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Ɋɟɲɟɧɢɟ. ɉɪɟɞɫɬɚɜɢɦ ɢɫɯɨɞɧɭɸ ɫɢɫɬɟɦɭ ɜ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɟ
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61
ɇɚɣɞɟɦ ɦɚɬɪɢɰɭ, ɨɛɪɚɬɧɭɸ ɦɚɬɪɢɰɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɫɢɫɬɟɦɵ
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ȼ ɜɟɤɬɨɪɧɨɣ ɮɨɪɦɟ ɧɚɯɨɞɢɦ ɫɬɨɥɛɟɰ ɩɟɪɟɦɟɧɧɵɯ
Ɂɚɞɚɱɚ 3.2. Ɋɟɲɢɬɶ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ ɩɪɨɫɬɵɯ ɢɬɟ-
ɪɚɰɢɣ
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Ɂɚɞɚɧɧɚɹ ɬɨɱɧɨɫɬɶ İ = 0,1.
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ɉɪɨɞɨɥɠɚɟɦ ɢɬɟɪɚɰɢɢ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɧɟ ɜɵɩɨɥɧɢɬɫɹ ɭɫɥɨɜɢɟ
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Ȼɥɨɤ-ɫɯɟɦɚ ɚɥɝɨɪɢɬɦɚ ɪɚɛɨɬɵ ɩɪɢɥɨɠɟɧɢɹ ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɛɥɨɤ-ɫɯɟɦɟ
(ɪɢɫ. 19)
Ɋɢɫ. 19. Ȼɥɨɤ-ɫɯɟɦɚ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɢɬɟɪɚɰɢɣ
ɉɪɢɦɟɪ ɤɨɞɚ ɩɪɢɥɨɠɟɧɢɹ
float e,x10,x20,x30,x11,x21,x31;
x10 = 2;
x20 = 3;
x30 = 5;
e = 0.01;
63
while ((fabs(x11-x10)>e) && (fabs(x21-x20)>e) && (fabs(x31-x30)>e))
{
x11 = 2-0.06*x20+0.02*x30;
x21 = 3-0.03*x10+0.05*x30;
x31 = 5-0.01*x10+0.02*x20;
x10 = x11;
x20 = x21;
x30 = x31;
};
Edit1->Text = FloatToStr(x10);
Edit2->Text = FloatToStr(x20);
Edit3->Text = FloatToStr(x30);
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
ɑɬɨ ɬɚɤɨɟ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ?
1.
Ʉɚɤ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɨɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ?
2.
Ʉɚɤ ɧɚɯɨɞɢɬɫɹ ɦɢɧɨɪ ɷɥɟɦɟɧɬɚ ɦɚɬɪɢɰɵ?
3.
Ʉɚɤ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɞɜɭɯ ɦɚɬɪɢɰ?
4.
Ʉɚɤɨɜɵ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɦɚɬɪɢɰ ɞɥɹ ɪɟɲɟɧɢɹ ɫɢ-
5.
ɫɬɟɦ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ?
Ʉɚɤɨɜɵ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɢɬɟɪɚɰɢɣ ɞɥɹ ɪɟɲɟɧɢɹ ɫɢ-
6.
ɫɬɟɦ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ?
64
ȽɅȺȼȺ 4
ɈɉɌɂɆɂɁȺɐɂə ɐȿɅȿȼɕɏ ɎɍɇɄɐɂɃ
Ɂɧɚɧɢɟ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ ɞɥɹ ɬɟɯɧɢɱɟɫɤɢɯ ɫɩɟɰɢɚɥɢɫɬɨɜ ɧɟɨɛɯɨɞɢɦɨ
ɬɚɤ ɠɟ, ɤɚɤ ɡɧɚɧɢɟ ɦɚɬɟɦɚɬɢɤɢ, ɮɢɡɢɤɢ, ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ ɢ ɞɪɭɝɢɯ
ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɧɚɭɤ. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɬɟɨɪɢɹ ɨɩɬɢɦɢɡɚɰɢɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɨɜɨɤɭɩɧɨɫɬɶ ɱɢɫɥɟɧɧɵɯ ɦɟɬɨɞɨɜ, ɩɨɡɜɨɥɹɸɳɢɯ ɧɚɣɬɢ ɧɚɢɥɭɱɲɢɣ ɜɚɪɢɚɧɬ
ɢɡ ɦɧɨɠɟɫɬɜɚ ɜɨɡɦɨɠɧɵɯ. ɉɪɨɰɟɫɫ ɨɩɬɢɦɢɡɚɰɢɢ ɥɟɠɢɬ ɜ ɨɫɧɨɜɟ ɩɪɨɮɟɫɫɢɨɧɚɥɶɧɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ ɬɟɯɧɢɱɟɫɤɨɝɨ ɫɩɟɰɢɚɥɢɫɬɚ, ɬɚɤ ɤɚɤ ɬɪɟɛɭɟɬɫɹ
ɪɨɜɚɬɶ ɛɨɥɟɟ ɷɮɮɟɤɬɢɜɧɵɟ, ɧɨ ɩɪɢ ɷɬɨɦ ɦɟɧɟɟ ɞɨɪɨɝɨɫɬɨɹɳɢɟ ɫɢɫɬɟɦɵ.
