численные методыА.Б. САМОХИН, В.В. ЧЕРДЫНЦЕВ, А.А. ВОРОНЦОВ
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ɋɪɚɜɧɢɬɶ ɟɝɨ ɫɨ ɡɧɚɱɟɧɢɹɦɢ ɩɨɥɭɱɟɧɧɵɦɢ ɦɟɬɨɞɨɦ ɬɪɚɩɟɰɢɣɦɟɬɨɞɨɦ ɩɚɪɚɛɨɥ ɦɟɬɨɞɨɦ Ƚɚɭɫɫɚ ɤɨɷɮɮɢ ɰɢɟɧɬɵ ɷɬɨɝɨ ɦɟɬɨɞɚ ɩɪɢɜɟɞɟɧɵ ɜ ɬɚɛɥ
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Ɍɚɛɥɢɰɚ |
n |
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ti |
Ai |
n=4 |
1,4 |
B 0,861136 |
0,347854 |
2,3 |
B 0,339981 |
0,652145 |
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n=6 |
1,6 |
B 0,932464 |
0,171324 |
2,5 |
B 0,661209 |
0,360761 |
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3,4 |
B 0,238619 |
0,467913 |
n=8 |
1,8 |
B 0,960289 |
0,101228 |
2,7 |
B 0,796666 |
0,222381 |
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3,6 |
B 0,525532 |
0,313706 |
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4,5 |
B 0,183434 |
0,362683 |
Ɋɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɨɜ ɫɜɟɫɬɢ ɜ ɬɚɛɥ
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Ɍɚɛɥɢɰɚ |
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Iɬɪ |
… |
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Iɩɚɪ |
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Ig |
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ɉɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɜɟɥɢɱɢɧɵ ɢɧɬɟɝɪɚɥɨɜ ɨɬ n, ɧɚ ɤɨɬɨɪɵɣ ɧɚɧɟɫɬɢ ɪɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɨɜ ɢ ɬɨɱɧɨɟ ɡɧɚɱɟɧɢɟ ɢɧɬɟɝɪɚɥɚ Ɉɰɟɧɢɬɶ ɤɚɱɟɫɬɜɟɧɧɨ ɫɤɨɪɨɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɚɡɥɢɱɧɵɯ ɦɟɬɨɞɨɜ
Ɍɚɛɥɢɰɚ
ʋ |
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K |
3,2 |
3,4 |
3,6 |
3,8 |
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2,2 |
2,4 |
2,6 |
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1,6 |
1,8 |
2,0 |
2,2 |
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2,4 |
1,2 |
1,4 |
1,6 |
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ʋ |
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K |
2,8 |
3,0 |
1,2 |
1,4 |
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1,6 |
1,8 |
4,2 |
4,4 |
L |
1,8 |
2,2 |
0,8 |
1,0 |
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1,4 |
3,2 |
3,4 |
32
5. ɑɂɋɅȿɇɇɕȿ ɆȿɌɈȾɕ ɅɂɇȿɃɇɈɃ ȺɅȽȿȻɊɕ
Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɱɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦ ɥɢɧɟɣ ɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɋɅȺɍ ɚ ɬɚɤɠɟ ɧɚɯɨɠɞɟɧɢɹ ɫɨɛ ɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɢ ɫɨɛɫɬɜɟɧɧɵɯ ɜɟɤɬɨɪɨɜ ɦɚɬɪɢɰ
ɑɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ ɋɅȺɍ
ɋɅȺɍ ɢɫɩɨɥɶɡɭɸɬɫɹ ɜɨ ɦɧɨɝɢɯ ɨɛɥɚɫɬɹɯ ɧɚɭɤɢ ɢ ɬɟɯɧɢɤɢ ɢ ɹɜɥɹɸɬɫɹ ɧɚɢɛɨɥɟɟ ɱɚɫɬɨ ɜɫɬɪɟɱɚɸɳɢɦɫɹ ɬɢɩɨɦ ɡɚɞɚɱ ɜɵɱɢɫɥɢ ɬɟɥɶɧɨɣ ɦɚɬɟɦɚɬɢɤɢ ȼ ɨɛɳɟɦ ɜɢɞɟ ɋɅȺɍ ɢɡ n ɭɪɚɜɧɟɧɢɣ ɫ n ɧɟ ɢɡɜɟɫɬɧɵɦɢ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ
G |
b . |
(5.1) |
Ax |
Ɂɞɟɫɶ x — ɧɟɢɡɜɟɫɬɧɵɣ ɜɟɤɬɨɪ ɪɟɲɟɧɢɹ b — ɡɚɞɚɧɧɵɣ ɜɟɤɬɨɪ ɜ
n -ɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɚ |
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¨ 11 |
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— ɥɢɧɟɣɧɵɣ ɨɩɟɪɚɬɨɪ ɜ ɷɬɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɚɞɚɧɧɚɹ ɦɚɬɪɢɰɚ ɪɚɡɦɟɪɨɦ n u n ɢɥɢ ɜ ɞɪɭɝɨɦ ɜɢɞɟ A ^ai j `, i, j 1, 2, ..., n.
