞ,
ɦɟɬɨɞɨɦ Ɋɭɧɝɟ – Ʉɭɬɬɚ p -ɝɨ ɫɬɟɩɟɧɹ ɬɨɱɧɨɫɬɿ, ɛɭɞɟ ɨɬɪɢɦɚɧɨ ɧɚɛɥɢɠɟɧɧɹ |
yW tn |
2W ɞɨ ɡɧɚɱɟɧɧɹ u(tn |
2 ) ɬɚɤɟ, ɳɨ |
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W |
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M p 1 0 Wp 1 |
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y |
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tn 2W |
u tn 2W |
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p 1! |
2 R W . |
əɤɳɨ ɬɟɩɟɪ ɡɚɫɬɨɫɭɜɚɬɢ ɦɟɬɨɞ Ɋɭɧɝɟ – Ʉɭɬɬɚ p -ɝɨ ɫɬɟɩɟɧɹ ɡ ɨɞɧɢɦ
ɤɪɨɤɨɦ ɞɨɜɠɢɧɢ 2 ɧɚ ɿɧɬɟɪɜɚɥɿ (tn ,tn 2 ), ɬɨ ɨɬɪɢɦɚɽɦɨ ɧɚɛɥɢɠɟɧɟ ɡɧɚɱɟɧɧɹ |
y2W tn |
2W , ɞɥɹ ɹɤɨɝɨ |
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M p 1 0 (2W)p 1 |
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2W |
tn 2W u tn |
2W |
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p 1 |
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y |
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p 1! |
R 2W 2 |
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R W . |
Ɂ ɰɢɯ ɫɩɿɜɜɿɞɧɨɲɟɧɶ ɜɢɩɥɢɜɚɽ ɩɪɟɞɫɬɚɜɥɟɧɧɹ ɝɨɥɨɜɧɨɝɨ ɱɥɟɧɚ ɩɨɯɢɛɤɢ ɧɚ
ɤɪɨɰɿ |
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2W |
W |
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$ |
y W tn 2W u tn 2W |
2 R W |
yn 1 |
yn 1 |
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p |
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ɉɪɢ ɧɟɨɛɯɿɞɧɨɫɬɿ ɦɨɠɧɚ ɭɬɨɱɧɢɬɢ ɨɬɪɢɦɚɧɟ ɧɚɛɥɢɠɟɧɟ ɡɧɚɱɟɧɧɹ, ɞɨɞɚɜɲɢ ɞɨ ɧɶɨɝɨ ɜɟɥɢɱɢɧɭ ɝɨɥɨɜɧɨɝɨ ɱɥɟɧɚ ɩɨɯɢɛɤɢ, ɬɨɛɬɨ ɩɨɤɥɚɫɬɢ
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u tn 2W |
yW |
yW |
y2W |
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g W |
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p |
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ɉɨɡɧɚɱɢɦɨ |
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yn2W1 |
ynW 1 |
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əɤɳɨ |
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g W |
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d H , ɞɟ |
– ɞɟɹɤɚ |
ɡɚɞɚɧɚ ɦɚɥɚ |
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p |
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u tn 2W . |
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– ɭɫɩɿɲɧɢɣ ɤɪɨɤ ɿ ynW 1 |
ɜɟɥɢɱɢɧɚ (ɩɨɯɢɛɤɚ ɧɚ ɨɞɧɨɦɭ ɤɪɨɰɿ), ɬɨ |
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əɤɳɨ |
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g |
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ɬɨ ɡɦɟɧɲɭɽɦɨ ɤɪɨɤ |
W: |
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. Ɍɪɟɛɚ ɳɟ ɩɟɪɟɞɛɚɱɢɬɢ ɭɦɨɜɭ |
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, ɬɨ ynW 1 |
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u tn 2W ɿ |
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ɡɛɿɥɶɲɟɧɧɹ ɤɪɨɤɭ. Ɂɚɞɚɽɦɨ ɞɟɹɤɟ |
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ɿ ɹɤɳɨ |
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ɞɚɥɿ ɛɟɪɟɦɨ |
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G |
D H 2 p , |
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2 . ɉɚɪɚɦɟɬɪ |
ɜɢɛɢɪɚɸɬɶ, |
ɧɚɩɪɢɤɥɚɞ, ɬɚɤ: |
01, ɞɟ p – ɩɨɪɹɞɨɤ ɬɨɱɧɨɫɬɿ ɦɟɬɨɞɭ.
ȱɧɲɢɣ ɩɿɞɯɿɞ ɜɢɛɨɪɭ ɤɪɨɤɭ ɿɧɬɟɝɪɭɜɚɧɧɹ ɡɚɤɥɸɱɚɽɬɶɫɹ ɜ ɜɢɤɨɪɢɫɬɚɧɧɿ
ɦɟɬɨɞɿɜ ɪɿɡɧɨɝɨ ɫɬɟɩɟɧɹ ɬɨɱɧɨɫɬɿ. Ɉɬɠɟ, ɹɤɳɨ ɜ ɧɚɫ ɽ ɞɜɚ ɦɟɬɨɞɢ ɫɬɟɩɟɧɹ |
ɬɨɱɧɨɫɬɿ ɧɚ ɤɪɨɰɿ p ɬɚ p 1: |
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O Wp 1 |
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yn1 1 |
u tn |
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yn21 |
u tn |
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O Wp 2 |
, |
ɬɨ ɝɨɥɨɜɧɢɣ ɱɥɟɧ ɩɨɯɢɛɤɢ ɩɟɪɲɨɝɨ ɦɟɬɨɞɭ |
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g W |
yn1 1 |
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yn21 O W p 1 . |
Ⱦɚɥɿ ɡ ɝɨɥɨɜɧɢɦ ɱɥɟɧɨɦ ɩɨɯɢɛɤɢ g |
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ɦɟɬɨɞɭ ɫɬɟɩɟɧɿ p ɨɩɟɪɭɽɦɨ ɬɚɤ ɹɤ ɿ ɜ |
ɩɟɪɲɨɦɭ ɩɿɞɯɨɞɿ.