Ⱦɨɤɚɡɵɜɚɟɬɫɹ ɱɬɨ ɟɫɥɢ ɨɩɪɟɞɟɥɢɬɟɥɶ ɦɚɬɪɢɰɵ ɧɟ ɪɚɜɟɧ ɧɭɥɸ ɬɨ ɋɅȺɍ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɇɢɠɟ ɛɭɞɟɦ ɩɨɥɚɝɚɬɶ ɱɬɨ ɷɬɨ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɹɟɬɫɹ Ɉɞɧɚɤɨ ɨɬɥɢɱɢɟ ɨɩɪɟɞɟɥɢɬɟɥɹ A ɨɬ ɧɭɥɹ ɧɟ ɦɨɝɭɬ ɫɥɭɠɢɬɶ ɝɚɪɚɧɬɢɟɣ ɬɨɝɨ ɱɬɨ ɪɟɲɟɧɢɟ ɋɅȺɍ ɛɭɞɟɬ ɧɚɣɞɟɧɨ ɱɢɫɥɟɧɧɨ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ ɉɪɢɱɢɧɨɣ ɷɬɨɝɨ ɦɨɠɟɬ ɛɵɬɶ ɤɚɤ ɩɥɨɯɚɹ ɨɛɭɫɥɨɜɥɟɧɧɨɫɬɶ ɫɚɦɨɣ ɫɢɫɬɟɦɵ ɬɚɤ ɢ ɜɵɛɪɚɧɧɨ ɝɨ ɚɥɝɨɪɢɬɦɚ Ɂɚɦɟɬɢɦ ɱɬɨ ɛɥɢɡɨɫɬɶ ɨɩɪɟɞɟɥɢɬɟɥɹ ɤ ɧɭɥɸ ɢ ɞɚɠɟ ɜɟɫɶɦɚ ɦɚɥɨɟ ɟɝɨ ɡɧɚɱɟɧɢɟ ɧɟ ɫɜɢɞɟɬɟɥɶɫɬɜɭɸɬ ɜɨɨɛɳɟ ɝɨɜɨɪɹ ɨ ɩɥɨɯɨɣ ɨɛɭɫɥɨɜɥɟɧɧɨɫɬɢ ɫɢɫɬɟɦɵ ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɦɚɬɪɢɰɭ ɫɢɫɬɟɦɵ ɭ ɤɨɬɨɪɨɣ ɩɪɢɫɭɬɫɬɜɭɟɬ ɬɨɥɶɤɨ ɝɥɚɜ ɧɚɹ ɞɢɚɝɨɧɚɥɶ ɫ ɜɟɫɶɦɚ ɦɚɥɵɦɢ ɧɨ ɨɬɥɢɱɧɵɦɢ ɨɬ ɧɭɥɹ ɤɨɷɮɮɢɰɢ ɟɧɬɚɦɢ Ɉɩɪɟɞɟɥɢɬɟɥɶ ɬɚɤɨɣ ɦɚɬɪɢɰɵ ɦɨɠɟɬ ɛɵɬɶ ɦɚɲɢɧɧɵɣ
33
ɧɭɥɶ ɜ ɬɨ ɠɟ ɜɪɟɦɹ ɫɜɨɣɫɬɜɚ ɬɚɤɨɣ ɦɚɬɪɢɰɵ ɛɥɢɡɤɢ ɤ ɟɞɢɧɢɱɧɨɣ ɚ ɨɲɢɛɤɚ ɜ ɪɟɲɟɧɢɢ ɩɨɪɹɞɤɚ ɨɲɢɛɤɢ ɜ ɡɚɞɚɧɢɢ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ
Ⱦɥɹ ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ ɩɥɨɯɨ ɨɛɭɫɥɨɜɥɟɧɧɵɯ ɡɚɞɚɱ ɢɯ ɪɟɲɟɧɢɟ ɩɪɢɧɰɢɩɢɚɥɶɧɨ ɧɟɥɶɡɹ ɩɨɥɭɱɢɬɶ ɫɨɜɟɪɲɟɧɧɨ ɬɨɱɧɨ Ⱦɥɹ ɧɢɯ ɦɚ ɥɵɟ ɢɡɦɟɧɟɧɢɹ ɜ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɚɯ ɦɚɬɪɢɰɵ ɢ ɜ ɜɟɤɬɨɪɟ ɩɪɚɜɨɣ ɱɚɫɬɢ ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɩɪɟɞɟɥɚɯ ɬɨɱ ɧɨɫɬɢ ɢɯ ɡɚɞɚɧɢɹ ɩɪɢɜɨɞɹɬ ɤ ɧɟɫɨɪɚɡɦɟɪɧɨ ɛɨɥɶɲɢɦ ɢɡɦɟɧɟɧɢɹɦ ɜ ɪɟɲɟɧɢɢ ȼ ɪɟɡɭɥɶɬɚɬɟ ɜ ɩɪɟɞɟɥɚɯ ɬɨɱɧɨɫɬɢ ɡɚɞɚɧɢɹ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ ɧɚɩɪɢɦɟɪ ɜ ɩɪɟɞɟɥɚɯ ɨɲɢɛɤɢ ɨɤɪɭɝɥɟɧɢɹ ɢɡ-ɡɚ ɨɝɪɚɧɢ ɱɟɧɧɨɝɨ ɮɨɪɦɚɬɚ ɱɢɫɥɨɜɵɯ ɞɚɧɧɵɯ ɗȼɆ ɦɨɠɟɬ ɫɭɳɟɫɬɜɨɜɚɬɶ ɦɧɨɠɟɫɬɜɨ ɪɚɡɥɢɱɧɵɯ ɪɟɲɟɧɢɣ ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɯ ɫɢɫɬɟɦɟ
ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɩɥɨɯɨ ɨɛɭɫɥɨɜɥɟɧɧɨɣ ɫɢɫɬɟɦɵ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɋɅȺɍ ɫ ɩɨɱɬɢ ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɵɦɢ ɫɬɪɨɤɚɦɢ ɫɬɨɥɛɰɚ ɦɢ ɜ ɦɚɬɪɢɰɟ ɉɥɨɯɨ ɨɛɭɫɥɨɜɥɟɧɧɵɦ ɚɥɝɨɪɢɬɦɨɦ ɞɥɹ ɪɟɲɟɧɢɹ ɋɅȺɍ ɦɨɠɧɨ ɧɚɡɜɚɬɶ ɦɟɬɨɞ Ƚɚɭɫɫɚ ɛɟɡ ɜɵɛɨɪɚ ɝɥɚɜɧɨɝɨ ɷɥɟɦɟɧɬɚ
Ⱦɥɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɨɛɭɫɥɨɜɥɟɧɧɨɫɬɢ ɡɚɞɚɱɢ ɜɜɨɞɹɬ ɬɚɤ ɧɚɡɵ ɜɚɟɦɨɟ ɱɢɫɥɨ ɨɛɭɫɥɨɜɥɟɧɧɨɫɬɢ K Ⱦɥɹ ɡɚɞɚɱɢ ɪɟɲɟɧɢɹ ɋɅȺɍ ɜ ɤɚɱɟɫɬɜɟ ɱɢɫɥɚ ɨɛɭɫɥɨɜɥɟɧɧɨɫɬɢ ɦɨɠɧɨ ɩɪɢɧɹɬɶ
K 
A
A 1
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Ɂɞɟɫɶ 
— ɤɚɤɚɹ-ɥɢɛɨ ɧɨɪɦɚ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ n -ɦɟɪɧɵɯ ɜɟɤɬɨ
ɪɨɜ ɤɨɬɨɪɚɹ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɧɨɪɦɭ ɜɟɤɬɨɪɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚ ɡɨɦ
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ɇɨɪɦɚ ɦɚɬɪɢɰɵ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɦɚɤɫɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɨɟ ɨɬɧɨ ɫɢɬɟɥɶɧɨɟ ɭɜɟɥɢɱɟɧɢɟ ɩɨ ɧɨɪɦɟ ɧɟɧɭɥɟɜɨɝɨ ɜɟɤɬɨɪɚ ɩɪɢ ɜɨɡɞɟɣ ɫɬɜɢɢ ɧɚ ɧɟɝɨ ɦɚɬɪɢɰɵ
ɉɭɫɬɶ ɪɟɲɟɧɢɟ x ɋɅȺɍ ɩɨɥɭɱɟɧɨ ɫ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɨɲɢɛɤɨɣ Gx Ɍɨɝɞɚ ɞɥɹ ɧɟɟ ɫɩɪɚɜɟɞɥɢɜɚ ɨɰɟɧɤɚ
Gx 
| KHɦɚɲ .
Ɂɞɟɫɶ Hɦɚɲ — ɦɚɲɢɧɧɚɹ ɤɨɧɫɬɚɧɬɚ ɬ ɟ ɧɚɢɦɟɧɶɲɟɟ ɱɢɫɥɨ ɤɨɬɨ
ɪɨɟ ɩɪɢ ɩɪɢɛɚɜɥɟɧɢɢ ɤ ɟɞɢɧɢɰɟ ɟɳɺ ɢɡɦɟɧɹɟɬ ɟɺ ɡɧɚɱɟɧɢɟ ɜ ɦɚ ɲɢɧɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ Ɉɬɦɟɬɢɦ ɱɬɨ ɨɰɟɧɤɚ ɫɩɪɚɜɟɞɥɢɜɚ ɞɥɹ ɦɚɥɵɯ ɨɲɢɛɨɤ ɜ ɡɚɞɚɧɧɨɣ ɦɚɬɪɢɰɟ K 
'A
/ 
A
1.
34
ȼɜɟɞɺɦ ɩɨɧɹɬɢɟ ɧɟɜɹɡɤɢ h ɪɟɲɟɧɢɹ
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ɇɨɪɦɚ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ ɞɥɹ ɩɥɨɯɨ ɨɛɭɫɥɨɜɥɟɧɧɨɣ ɋɅȺɍ ɜɟ ɥɢɤɚ ɬɚɤɠɟ ɤɚɤ ɢ ɱɢɫɥɨ ɨɛɭɫɥɨɜɥɟɧɧɨɫɬɢ K ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɛɥɢɡɨɫɬɶ ɦɚɬɪɢɰɵ ɤ ɜɵɪɨɠɞɟɧɧɨɣ ɫɢɧɝɭɥɹɪɧɨɣ ɞɥɹ ɤɨɬɨɪɨɣ 
A 1
o f.
ɋɭɳɟɫɬɜɭɸɬ ɞɜɚ ɨɫɧɨɜɧɵɯ ɤɥɚɫɫɚ ɦɟɬɨɞɨɜ ɞɥɹ ɪɟɲɟɧɢɹ ɋɅȺɍ
– ɩɪɹɦɵɟ ɢ ɢɬɟɪɚɰɢɨɧɧɵɟ ɉɪɹɦɵɟ ɦɟɬɨɞɵ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɬɟɦ ɱɬɨ ɩɪɢ ɚɛɫɨɥɸɬɧɨɣ ɬɨɱɧɨɫɬɢ ɜɵɱɢɫɥɟɧɢɣ ɧɚ ɝɢɩɨɬɟɬɢɱɟɫɤɨɣ ɛɟɫɤɨɧɟɱɧɨɪɚɡɪɹɞɧɨɣ ɗȼɆ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɋɅȺɍ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ ɫ ɩɨɦɨɳɶɸ ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɚɪɢɮɦɟɬɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɣ ɂɬɟɪɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɬɟɦ ɱɬɨ ɞɚɠɟ ɩɪɢ ɚɛɫɨ ɥɸɬɧɨɣ ɬɨɱɧɨɫɬɢ ɜɵɱɢɫɥɟɧɢɣ ɡɚ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɚɪɢɮɦɟɬɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɣ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ ɥɢɲɶ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɫɢɫ ɬɟɦɵ ɯɨɬɹ ɜɨɡɦɨɠɧɨ ɢ ɤɚɤ ɭɝɨɞɧɨ ɛɥɢɡɤɨɟ ɤ ɬɨɱɧɨɦɭ Ɉɞɧɚɤɨ ɩɪɢ ɪɟɚɥɶɧɵɯ ɜɵɱɢɫɥɟɧɢɹɯ ɧɚ ɗȼɆ ɭɤɚɡɚɧɧɨɟ ɪɚɡɥɢɱɢɟ ɬɟɪɹɟɬ ɫɜɨɣ ɫɦɵɫɥ ɢ ɞɥɹ ɦɧɨɝɢɯ ɡɚɞɚɱ ɢɬɟɪɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɨɤɚɡɵɜɚɸɬɫɹ ɛɨ ɥɟɟ ɩɪɟɞɩɨɱɬɢɬɟɥɶɧɵɦɢ ɱɟɦ ɩɪɹɦɵɟ ɜ ɫɢɥɭ ɨɬɫɭɬɫɬɜɢɹ ɧɚɤɨɩɥɟ ɧɢɹ ɨɲɢɛɨɤ ɞɥɹ ɫɯɨɞɹɳɟɝɨɫɹ ɩɪɨɰɟɫɫɚ ɢ ɜɨɡɦɨɠɧɨɫɬɢ ɩɪɢɛɥɢ ɡɢɬɶɫɹ ɤ ɪɟɲɟɧɢɸ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ
Ɋɚɫɫɦɨɬɪɢɦ ɫɧɚɱɚɥɚ ɩɪɹɦɵɟ ɦɟɬɨɞɵ ɇɚɢɛɨɥɟɟ ɢɡɜɟɫɬɧɵɦ ɹɜ ɥɹɟɬɫɹ ɦɟɬɨɞ Ƚɚɭɫɫɚ ɩɨɫɤɨɥɶɤɭ ɞɪɭɝɢɟ ɦɟɬɨɞɵ ɹɜɥɹɸɬɫɹ ɤɚɤ ɩɪɚ ɜɢɥɨ ɟɝɨ ɦɨɞɢɮɢɤɚɰɢɟɣ.