Ȼɚɠɚɧɨ ɦɚɬɢ ɦɨɠɥɢɜɿɫɬɶ ɡɞɿɣɫɧɸɜɚɬɢ ɤɪɨɤ ɿɧɬɟɝɪɭɜɚɧɧɹ ɿ ɨɰɿɧɸɜɚɬɢ ɩɨɯɢɛɤɭ ɩɪɢ ɦɟɧɲɿɣ ɤɿɥɶɤɨɫɬɿ ɨɛɱɢɫɥɟɧɧɹ ɡɧɚɱɟɧɶ ɩɪɚɜɢɯ ɱɚɫɬɢɧ. ȼɢɝɪɚɲ ɞɨɫɹɝɚɽɬɶɫɹ, ɹɤɳɨ ɜɢɤɨɪɢɫɬɨɜɭɸɬɶ ɦɟɬɨɞɢ, ɹɤɿ ɧɚɡɢɜɚɸɬɶɫɹ ɜɤɥɚɞɟɧɢɦɢ. Ɍɚɛɥɢɰɹ Ȼɚɬɱɟɪɚ ɞɥɹ ɧɢɯ ɦɚɽ ɜɢɝɥɹɞ:
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Dm |
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Emm 1 |
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p1 |
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pm 1 |
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p |
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Ɇɟɬɨɞ ɡ ɩɚɪɚɦɟɬɪɚɦɢ |
p1,..., pm 1, pm ɦɚɽ ɩɨɪɹɞɨɤ ɬɨɱɧɨɫɬɿ p , |
ɚ ɡ ɩɚɪɚɦɟɬɪɚɦɢ |
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m - p 1. Ʉɨɟɮɿɰɿɽɧɬɢ |
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ɭ ɨɛɨɯ ɦɟɬɨɞɿɜ ɨɞɧɚɤɨɜɿ. |
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ɇɚɣɩɪɨɫɬɿɲɢɣ ɩɪɢɤɥɚɞ ɜɤɥɚɞɟɧɢɯ ɦɟɬɨɞɿɜ ɞɥɹ m |
2 |
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ɦɚɽ |
ɬɚɛɥɢɰɸ |
Ȼɚɬɱɟɪɚ: |
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ɉɟɪɲɢɣ ɦɟɬɨɞ, ɹɤɨɦɭ ɜɿɞɩɨɜɿɞɚɸɬɶ ɤɨɟɮɿɰɿɽɧɬɢ |
p1 |
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ɰɟ |
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ȿɣɥɟɪɚ, |
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1. Ⱦɪɭɝɢɣ - p1 |
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1 - ɦɟɬɨɞ ȿɣɥɟɪɚ – Ʉɨɲɿ, |
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ȱɧɲɢɦ ɩɪɢɤɥɚɞɨɦ ɦɨɠɟ ɫɥɭɠɢɬɢ ɫɭɤɭɩɧɿɫɬɶ ɮɨɪɦɭɥ ɲɟɫɬɿɫɬɚɞɿɣɧɨɝɨ |
m 6 ɦɟɬɨɞɭ Ɋɭɧɝɟ – Ʉɭɬɬɚ – Ɏɟɥɶɛɟɪɝɚ: |
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k1 |
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Wf tn , yn , k2 |
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¸ , |
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2W |
© |
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Wf tn W, yn |
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k2 2k3 , k5 |
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Wf ©¨tn |
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10k2 k4 ¹¸ , |
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k6 |
Wf ©¨tn |
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28k1 |
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125k2 546k3 54k4 378k5 ¹¸ , |
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5 |
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'yn |
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k1 4k3 k4 , |
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ɡ ɝɨɥɨɜɧɢɦ ɱɥɟɧɨɦ ɩɨɯɢɛɤɢ |
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g(W) O W5 , |
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yn tn |
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g(W) |
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42k1 |
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366 |
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Ɇɟɬɨɞɢ Ɋɭɧɝɟ – Ʉɭɬɬɚ – Ɏɟɥɶɛɟɪɝɚ ɦɚɸɬɶ ɱɟɬɜɟɪɬɢɣ ɬɚ ɩ’ɹɬɢɣ ɫɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ. ɉɨɪɿɜɧɹɽɦɨ ɤɿɥɶɤɿɫɬɶ ɨɛɱɢɫɥɟɧɶ ɩɪɚɜɢɯ ɱɚɫɬɢɧ ɜ ɦɟɬɨɞɚɯ Ɋɭɧɝɟ- Ʉɭɬɬɚ ɬɚ Ɋɭɧɝɟ – Ʉɭɬɬɚ – Ɏɟɥɶɛɟɪɝɚ ( p 4 ɞɥɹ ɨɛɨɯ). Ɂɝɿɞɧɨ ɫɯɟɦɢ ɤɪɨɤɿɜ ɩɨ
ɡɦɿɧɧɿɣ t ɬɚ ɩɨ ɫɬɚɞɿɹɯ i 1,4 ɞɥɹ ɦɟɬɨɞɭ Ɋɭɧɝɟ – Ʉɭɬɬɚ ɧɟɨɛɯɿɞɧɨ ɨɛɱɢɫɥɢɬɢ
ɞɥɹ ɨɰɿɧɤɢ ɩɨɯɢɛɤɢ 11 ɡɧɚɱɟɧɶ ɮɭɧɤɰɿʀ, ɚ ɞɥɹ ɦɟɬɨɞɭ Ɋɭɧɝɟ – Ʉɭɬɬɚ – Ɏɟɥɶɛɟɪɝɚ 6ɡɧɚɱɟɧɶ ɮɭɧɤɰɿʀ.
Ɉɫɧɨɜɧɢɣ ɧɟɞɨɥɿɤ ɦɟɬɨɞɿɜ Ɋɭɧɝɟ-Ʉɭɬɬɚ: ɳɨɛ ɨɬɪɢɦɚɬɢ ɞɨɫɢɬɶ ɜɢɫɨɤɢɣ ɫɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ ɩɨɬɪɿɛɧɨ ɛɚɝɚɬɨ ɪɚɡ ɨɛɱɢɫɥɸɜɚɬɢ ɡɧɚɱɟɧɧɹ ɮɭɧɤɰɿʀ.
Ⱦɨɫɬɨɣɧɿɫɬɸ ɦɟɬɨɞɿɜ Ɋɭɧɝɟ – Ʉɭɬɬɚ ɽ ɦɨɠɥɢɜɿɫɬɶ ɡɦɿɧɢ ɤɪɨɤɭ ɿɧɬɟɝɪɭɜɚɧɧɹ n ɿ ɡɚ ɪɚɯɭɧɨɤ ɰɶɨɝɨ ɡɚɞɨɜɨɥɶɧɹɬɢ ɭɦɨɜɭ ɬɨɱɧɨɫɬɿ ɧɚ ɤɪɨɰɿ.