35
ɉɪɹɦɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɋɅȺɍ
Ʉɨɥɢɱɟɫɬɜɨ ɨɩɟɪɚɰɢɣ ɞɥɹ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ~ n3 Ɇɚɬɪɢɰɚ A ɥɢɛɨ ɧɟɹɜɧɨ ɨɛɪɚɳɚɟɬɫɹ ɥɢɛɨ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟ ɧɢɹ ɦɚɬɪɢɰ ɭɞɨɛɧɵɯ ɞɥɹ ɨɛɪɚɳɟɧɢɹ
ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɦɚɬɪɢɰɚ Aɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɷɥɟɦɟɧɬɚɪɧɵɯ ɷɤɜɢɜɚɥɟɧɬɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ
1.ɉɟɪɟɫɬɚɧɨɜɤɚ ɫɬɨɥɛɰɨɜ ɢ ɫɬɪɨɤ
2.ɍɦɧɨɠɟɧɢɟ ɫɬɨɥɛɰɨɜ ɢ ɫɬɪɨɤ ɧɚ ɱɢɫɥɨ
3.ɉɪɢɛɚɜɥɟɧɢɟ ɤ ɫɬɪɨɤɟ ɫɬɨɥɛɰɭ ɞɪɭɝɨɣ ɫɬɪɨɤɢ ɭɦɧɨɠɟɧɧɨɣ
ɧɚ ɱɢɫɥɨ Ʉɚɠɞɨɟ ɷɥɟɦɟɧɬɚɪɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ
ɜɢɞɟ ɦɚɬɪɢɰɵ Li ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɭɦɧɨɠɟɧɢɹ A ɧɚ Li ɨɧɚ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ ɟɞɢɧɢɱɧɭɸ ɦɚɬɪɢɰɭ
Ln ..L2 L1A x Ln ..L1b .
Ɇɟɬɨɞ Ƚɚɭɫɫɚ ɦɟɬɨɞ ɢɫɤɥɸɱɟɧɢɣ
Ɏɨɪɦɚɥɶɧɨ ɦɟɬɨɞ Ƚɚɭɫɫɚ ɨɫɧɨɜɚɧ ɧɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɦ ɩɪɢ ɦɟɧɟɧɢɢ ɦɚɬɪɢɰ
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ɉɪɢɦɟɪ ɞɥɹ ɦɚɬɪɢɰɵ u3): |
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Ⱦɟɣɫɬɜɢɟ ɦɚɬɪɢɰɵ Li |
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ɩɪɟɨɛɪɚɡɭɸɬ ɷɥɟɦɟɧɬɵ i -ɝɨ ɫɬɨɥɛɰɚ |
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ɦɚɬɪɢɰɵ A ɧɢɠɟ ɞɢɚɝɨɧɚɥɢ ɜ ɧɭɥɟɜɵɟ ɬ ɟ ɢɫɤɥɸɱɚɸɬ ɢɯ
ȼɵɱɢɫɥɢɬɟɥɶɧɚɹ ɫɯɟɦɚ ɦɟɬɨɞɚ Ƚɚɭɫɫɚ
ȼ ɤɚɠɞɨɦ ɭɪɚɜɧɟɧɢɢ ɜɵɞɟɥɹɟɬɫɹ ɜɟɞɭɳɢɣ ɷɥɟɦɟɧɬ ɧɚ ɤɨɬɨ ɪɵɣ ɩɪɨɢɡɜɨɞɢɬɫɹ ɞɟɥɟɧɢɟ ɩɭɫɬɶ ɷɬɨ ɛɭɞɟɬ a11 Ⱦɟɥɢɦ ɩɟɪɜɨɟ
ɭɪɚɜɧɟɧɢɟ ɧɚ a11:
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ȼɫɟ ɨɫɬɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɩɨ ɫɯɟɦɟ
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ɇɚ ɜɬɨɪɨɦ ɲɚɝɟ ɜɟɞɭɳɢɦ ɷɥɟɦɟɧɬɨɦ ɜɵɛɢɪɚɟɬɫɹ a22(1) ɧɚ ɧɟɝɨ
ɞɟɥɢɬɫɹ ɜɬɨɪɚɹ ɫɬɪɨɤɚ ɚ ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ
37
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ɗɥɟɦɟɧɬɵ ɜɨ ɜɬɨɪɨɦ ɫɬɨɥɛɰɟ ɫ i ! ɫɬɚɧɨɜɹɬɫɹ ɪɚɜɧɵ ȼ ɪɟ ɡɭɥɶɬɚɬɟ ɬɚɤɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɜɟɪɯɧɟɣ ɬɪɟɭɝɨɥɶ ɧɨɣ ɦɚɬɪɢɰɟ ɫ ɟɞɢɧɢɱɧɨɣ ɞɢɚɝɨɧɚɥɶɸ
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ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɤ ɜɟɪɯɧɟɣ ɬɪɟɭɝɨɥɶɧɨɣ ɦɚɬɪɢɰɟ ɧɚɡɵɜɚɟɬɫɹ
ɩɪɹɦɵɦ ɯɨɞɨɦ |
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xn ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ |
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Ⱦɚɥɟɟ ɫɥɟɞɭɟɬ ɨɛɪɚɬɧɵɣ ɯɨɞ ɧɚɱɢɧɚɹ ɫ |
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ɜɵɱɢɫɥɹɸɬɫɹ ɤɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ |
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xn |
gn ; xn 1 gn 1 cn 1, n xn ; |
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xk |
gk |
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¦ cki xi , k n, n 1, |
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(5.2.2.3) |
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k 1 |