10.5. Ȼɚɝɚɬɨɤɪɨɤɨɜɿ ɦɟɬɨɞɢ ɪɨɡɜ’ɹɡɚɧɧɹ ɡɚɞɚɱɿ Ʉɨɲɿ. Ɇɟɬɨɞɢ ɞɚɦɫɚ [ɋȽ,
230-231], [ȻɀɄ, 372-375]
ɇɟɞɨɥɿɤ ɦɟɬɨɞɿɜ Ɋɭɧɝɟ-Ʉɭɬɬɚ: ɜɟɥɢɤɚ ɤɿɥɶɤɿɫɬɶ ɨɛɱɢɫɥɟɧɶ ɡɧɚɱɟɧɶ ɮɭɧɤɰɿɣ ɧɚ ɨɞɧɨɦɭ ɤɪɨɰɿ (ɨɫɨɛɥɢɜɨ, ɱɭɬɥɢɜɨ ɰɟ ɞɥɹ ɫɢɫɬɟɦ). ȼɢɫɨɤɚ ɬɨɱɧɿɫɬɶ
ɜ ɰɢɯ ɦɟɬɨɞɚɯ ɞɨɫɹɝɚɽɬɶɫɹ ɡɚ ɪɚɯɭɧɨɤ ɨɛɱɢɫɥɟɧɶ ɞɥɹ m- ɫɬɚɞɿɣ ɤɨɟɮɿɰɿɽɧɬɿɜ |
ki W |
Wf [i ,Ki ɜ ɩɪɨɦɿɠɧɢɯ ɬɨɱɤɚɯ ɦɿɠ tn |
ɬɚ tn |
. |
ɱɢ ɧɟ ɦɨɠɧɚ ɞɥɹ |
ɰɶɨɝɨ ɜɢɤɨɪɢɫɬɚɬɢ ɩɨɩɟɪɟɞɧɿ ɡɧɚɱɟɧɧɹ f |
tn , yn |
, f |
tn 1, yn 1 ,..., f tn |
m , yn m ? |
Ⱦɥɹ ɪɨɡɜ’ɹɡɚɧɧɹ ɡɚɞɚɱɿ Ʉɨɲɿ |
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f t,u , t |
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du |
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ɜɜɟɞɟɦɨ ɫɿɬɤɭ |
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n , n 0,1,...` |
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f tn , yn ɮɭɧɤɰɿʀ, |
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ɡ ɩɨɫɬɿɣɧɢɦ ɤɪɨɤɨɦ |
0 . ɉɨɡɧɚɱɢɦɨ ɱɟɪɟɡ yn |
y tn , |
fn |
ɜɢɡɧɚɱɟɧɿ ɧɚ ɫɿɬɰɿ |
W. |
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Ʌɿɧɿɣɧɢɦ m - ɤɪɨɤɨɜɢɦ ɪɿɡɧɢɰɟɜɢɦ ɦɟɬɨɞɨɦ ɧɚɡɢɜɚɽɬɶɫɹ ɪɿɜɧɹɧɧɹ: |
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a0 yn a1 yn 1 |
... am yn m |
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b1 fn 1 ... bm fn m , n |
m,m 1,... (2) |
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ɞɟ ak ,bk – ɱɢɫɥɨɜɿ ɤɨɟɮɿɰɿɽɧɬɢ, ɹɤɿ ɧɟ ɡɚɥɟɠɚɬɶ ɜɿɞ n , ɩɪɢɱɨɦɭ a0 |
0. |
Ɋɿɜɧɹɧɧɹ (2) ɫɥɿɞ ɪɨɡɝɥɹɞɚɬɢ ɹɤ ɪɟɤɭɪɟɧɬɧɟ ɫɩɿɜɜɿɞɧɨɲɟɧɧɹ, ɹɤɟ ɜɢɪɚɠɚɽ |
ɧɨɜɟ ɡɧɚɱɟɧɧɹ yn |
y tn ɱɟɪɟɡ ɡɧɚɣɞɟɧɿ ɪɚɧɿɲɟ ɡɧɚɱɟɧɧɹ yn 1, yn |
2 ,..., yn m . |
Ɋɨɡɪɚɯɭɧɨɤ ɩɨɱɢɧɚɽɬɶɫɹ ɡ n |
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m, ɬɨɛɬɨ ɡ ɪɿɜɧɹɧɧɹ |
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a0 ym |
a1 ym 1 ... |
am y0 |
b0 fm b1 fm 1 ... bm f |
0 . |
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əɤ ɛɚɱɢɦɨ ɡ ɪɿɜɧɹɧɧɹ, ɞɥɹ ɩɨɱɚɬɤɭ ɪɨɡɪɚɯɭɧɤɿɜ ɧɟɨɛɯɿɞɧɨ ɡɚɞɚɬɢ m ɩɨɱɚɬɤɨɜɢɯ ɡɧɚɱɟɧɶ. Ɂɧɚɱɟɧɧɹ y0 ɜɢɡɧɚɱɚɽɬɶɫɹ ɩɨɱɚɬɤɨɜɨɸ ɭɦɨɜɨɸ ɡɚɞɚɱɿ
(1), ɚ ɫɚɦɟ ɩɨɤɥɚɞɚɸɬɶ y0 u0 . ȼɟɥɢɱɢɧɢ y1,..., ym 1 ɦɨɠɧɚ ɨɛɱɢɫɥɢɬɢ,
ɧɚɩɪɢɤɥɚɞ, ɡɚ ɦɟɬɨɞɨɦ Ɋɭɧɝɟ-Ʉɭɬɬɚ, ɚɛɨ ɡɚ ɦɟɬɨɞɨɦ ɪɹɞɿɜ Ɍɟɣɥɨɪɚ. ȼ ɩɨɞɚɥɶɲɨɦɭ ɛɭɞɟɦɨ ɜɜɚɠɚɬɢ, ɳɨ ɩɨɱɚɬɤɨɜɿ ɡɧɚɱɟɧɧɹ y0 , y1,..., ym 1 ɡɚɞɚɧɿ.
Ɂ ɪɿɜɧɹɧɧɹ (2) ɜɢɞɧɨ, ɳɨ ɧɚ ɜɿɞɦɿɧɭ ɜɿɞ ɦɟɬɨɞɭ Ɋɭɧɝɟ-Ʉɭɬɬɚ
ɛɚɝɚɬɨɤɪɨɤɨɜɿ ɪɿɡɧɢɰɟɜɿ ɦɟɬɨɞɢ ɞɨɩɭɫɤɚɸɬɶ ɨɛɱɢɫɥɟɧɧɹ ɩɪɚɜɢɯ ɱɚɫɬɢɧ ɬɿɥɶɤɢ |
ɜ ɬɨɱɤɚɯ ɨɫɧɨɜɧɨʀ ɫɿɬɤɢ |
W. |
115
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Ɇɟɬɨɞ (2) ɧɚɡɢɜɚɽɬɶɫɹ ɹɜɧɢɦ, ɹɤɳɨ b0 |
0, ɿ ɜɿɞɩɨɜɿɞɧɨ, ɲɭɤɚɧɟ ɡɧɚɱɟɧɧɹ |
yn |
ɜɢɪɚɠɚɽɬɶɫɹ ɹɜɧɨ ɱɟɪɟɡ yn 1,..., yn m . əɤɳɨ b0 |
0, ɬɨ ɦɟɬɨɞ (2) ɧɚɡɢɜɚɽɬɶɫɹ |
ɧɟɹɜɧɢɦ. |
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ȼ ɧɟɹɜɧɨɦɭ ɦɟɬɨɞɿ ɞɥɹ ɩɨɲɭɤɭ |
yn |
ɩɨɬɪɿɛɧɨ |
ɪɨɡɜ’ɹɡɭɜɚɬɢ |
ɧɟɥɿɧɿɣɧɟ |
ɪɿɜɧɹɧɧɹ: |
F>yn 1, yn |
2 ,..., yn m @, |
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aW0 |
yn b0 f tn , yn |
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F>yn 1, yn 2 ,..., yn m @ |
m § |
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k 1 © |
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əɤ |
ɩɪɚɜɢɥɨ ɰɟ ɪɿɜɧɹɧɧɹ ɪɨɡɜ’ɹɡɭɸɬɶ |
ɦɟɬɨɞɨɦ |
ɇɶɸɬɨɧɚ, |
ɨɛɢɪɚɸɱɢ, |
ɧɚɩɪɢɤɥɚɞ, ɩɨɱɚɬɤɨɜɟ ɧɚɛɥɢɠɟɧɧɹ yn(0) ɪɿɜɧɢɦ yn 1 .
Ʉɨɟɮɿɰɿɽɧɬɢ ɪɿɜɧɹɧɧɹ (2) ɜɢɡɧɚɱɟɧɿ ɡ ɬɨɱɧɿɫɬɸ ɞɨ ɦɧɨɠɧɢɤɚ. Ⱦɥɹ ɭɧɢɤɧɟɧɧɹ ɰɿɽʀ ɧɟɨɞɧɨɡɧɚɱɧɨɫɬɿ, ɛɭɞɟɦɨ ɜɜɚɠɚɬɢ, ɳɨ ɜɢɤɨɧɭɽɬɶɫɹ ɭɦɨɜɚ:
k 0
ɹɤɚ ɨɡɧɚɱɚɽ, ɳɨ ɩɪɚɜɚ ɱɚɫɬɢɧɚ ɪɿɡɧɢɰɟɜɨɝɨ ɪɿɜɧɹɧɧɹ (2) ɚɩɪɨɤɫɢɦɭɽ ɩɪɚɜɭ ɱɚɫɬɢɧɭ ɞɢɮɟɪɟɧɰɿɚɥɶɧɨɝɨ ɪɿɜɧɹɧɧɹ (1).
ȼ ɩɪɚɤɬɢɱɧɨɦɭ ɜɢɤɨɪɢɫɬɚɧɧɿ ɧɚɣɛɿɥɶɲ ɩɨɲɢɪɟɧɿ ɦɟɬɨɞɢ ɞɚɦɫɚ, ɹɤɿ ɹɜɥɹɸɬɶc ɫɨɛɨɸ ɱɚɫɬɢɧɧɢɣ ɜɢɩɚɞɨɤ ɛɚɝɚɬɨɤɪɨɤɨɜɢɯ ɦɟɬɨɞɿɜ (2), ɤɨɥɢ ɩɨɯɿɞɧɚ u t ɚɩɪɨɤɫɢɦɭɽɬɶɫɹ ɬɿɥɶɤɢ ɩɨ ɞɜɨɯ ɬɨɱɤɚɯ, tn ɿ tn 1 , ɬɨɛɬɨ
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0, k 2,m. |
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Ɍɚɤɢɦ ɱɢɧɨɦ, ɦɟɬɨɞɢ Ⱥɞɚɦɫɚ ɦɚɸɬɶ ɜɢɝɥɹɞ: |
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yn |
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yn 1 |
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k 0 |
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ɍ ɜɢɩɚɞɤɭ, |
ɤɨɥɢ b0 |
0, ɦɟɬɨɞɢ |
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ɞɚɦɫɚ ɧɚɡɢɜɚɸɬɶ ɹɜɧɢɦɢ, |
ɩɪɢ |
b0 0, |
ɦɟɬɨɞɢ |
ɞɚɦɫɚ ɧɚɡɢɜɚɸɬɶ ɧɟɹɜɧɢɦɢ. |
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Ɋɨɡɝɥɹɧɟɦɨ ɞɟɬɚɥɶɧɿɲɟ ɩɪɨɰɟɞɭɪɭ ɩɨɛɭɞɨɜɢ ɦɟɬɨɞɿɜ |
ɞɚɦɫɚ. |
ȱɧɬɟɝɪɭɽɦɨ ɪɿɜɧɹɧɧɹ (1) ɩɨ t [tn 1,tn |
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u tn 1 |
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Ɂɚɦɿɧɢɦɨ |
ɧɚ |
ɿɧɬɟɪɩɨɥɹɰɿɣɧɢɣ |
ɩɨɥɿɧɨɦ. ȼɢɛɟɪɟɦɨ ɜɭɡɥɚɦɢ |
ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ ɬɨɱɤɢ tn ,tn 1,...tn m . |
ȼɢɤɨɪɢɫɬɚɽɦɨ ɛɚɝɚɬɨɱɥɟɧ ɫɬɟɩɟɧɹ m ɡɚ |
ɮɨɪɦɭɥɨɸ ɇɶɸɬɨɧɚ ɩɨ ɪɿɜɧɨɜɿɞɞɚɥɟɧɢɯ ɜɭɡɥɚɯ, ɬɨɛɬɨ |
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f |
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Lm t |
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fn s'fn 1 ... |
s s |
1 ... |
s m 1 |
'm fn m , |
f t,u t |
Lm t |
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m! |
ɞɟ s |
t |
Wtn |
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ɫɬɚɥɢɣ. ȼ ɪɟɡɭɥɶɬɚɬɿ ɩɿɞɫɬɚɧɨɜɤɢ ɜ (4) |
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ɨɬɪɢɦɭɽɦɨ ɦɟɬɨɞ
k 0
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yn 1 |
fn |
E1'fn 1 ... Em'm fn m , |
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ɞɟ |
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s k 1 |
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ds ,k |
1,m |
(4) |
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fn 1, '2 fn 2 |
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2 fn 1 fn 2 . |
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fn 1 |
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ɉɨɯɢɛɤɚ ɦɟɬɨɞɭ ɧɚ ɨɞɧɨɦɭ ɤɪɨɰɿ |
³n |
rm t dt O Wm 2 , |
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R W |
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tn 1 |
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f t Lm t , |
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rm t |
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ɡɜɿɞɤɢ ɫɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ ɧɚ ɨɞɧɨɦɭ ɤɪɨɰɿ p |
m |
1. |
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Ɏɨɪɦɭɥɚ (4) ɧɚɡɢɜɚɽɬɶɫɹ ɮɨɪɦɭɥɨɸ Ⱥɞɚɦɫɚ-Ɇɭɥɬɨɧɚ (ɧɟɹɜɧɢɣ ɦɟɬɨɞ
ɞɚɦɫɚ, ɿɧɬɟɪɩɨɥɹɰɿɣɧɢɣ ɦɟɬɨɞ |
ɞɚɦɫɚ). |
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ȼɢɛɟɪɟɦɨ ɜɭɡɥɢ ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ ɬɨɱɤɢ |
tn 1,...tn m . Ɉɬɪɢɦɚɽɦɨ ɛɚɝɚɬɨɱɥɟɧ |
ɫɬɟɩɟɧɹ m |
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v v 1 ... v m |
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m 1 |
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t,u t Lm 1 t |
fn 1 v'fn 2 ... |
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m 1! |
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fn m , |
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ɞɟ v |