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ȼ ɦɚɲɢɧɧɵɯ ɪɚɫɱɟɬɚɯ ɜ ɤɚɱɟɫɬɜɟ ɜɟɞɭɳɟɝɨ ɷɥɟɦɟɧɬɚ ɨɛɵɱɧɨ ɜɵɛɢɪɚɟɬɫɹ ɦɚɤɫɢɦɚɥɶɧɵɣ ɷɥɟɦɟɧɬ i -ɝɨ ɫɬɨɥɛɰɚ ɫ j ! i ɢɥɢ ɫɬɪɨɤɢ
aij ɫ i ! j .
ɗɬɚ ɫɬɪɨɤɚ ɢɥɢ ɫɬɨɥɛɟɰ ɩɟɪɟɫɬɚɜɥɹɸɬɫɹ ɧɚ ɦɟɫɬɨ i -ɣ ɫɬɪɨɤɢɫɬɨɥɛɰɚ Ɍɚɤɨɣ ɜɵɛɨɪ ɭɦɟɧɶɲɚɟɬ ɨɲɢɛɤɢ ɨɤɪɭɝɥɟɧɢɹ ɉɪɢ ɪɭɱ ɧɵɯ ɪɚɫɱɟɬɚɯ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ ɡɚɩɢɫɵɜɚɸɬɫɹ ɜɦɟɫɬɟ ɫ ɷɥɟɦɟɧ
ɬɚɦɢ ɜɟɤɬɨɪɚ b ɜ ɪɚɫɲɢɪɟɧɧɭɸ ɦɚɬɪɢɰɭ
38
§ a11 |
a12 |
a13 |
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b1 |
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¨ a |
a |
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¸ |
, |
¨ 21 |
22 |
23 |
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¨ a |
a |
a |
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b |
¸ |
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© 31 |
32 |
33 |
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3 |
¹ |
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ɞɚɥɟɟ ɢɡ ɫɨɨɛɪɚɠɟɧɢɣ ɭɞɨɛɫɬɜɚ ɜɵɛɢɪɚɸɬ ɜɟɞɭɳɢɣ ɷɥɟɦɟɧɬ ɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɨɫɬɚɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ ɧɚ ɨɞɧɨɦ ɲɚɝɟ ɩɪɹɦɨɝɨ ɯɨɞɚ ɦɟɬɨɞɚ Ƚɚɭɫɫɚ ɩɪɨɜɨɞɹɬ ɩɨ ɩɪɚɜɢɥɭ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ ȼ ɦɚɬɪɢɰɟ ɜɵɞɟɥɹɟɬɫɹ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɧɚ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɤɨɬɨɪɨɝɨ ɪɚɫɩɨ ɥɨɠɟɧɵ ɜɟɞɭɳɢɣ ɢ ɩɪɟɨɛɪɚɡɭɟɦɵɣ ɷɥɟɦɟɧɬɵ
ɉɭɫɬɶ aii — ɜɟɞɭɳɢɣ ɷɥɟɦɟɧɬ ɬɨɝɞɚ
a(1) |
a |
aki ait |
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(5.2.2.4) |
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aii |
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ɂɡ ɩɪɟɨɛɪɚɡɭɟɦɨɝɨ ɷɥɟɦɟɧɬɚ ɜɵɱɢɬɚɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɷɥɟ ɦɟɧɬɨɜ ɫɬɨɹɳɢɯ ɧɚ ɩɨɛɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢ ɞɟɥɟɧɧɨɟ ɧɚ ɜɟɞɭɳɢɣ ɷɥɟɦɟɧɬ
Ɉɪɬɨɝɨɧɚɥɢɡɚɰɢɹ ɦɚɬɪɢɰ |
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Ɇɚɬɪɢɰɚ ɧɚɡɵɜɚɟɬɫɹ ɨɪɬɨɝɨɧɚɥɶɧɨɣ ɟɫɥɢ A AT |
D , ɝɞɟ D – |
ɞɢɚɝɨɧɚɥɶɧɚɹ ɦɚɬɪɢɰɚ ɬ ɟ ɜ ɧɟɣ ɨɬɥɢɱɧɵ ɨɬ ɧɭɥɹ ɬɨɥɶɤɨ ɞɢɚɝɨ
ɧɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɟɫɥɢ A AT |
E ɬɨ A – ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɧɚɹ |
ɦɚɬɪɢɰɚ Ʌɸɛɚɹ ɧɟɨɫɨɛɟɧɧɚɹ ɦɚɬɪɢɰɚ A ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ |
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ɜ ɜɢɞɟ: A R T , R – ɨɪɬɨɝɨɧɚɥɶɧɚɹ ɚ T – ɜɟɪɯɧɹɹ ɬɪɟɭɝɨɥɶɧɚɹ |
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ɦɚɬɪɢɰɚ ɫ ɟɞɢɧɢɱɧɨɣ ɞɢɚɝɨɧɚɥɶɸ |
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Ɋɚɫɫɦɨɬɪɢɦ ɦɚɬɪɢɰɭ Ⱥ |
ɤɚɤ |
ɧɚɛɨɪ |
ɜɟɤɬɨɪ-ɫɬɨɥɛɰɨɜ |
ai , |
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A |
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ɜɟɤɬɨɪɚ ai |
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ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ ɬɚɤ ɤɚɤ |
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[a1 |