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Wtn 1 |
. ɉɿɞɫɬɚɜɥɹɸɱɢ ɜ (4), ɦɚɽɦɨ ɦɟɬɨɞ |
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fn 1 J1'fn 1 ... Jm 1'm 1 fn m |
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ɞɟ |
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... v k |
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Jk |
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dv , k 1,m 1. |
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k! |
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Ɏɨɪɦɭɥɚ (5) ɧɚɡɢɜɚɽɬɶɫɹ ɮɨɪɦɭɥɨɸ Ⱥɞɚɦɫɚ-Ȼɚɲɮɨɪɬɚ (ɹɜɧɢɣ ɦɟɬɨɞ |
ɞɚɦɫɚ, |
ɟɤɫɬɪɚɩɨɥɹɰɿɣɧɢɣ ɦɟɬɨɞ |
ɞɚɦɫɚ). ɉɨɯɢɛɤɚ ɰɶɨɝɨ ɦɟɬɨɞɭ ɧɚ ɨɞɧɨɦɭ |
ɤɪɨɰɿ R W |
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O Wm 1 . ɋɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ ɧɚ ɨɞɧɨɦɭ ɤɪɨɰɿ p |
m . |
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Ɂɚɞɚɱɚ 39 ɉɨɛɭɞɭɜɚɬɢ ɹɜɧɢɣ ɬɚ ɧɟɹɜɧɢɣ ɞɜɨɤɪɨɤɨɜɿ ɦɟɬɨɞɢ |
|
ɞɚɦɫɚ. |
əɤɢɣ |
ɫɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ ɜɨɧɢ ɦɚɸɬɶ? |
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10.6. Ɇɟɬɨɞ ɧɟɜɢɡɧɚɱɟɧɢɯ ɤɨɟɮɿɰɿɽɧɬɿɜ ɩɨɛɭɞɨɜɢ ɛɚɝɚɬɨɤɪɨɤɨɜɢɯ ɦɟɬɨɞɿɜ ɞɥɹ
ɪɨɡɜ’ɹɡɤɭ ɡɚɞɚɱɿ Ʉɨɲɿ [ȻɀɄ, 375-379], [ɋȽ, 230-231] |
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Ɋɨɡɝɥɹɧɟɦɨ ɡɚɞɚɱɭ Ʉɨɲɿ |
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f t,u , t 0, u t0 u0 . |
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du |
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(1) |
ɿ ɛɚɝɚɬɨɤɪɨɤɨɜɢɣ ɦɟɬɨɞ |
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ɉɿɞɛɟɪɟɦɨ ɤɨɟɮɿɰɿɽɧɬɢ ak , bk ɬɚɤ, ɳɨɛ ɞɨɫɹɝɬɢ ɧɚɣɜɢɳɨʀ ɬɨɱɧɨɫɬɿ ɦɟɬɨɞɭ (2). 116
ȼɜɟɞɟɦɨ ɩɨɯɢɛɤɭ ɚɩɪɨɤɫɢɦɚɰɿʀ ɞɥɹ ɮɨɪɦɭɥɢ (2). ɉɨɯɢɛɤɨɸ
ɚɩɪɨɤɫɢɦɚɰɿʀ ɧɚ ɪɨɡɜ’ɹɡɤɭ ɚɛɨ ɧɟɜ’ɹɡɤɨɸ ɪɿɡɧɢɰɟɜɨɝɨ ɦɟɬɨɞɭ (2) ɧɚɡɢɜɚɽɬɶɫɹ
ɮɭɧɤɰɿɹ
m |
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un k |
¦bk f tn k ,un k |
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(3) |
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k 0 |
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ɡɚɞɚɱɿ |
(1) ɜ ɪɿɡɧɢɰɟɜɟ ɪɿɜɧɹɧɧɹ (2). Ɋɨɡɝɥɹɧɟɦɨ ɩɢɬɚɧɧɹ ɩɪɨ ɩɨɪɹɞɨɤ ɩɨɯɢɛɤɢ |
ɚɩɪɨɤɫɢɦɚɰɿʀ ɩɪɢ |
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ɜ ɡɚɥɟɠɧɨɫɬɿ ɜɿɞ ɜɢɛɨɪɭ ɤɨɟɮɿɰɿɽɧɬɿɜ ak , bk , k |
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Ɋɨɡɤɥɚɞɟɦɨ ɮɭɧɤɰɿʀ un k |
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Ɍɭɬ u( j) |
– j -ɬɚ ɩɨɯɿɞɧɚ. Ⱦɚɥɿ |
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ɉɿɞɫɬɚɜɥɹɸɱɢ ɰɿ ɜɢɪɚɡɢ ɜ ɮɨɪɦɭɥɭ |
(3), ɨɬɪɢɦɚɽɦɨ: |
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ɉɿɫɥɹ ɩɟɪɟɬɜɨɪɟɧɶ, ɩɪɢɯɨɞɢɦɨ ɞɨ ɪɨɡɤɥɚɞɭ: |
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Ɂɜɿɞɤɢ ɜɢɞɧɨ, ɳɨ ɩɨɯɢɛɤɚ ɚɩɪɨɤɫɢɦɚɰɿʀ |
ɦɚɽ |
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ɩɨɪɹɞɨɤ |
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p , |
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ɜɢɤɨɧɭɸɬɶɫɹ ɭɦɨɜɢ: |
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l 1 |
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(6) |
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Ɋɚɡɨɦ ɡ ɭɦɨɜɨɸ ɧɨɪɦɭɜɚɧɧɹ |
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¦bk |
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1, |
ɪɿɜɧɹɧɧɹ (6) |
ɭɬɜɨɪɸɸɬɶ ɫɢɫɬɟɦɭ ɡ |
p 2 |
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2 m 1 |
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ɥɿɧɿɣɧɢɯ |
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ɚɥɝɟɛɪɚʀɱɧɢɯ |
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ɪɿɜɧɹɧɶ |
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ɜɿɞɧɨɫɧɨ |
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ɧɟɜɿɞɨɦɢɯ |
a0 ,a1,...,am ,b0 ,b1,...,bm .