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a2 |
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an ] – |
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det A z 0. |
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ȼɵɛɟɪɟɦ ɩɟɪɜɵɣ ɫɬɨɥɛɟɰ ɦɚɬɪɢɰɵ R – r1 ɪɚɜɧɵɦ a1; |
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ri |
a1. |
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Ɂɚɩɢɲɟɦ a2 |
t12r1 r2 , |
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ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ |
R |
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r1, r2 |
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0 ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ t12 : |
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r , a |
t r , r , t |
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ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɢɡɜɟɫɬɟɧ ɢ ɜɟɤɬɨɪ r2 |
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a2 t12r1 Ⱥɧɚɥɨɝɢɱɧɵɦ ɨɛ |
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ɪɚɡɨɦ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɢ a3 |
t13 r1 t23 r2 r3 ɝɞɟ |
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39
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r1, a3 |
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ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɹ
G |
ak |
k 1 |
ri , ak |
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(5.2.3.1) |
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¦ tik ri , tik |
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ɉɨɤɚɠɟɦ ɱɬɨ tik – ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ Ɍ Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ
a1 |
r1; |
G |
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t12 |
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a2 |
r1 |
r2 |
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t13 r1 |
t23 r2 |
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ɢɥɢ ɢɧɚɱɟ
ªa11 a12 a13 ««a21 a22 a23
««a31 a32 a33
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ªr11 |
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« 21 |
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«r |
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« 31 |
¼ |
¬ . |
r12 |
r13 |
º |
ª |
1 |
t12 |
t13 |
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r |
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t |
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« 0 |
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32 |
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¼ |
¬ . |
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Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ ɨɪɬɨɝɨɧɚɥɢɡɚɰɢɢ
Ɉɩɬɢɦɚɥɶɧɨɣ ɹɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɚɹ ɫɯɟɦɚ ɨɫɧɨɜɚɧɧɚɹ ɧɚ ɫɜɨɣ
ɫɬɜɚɯ ɜɟɤɬɨɪɚ |
r . |
Ɂɚɩɢɲɟɦ |
ɫɢɫɬɟɦɭ |