ɍɦɨɜɢ ɧɨɪɦɭɜɚɧɧɹ ɡɚɩɢɲɭɬɶɫɹ ɭ ɜɢɝɥɹɞɿ: |
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ak |
un k |
du |
tn |
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m |
bk f tn k ,un k |
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Ɂ ɭɪɚɯɭɜɚɧɧɹɦ (4), (5) ɦɚɽɦɨ |
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Ɂ ɪɿɜɧɹɧɧɹ (6) ɩɪɢ l |
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0 ɨɬɪɢɦɭɽɦɨ |
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Ɍɨɦɭ ɭɦɨɜɢ ɧɨɪɦɭɜɚɧɧɹ ɞɨɞɚɬɤɨɜɨ ɞɨ (6) ɞɚɸɬɶ ɬɿɥɶɤɢ ɪɿɜɧɹɧɧɹ |
m |
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Ⱦɥɹ ɬɨɝɨ, ɳɨɛ ɫɢɫɬɟɦɚ (5) ɧɟ ɛɭɥɚ ɩɟɪɟɜɢɡɧɚɱɟɧɨɸ, ɧɟɨɛɯɿɞɧɨ ɜɢɦɚɝɚɬɢ, |
ɳɨɛ p d 2m . ɐɹ ɜɢɦɨɝɚ ɨɡɧɚɱɚɽ, |
ɳɨ |
ɩɨɪɹɞɨɤ ɚɩɪɨɤɫɢɦɚɰɿʀ ɥɿɧɿɣɧɢɯ m - |
ɤɪɨɤɨɜɢɯ ɪɿɡɧɢɰɟɜɢɯ ɦɟɬɨɞɿɜ ɧɟ ɦɨɠɟ ɩɟɪɟɜɢɳɭɜɚɬɢ 2m (ɧɟɹɜɧɿ ɦɟɬɨɞɢ).
ɇɚɣɜɢɳɢɣ ɩɨɪɹɞɨɤ ɚɩɪɨɤɫɢɦɚɰɿʀ ɹɜɧɢɯ ɦɟɬɨɞɿɜ – |
2m |
1. |
Ⱦɥɹ ɦɟɬɨɞɿɜ ɞɚɦɫɚ ɭɦɨɜɢ p -ɝɨ ɩɨɪɹɞɤɭ ɚɩɪɨɤɫɢɦɚɰɿʀ ɦɚɸɬɶ ɜɢɝɥɹɞ: |
m |
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¦kl 1bk 1, l 2,3,..., p, b0 |
1 |
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Ɂɜɿɞɤɢ ɛɚɱɢɦɨ, ɳɨ ɧɚɣɜɢɳɢɣ ɩɨɪɹɞɨɤ ɚɩɪɨɤɫɢɦɚɰɿʀ ɧɟɹɜɧɨɝɨ m- ɤɪɨɤɨɜɨɝɨ ɦɟɬɨɞɭ ɞɚɦɫɚ ɞɨɪɿɜɧɸɽ m 1, ɚ ɧɚɣɜɢɳɢɣ ɩɨɪɹɞɨɤ ɚɩɪɨɤɫɢɦɚɰɿʀ ɹɜɧɨɝɨ ɦɟɬɨɞɭ ɞɚɦɫɚ (b0 0) ɞɨɪɿɜɧɸɽ m.
Ɂɚɞɚɱɚ 40 ɉɨɛɭɞɭɜɚɬɢ ɹɜɧɢɣ ɬɚ ɧɟɹɜɧɢɣ ɞɜɨɤɪɨɤɨɜɿ ɦɟɬɨɞɢ ɧɚɣɜɢɳɨɝɨ ɫɬɟɩɟɧɹ ɬɨɱɧɨɫɬɿ.
10.7. ɉɢɬɚɧɧɹ ɪɟɚɥɿɡɚɰɿʀ ɛɚɝɚɬɨɤɪɨɤɨɜɢɯ ɦɟɬɨɞɿɜ 1). ɉɟɪɲɚ ɩɪɨɛɥɟɦɚ, ɹɤɚ ɜɢɧɢɤɚɽ ɩɪɢ ɡɚɫɬɨɫɭɜɚɧɧɹ ɛɚɝɚɬɨɤɪɨɤɨɜɢɯ
ɦɟɬɨɞɿɜ ɰɟ ɜɢɛɿɪ ɞɨɞɚɬɤɨɜɢɯ ɩɨɱɚɬɤɨɜɢɯ ɭɦɨɜ. Ɍɪɟɛɚ ɡɧɚɣɬɢ m 1 ɞɨɞɚɬɤɨɜɟ ɩɨɱɚɬɤɨɜɟ ɡɧɚɱɟɧɧɹ y1,..., ym 1. ɒɥɹɯɢ ɜɢɪɿɲɟɧɧɹ ɩɪɨɛɥɟɦɢ ɬɚɤɿ.
ɚ) Ɇɨɠɧɚ ɜɢɤɨɪɢɫɬɨɜɭɜɚɬɢ ɦɟɬɨɞɢ Ɋɭɧɝɟ-Ʉɭɬɬɚ ɞɥɹ ɩɨɲɭɤɭ ɰɢɯ ɩɨɱɚɬɤɨɜɢɯ ɡɧɚɱɟɧɶ. ɥɟ ɬɪɟɛɚ, ɳɨɛ ɰɿ ɦɟɬɨɞɢ ɦɚɥɢ ɚɛɨ ɬɨɱɧɿɫɬɶ ɚɩɪɨɤɫɢɦɚɰɿʀ p , ɚɛɨ ɬɨɱɧɿɫɬɶ ɩɨɯɢɛɤɢ ɧɚ ɤɪɨɰɿ p . ɇɟɞɨɥɿɤ ɬɚɤɨɝɨ ɫɩɨɫɨɛɭ:
ɫɚɦɚ ɩɪɨɰɟɞɭɪɚ ɨɛɱɢɫɥɟɧɧɹ ɡɚ ɦɟɬɨɞɚɦɢ Ɋɭɧɝɟ-Ʉɭɬɬɚ ɡɚɣɦɚɽ ɜɟɥɢɤɢɣ ɨɛ’ɽɦ, ɬɚ ɣ ɛɚɝɚɬɨɤɪɨɤɨɜɿ ɦɟɬɨɞɢ ɨɛ’ɽɦɧɿ.