G |
b |
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A x |
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G |
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x2 |
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b . ɂɡ ɫɬɪɭɤɬɭɪɵ ɜɟɤɬɨɪɨɜ r |
ɫɥɟɞɭɟɬ |
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a1 x1 |
a2 |
an xn |
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G |
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0, (i > j). ɍɦɧɨɠɚɟɦ ɫɢɫɬɟɦɭ ɫɥɟɜɚ ɧɚ rn : |
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ɱɬɨ ri |
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G G |
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rn a1 |
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an xn |
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ɜ ɭɪɚɜɧɟɧɢɢ ɨɫɬɚɟɬɫɹ ɨɞɧɨ ɫɥɚɝɚɟɦɨɟ rn |
an xn |
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rGn ,b |
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G(1) |
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an 1 |
xn 1 |
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ɉɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɦɧɨɠɢɦ ɧɚ rn 1 ɧɚɯɨɞɢɦ xn 1 |
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G 2 |
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ɥɹɟɦ b |
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xn i 1 an i 1. |
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ɢ ɜɵɱɢɫ
(5.2.4.1)
ɂɬɟɪɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɋɅȺɍ
Ɇɟɬɨɞ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ
Ɇɧɨɝɢɟ ɢɬɟɪɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɦɨɝɭɬ ɛɵɬɶ ɫɜɟɞɟɧɵ ɤ ɩɪɨɰɟɫɫɭ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɉɪɢ ɷɬɨɦ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɦ ɢɥɢ ɢɧɵɦ ɫɩɨɫɨɛɨɦ ɞɨɥɠɧɨ ɛɵɬɶ ɫɜɟɞɟɧɨ ɤ ɭɪɚɜɧɟɧɢɸ
G |
G |
b . |
(5.3.1.1) |
x |
Bx |
Ɂɞɟɫɶ x – ɧɟɢɡɜɟɫɬɧɵɣ ɜɟɤɬɨɪ b – ɡɚɞɚɧɧɵɣ ɜɟɤɬɨɪ ɩɪɚɜɨɣ ɱɚɫɬɢ B – ɡɚɞɚɧɧɚɹ ɦɚɬɪɢɰɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɨɩɟɪɚɬɨɪ ɇɚɩɪɢ ɦɟɪ ɟɫɥɢ ɡɚɞɚɧɚ ɋɅȺɍ ɬɨ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɪɢɧɢɦɚɹ
B I A, |
(5.3.1.2) |
ɝɞɟ I – ɟɞɢɧɢɱɧɚɹ ɦɚɬɪɢɰɚ ɩɪɢɯɨɞɢɦ ɤ ɉɪɨɰɟɫɫ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɫɬɪɨɢɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
G x
(k 1) |
G(k ) |
b , |
k 0,1, 2,... . |
(5.3.1.3) |
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Bx |
ȼ |
ɤɚɱɟɫɬɜɟ ɧɚɱɚɥɶɧɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ |
G(o) |
ɦɨɠɧɨ ɩɪɢɧɹɬɶ |
x |
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G(o) |
b . |
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Ɂɚɦɟɬɢɦ ɱɬɨ ɩɟɪɟɯɨɞ ɨɬ ɤ ɦɨɠɟɬ ɛɵɬɶ ɜɵɩɨɥɧɟɧ ɧɟ ɟɞɢɧɫɬɜɟɧɧɵɦ ɫɩɨɫɨɛɨɦ ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɪɚɡɥɢɱɧɵɦ ɦɨɞɢɮɢ ɤɚɰɢɹɦ ɦɟɬɨɞɚ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ Ɍɚɤ ɦɟɬɨɞ ɫ ɩɪɟɨɛɪɚ ɡɨɜɚɧɢɟɦ ɢɡɜɟɫɬɟɧ ɜ ɥɢɬɟɪɚɬɭɪɟ ɤɚɤ ɦɟɬɨɞ Ɋɢɱɚɪɞɫɨɧɚ