ɛ) Ɇɨɠɧɚ ɜɢɤɨɪɢɫɬɨɜɭɜɚɬɢ ɦɟɬɨɞ ɪɹɞɿɜ Ɍɟɣɥɨɪɚ. Ɂɧɨɜɭ ɠ ɬɚɤɢ ɩɨɬɪɿɛɧɨ
ɭɡɝɨɞɠɭɜɚɬɢ ɬɨɱɧɿɫɬɶ p . Ɏɭɧɤɰɿɸ ɪɨɡɤɥɚɞɚɸɬɶ ɜ ɪɹɞ: |
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ɞɟ ɿɧɞɟɤɫ p 1 ɧɚɞ u ɜɤɚɡɭɽ ɤɿɥɶɤɿɫɬɶ ɱɥɟɧɿɜ ɪɹɞɭ. ɉɨɯɢɛɤɚ |
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O Wp . |
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2). Ⱦɪɭɝɚ ɩɪɨɛɥɟɦɚ: ɪɟɚɥɿɡɚɰɿɹ ɧɟɹɜɧɢɯ ɦɟɬɨɞɿɜ |
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tn , yn ) yn 1,..., yn |
m M yn |
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ɚ) Ɇɨɠɧɚ ɜɢɤɨɪɢɫɬɨɜɭɜɚɬɢ ɞɥɹ ɡɧɚɯɨɞɠɟɧɧɹ ɪɨɡɜ‘ɹɡɤɭ ɧɟɥɿɧɿɣɧɨɝɨ |
ɪɿɜɧɹɧɧɹ ɦɟɬɨɞ ɩɪɨɫɬɨʀ ɿɬɟɪɚɰɿʀ: |
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-ɬɭ ɿɬɟɪɚɰɿɸ. ɍɦɨɜɚ ɡɛɿɠɧɨɫɬɿ ɦɟɬɨɞɭ ɩɪɨɫɬɨʀ ɿɬɟɪɚɰɿʀ |
ɞɟ |
ɿɧɞɟɤɫ k |
c u |
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t,u |
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L – ɦɚɤɫɢɦɭɦ ɩɨɯɿɞɧɨʀ ɱɢ ɫɬɚɥɚ Ʌɿɩɲɢɰɹ. |
Ɍɨɛɬɨ ɞɥɹ ɡɛɿɠɧɨɫɬɿ ɦɟɬɨɞɭ |
ɧɟɨɛɯɿɞɧɨ, ɳɨɛ ɤɪɨɤ |
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ɛɭɜ ɞɨɫɢɬɶ ɦɚɥɢɦ. |
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ɛ) Ɇɨɠɧɚ ɜɢɤɨɪɢɫɬɨɜɭɜɚɬɢ ɦɟɬɨɞ ɇɶɸɬɨɧɚ. əɤ ɜɿɞɨɦɨ, ɭɦɨɜɢ ɡɛɿɠɧɨɫɬɿ ɦɟɬɨɞɭ ɇɶɸɬɨɧɚ ɡɚɥɟɠɚɬɶ ɜɿɞ ɜɞɚɥɨɝɨ ɜɢɛɨɪɭ ɩɨɱɚɬɤɨɜɢɯ ɭɦɨɜ. ɏɨɪɨɲɟ
ɩɨɱɚɬɤɨɜɟ ɧɚɛɥɢɠɟɧɧɹ ɬɚɤɟ: ɚɛɨ yn |
(0) yn 1, |
ɚɛɨ yn |
(0) ɨɛɱɢɫɥɸɽɬɶɫɹ ɡɚ ɹɜɧɨɸ |
m- ɤɪɨɤɨɜɨɸ ɮɨɪɦɭɥɨɸ. |
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ȼ ɪɟɡɭɥɶɬɚɬɿ ɭɦɨɜɢ ɡɛɿɠɧɨɫɬɿ ɦɟɬɨɞɭ ɇɶɸɬɨɧɚ ɦɟɧɲ ɠɨɪɫɬɤɿ, ɧɿɠ ɭ |
ɦɟɬɨɞɭ ɩɪɨɫɬɨʀ ɿɬɟɪɚɰɿʀ. |
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ɜ) ɦɨɠɧɚ ɜɢɤɨɪɢɫɬɨɜɭɜɚɬɢ ɦɟɬɨɞ ɩɪɟɞɢɤɬɨɪ-ɤɨɪɟɤɬɨɪ. Ɂɚɩɢɲɟɦɨ |
ɮɨɪɦɭɥɢ ɧɟɹɜɧɨɝɨ ɦɟɬɨɞɭ (ɋ – ɤɨɪɟɤɬɨɪ, Ɋ – ɩɪɟɞɢɤɬɨɪ ): |
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ɤɪɨɤɨɜɢɣ ɦɟɬɨɞ, |
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yn 1,..., fn 1,... – ɹɜɧɢɣ m- ɤɪɨɤɨɜɢɣ ɦɟɬɨɞ, |
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Ⱦɚɥɿ ɜɢɤɨɧɭɽɬɶɫɹ ɬɚɤɚ ɩɪɨɰɟɞɭɪɚ: |
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Ɋ: |
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yn 1,..., fn 1,... ; |
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ɞɟ ȿ – ɩɿɞɪɚɯɭɜɚɧɧɹ ɩɪɚɜɨʀ ɱɚɫɬɢɧɢ ɪɿɜɧɹɧɧɹ (Equation). ɋɯɟɦɚ Ɋȿɋȿ – ɰɟ ɦɟɬɨɞ ɩɪɟɞɢɤɬɨɪ-ɤɨɪɟɤɬɨɪ.
ȱɧɨɞɿ, ɳɨɛ ɩɿɞɜɢɳɢɬɢ ɬɨɱɧɿɫɬɶ, ɜɢɤɨɪɢɫɬɨɜɭɸɬɶ ɫɯɟɦɭ Ɋȿ(ɋȿ)j. ɐɹ ɫɯɟɦɚ ɚɧɚɥɨɝɿɱɧɚ ɦɟɬɨɞɭ ɩɪɨɫɬɨʀ ɿɬɟɪɚɰɿʀ, ɞɟ j – ɰɟ ɦɚɤɫɢɦɚɥɶɧɚ ɤɿɥɶɤɿɫɬɶ
ɿɬɟɪɚɰɿɣ.
10.8. ɋɬɿɣɤɿɫɬɶ ɦɟɬɨɞɿɜ ɪɨɡɜ’ɹɡɚɧɧɹ ɡɚɞɚɱɿ Ʉɨɲɿ [ɅɆɋ,273,281-282],
[ȻɀɄ,379-382], [ɋȽ,235-238]
ȼɚɠɥɢɜɢɦɢ ɽ ɩɢɬɚɧɧɹ ɫɬɿɣɤɨɫɬɿ ɱɢɫɟɥɶɧɢɯ ɦɟɬɨɞɿɜ ɞɥɹ ɡɚɞɚɱ Ʉɨɲɿ, ɬɨɛɬɨ ɧɟɩɟɪɟɪɜɧɚ ɡɚɥɟɠɧɿɫɬɶ ɪɨɡɜ’ɹɡɤɿɜ ɜɿɞ ɡɛɭɪɟɧɶ ɩɨɱɚɬɤɨɜɢɯ ɞɚɧɢɯ, ɩɨɯɢɛɨɤ ɡɚɨɤɪɭɝɥɟɧɧɹ, ɬɨɳɨ.
ɑɢɫɟɥɶɧɢɣ ɦɟɬɨɞ ɪɨɡɜ’ɹɡɚɧɧɹ ɡɚɞɚɱɿ Ʉɨɲɿ |
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ɧɚɡɢɜɚɽɬɶɫɹ ɫɬɿɣɤɢɣ ɡɚ ɩɨɱɚɬɤɨɜɢɦɢ ɞɚɧɢɦɢ, ɹɤɳɨ ɞɥɹ ɞɨɜɿɥɶɧɨʀ ɡɚɞɚɱɿ Ʉɨɲɿ
(ɬɨɛɬɨ f (t,u) )
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0 V |
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0 ɩɪɢ |
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V, ɦɚɽɦɨ |
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H, |
n . |
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ɑɢɫɟɥɶɧɢɣ |
ɦɟɬɨɞ |
ɧɚɡɢɜɚɽɬɶɫɹ |
ɧɟɫɬɿɣɤɢɦ, |
ɹɤɳɨ |
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ɳɨ |
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V , |
ɦɚɽɦɨ |
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f (t,u), |
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u u0 |
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H, n t N H . |
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ɉɟɪɲɟ ɨɡɧɚɱɟɧɧɹ ɫɤɥɚɞɧɿɲɟ ɩɟɪɟɜɿɪɹɬɢ ɧɿɠ ɞɪɭɝɟ. ɉɟɪɟɜɿɪɤɨɸ ɭɦɨɜɢ
ɧɟɫɬɿɣɤɨɫɬɿ ɦɨɠɧɚ ɜɢɛɪɚɤɨɜɭɜɚɬɢ ɩɨɝɚɧɿ ɦɟɬɨɞɢ ɧɚ ɬɚɤɢɯ ɤɥɚɫɚɯ ɩɪɚɜɢɯ ɱɚɫɬɢɧ:
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ɚ) f |
0; ɛ) f |
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u, |
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0; ɜ) f |
u, Re |
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Ɋɨɡɝɥɹɧɟɦɨ ɤɨɠɟɧ ɜɢɩɚɞɨɤ. |
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ɚ) |
f 0. əɤɳɨ ɡɚɫɬɨɫɭɜɚɬɢ ɦɟɬɨɞ Ɋɭɧɝɟ-Ʉɭɬɬɚ, |
ɬɨ |
yn 1 |
yn ... |
y0 , ɚ |
ɰɟ ɨɡɧɚɱɚɽ, ɳɨ ɦɚɥɿ ɡɛɭɪɟɧɧɹ |
y0 |
ɩɪɢɜɨɞɹɬɶ ɞɨ ɬɚɤɢɯ ɫɚɦɢɯ ɡɛɭɪɟɧɶ ɜ |
yn 1 : |
G H . |
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Ɋɨɡɝɥɹɧɟɦɨ ɬɟɩɟɪ ɛɚɝɚɬɨɤɪɨɤɨɜɢɣ ɦɟɬɨɞ, ɳɨ ɩɪɢɜɨɞɢɬɶ ɞɨ ɪɿɡɧɢɰɟɜɨɝɨ |
ɪɿɜɧɹɧɧɹ n -ɝɨ ɩɨɪɹɞɤɭ |
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m |
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k 0 |
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¦ |
ak |
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yn k |
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(2) |
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ɑɚɫɬɤɨɜɿ ɪɨɡɜ’ɹɡɤɢ ɲɭɤɚɽɦɨ ɭ ɜɢɝɥɹɞɿ |
yn |
qn , |
yn k |
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qn k . ɉɿɞɫɬɚɜɢɦɨ ʀɯ ɿ |
ɨɬɪɢɦɚɽɦɨ ɯɚɪɚɤɬɟɪɢɫɬɢɱɧɟ ɪɿɜɧɹɧɧɹ |
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m |
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¦ak qm k |
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(3) |
ɡ ɤɨɪɟɧɹɦɢ q1, |
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k 0 |
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,qm . |
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Ɋɨɡɜ’ɹɡɨɤ ɪɿɜɧɹɧɧɹ (2) ɡɧɚɯɨɞɢɦɨ ɭ ɜɢɝɥɹɞɿ: |
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m |
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ɹɤɳɨ q – ɞɿɣɫɧɿ ɪɿɡɧɿ, ɬɨ yn |
¦ck qkn ; |
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ɹɤɳɨ q – ɞɿɣɫɧɿ ɤɪɚɬɧɿ, ɬɨ y |
k |
0 |
cj qj |
cj nqj |
; |
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n |
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~ |
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sin(nImqj ) .., |
ɹɤɳɨ q – ɤɨɦɩɥɟɤɫɧɿ, ɬɨ y |
n |
.. cjU j cos(nImqj ) cjU j |
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~ |
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ɞɟ U j |
modqj |
, ɹɤɳɨ |
U j 1 |
n o |
, yn |
o , |
ck |
ɡɧɚɯɨɞɹɬɶ ɡ ɩɨɱɚɬɤɨɜɢɯ |
ɭɦɨɜ. |
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ɉɪɢ |
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qk |
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1 ɛɚɝɚɬɨɤɪɨɤɨɜɢɣ ɦɟɬɨɞ – ɧɟɫɬɿɣɤɢɣ. |
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Ʉɚɠɭɬɶ, |
ɳɨ ɛɚɝɚɬɨɤɪɨɤɨɜɢɣ |
ɦɟɬɨɞ |
ɪɨɡɜ’ɹɡɚɧɧɹ |
ɡɚɞɚɱ |
ɡɚɞɨɜɨɥɶɧɹɽ |
ɤɨɪɟɧɟɜɿɣ ɭɦɨɜɿ, ɹɤɳɨ ɜɫɿ ɤɨɪɟɧɿ ɯɚɪɚɤɬɟɪɢɫɬɢɱɧɨɝɨ ɪɿɜɧɹɧɧɹ (3) ɥɟɠɚɬɶ ɜ |
ɨɞɢɧɢɱɧɨɦɭ |
ɤɪɭɡɿ, |
ɚ ɧɚ ɤɨɥɿ |
ɧɟɦɚɽ |
ɤɪɚɬɧɢɯ |
ɤɨɪɟɧɿɜ, ɬɨɛɬɨ |
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qj |
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d 1 , |
ɞɥɹ |
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qj |
qm , |
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1. |
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qj |
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qm |
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ɓɨɛ ɩɟɪɟɜɿɪɢɬɢ, ɹɤ ɩɪɚɰɸɽ ɤɨɪɟɧɟɜɚ ɭɦɨɜɚ ɪɨɡɝɥɹɧɟɦɨ ɡɚɞɚɱɭ. |
ɉɨɛɭɞɭɽɦɨ ɹɜɧɢɣ ɞɜɨɤɪɨɤɨɜɢɣ ɦɟɬɨɞ ɧɚɣɜɢɳɨɝɨ ɫɬɟɩɟɧɹ ɬɨɱɧɨɫɬɿ. m |
2, |
p |
3: |
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yn 4yn 1 |
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5yn 2 |
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2 |
fn 1 |
1 |
fn 2 . |
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6W |
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3 |
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3 |
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