4.1. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ
Ɉɫɧɨɜɧɚɹ ɰɟɥɶ ɥɸɛɨɝɨ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ – ɨɛɟɫɩɟɱɟɧɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɤɚɱɟɫɬɜɚ ɢɡɞɟɥɢɹ ɧɚɢɛɨɥɟɟ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɵɦ ɩɭɬɟɦ ɩɪɢ ɦɢɧɢɦɚɥɶɧɵɯ ɡɚɬɪɚɬɚɯ. ȼɵɛɨɪ ɧɚɢɥɭɱɲɟɝɨ ɜɚɪɢɚɧɬɚ ɢɡ ɦɧɨɠɟɫɬɜɚ ɜɨɡɦɨɠɧɵɯ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ.
ɉɪɢ ɤɨɦɩɥɟɤɫɧɨɦ ɩɨɞɯɨɞɟ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱ ɨɩɬɢɦɢɡɚɰɢɢ ɧɟɨɛɯɨɞɢɦɨ
ɭɱɢɬɵɜɚɬɶ ɞɜɚ ɜɢɞɚ ɨɩɬɢɦɢɡɚɰɢɢ: ɫɬɪɭɤɬɭɪɧɭɸ ɢ ɩɚɪɚɦɟɬɪɢɱɟɫɤɭɸ. ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɩɟɪɜɨɦɭ ɜɢɞɭ ɨɩɬɢɦɢɡɚɰɢɢ ɨɩɪɟɞɟɥɹɸɬ
ɰɟɫɫɚ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɟɪɟɯɨɞɨɜ, ɨɩɟɪɚɰɢɣ ɢ ɬ. ɞ. ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɨɩɪɟɞɟɥɹɸɬ ɤɨɥɢɱɟɫɬɜɟɧɧɵɟ ɩɚɪɚɦɟɬɪɵ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ.
ɉɪɨɢɡɜɨɞɢɬɶ ɨɩɬɢɦɢɡɚɰɢɸ ɦɨɠɧɨ ɩɨ ɪɚɡɥɢɱɧɵɦ ɤɪɢɬɟɪɢɹɦ, ɨɞɧɚɤɨ ɜɫɟ
ɨɧɢ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɨɩɪɟɞɟɥɟɧɧɵɦ ɬɪɟɛɨɜɚɧɢɹɦ:
ɨɛɥɚɞɚɬɶ ɞɨɫɬɚɬɨɱɧɨɣ ɩɨɥɧɨɬɨɣ ɨɩɢɫɚɧɢɹ ɨɛɴɟɤɬɚ;
1)
ɢɦɟɬɶ ɨɩɪɟɞɟɥɟɧɧɵɣ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ;
2)
ɛɵɬɶ ɤɨɥɢɱɟɫɬɜɟɧɧɵɦɢ ɢ ɜɵɪɚɠɚɬɶɫɹ ɨɞɧɨɡɧɚɱɧɨ ɧɟɤɨɬɨɪɵɦ ɱɢɫɥɨɦ;
3)
ɢɦɟɬɶ ɩɪɨɫɬɨɣ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɜɢɞ;
4)
ɨɩɪɟɞɟɥɹɬɶɫɹ ɫ ɞɨɩɭɫɬɢɦɨɣ ɬɨɱɧɨɫɬɶɸ.
5)
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɢɞɚ ɢ ɭɪɨɜɧɹ ɡɚɞɚɱ ɨɩɬɢɦɢɡɚɰɢɢ (ɪɚɫɱɟɬ ɪɟɠɢɦɨɜ ɪɟɡɚɧɢɹ, ɩɪɨɟɤɬɢɪɨɜɚɧɢɟ ɨɩɟɪɚɰɢɢ ɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɢɥɢ ɨɰɟɧɤɚ ɪɚɛɨɬɵ ɩɪɟɞɩɪɢɹɬɢɹ ɜ ɰɟɥɨɦ) ɨɫɧɨɜɧɵɟ ɩɪɢɦɟɧɹɟɦɵɟ ɤɪɢɬɟɪɢɢ ɨɩɬɢɦɚɥɶɧɨɫɬɢ
ɦɨɠɧɨ ɩɨɞɪɚɡɞɟɥɢɬɶ ɧɚ ɫɥɟɞɭɸɳɢɟ ɜɢɞɵ [3, 17–20, 28].
1. ɋɬɨɢɦɨɫɬɧɵɟ (ɷɤɨɧɨɦɢɱɟɫɤɢɟ): ɦɢɧɢɦɚɥɶɧɚɹ ɫɟɛɟɫɬɨɢɦɨɫɬɶ; ɧɚɢɦɟɧɶɲɢɟ ɧɚɪɨɞɧɨɯɨɡɹɣɫɬɜɟɧɧɵɟ ɩɪɢɜɟɞɟɧɧɵɟ ɡɚɬɪɚɬɵ; ɧɚɢɦɟɧɶɲɢɟ ɩɪɢɜɟɞɟɧɧɵɟ
ɯɨɡɪɚɫɱɟɬɧɵɟ ɡɚɬɪɚɬɵ;
ɭɪɨɜɟɧɶ ɡɚɬɪɚɬ ɧɚ ɩɪɨɢɡɜɨɞɫɬɜɨ (ɦɢɧɢɦɚɥɶɧɵɟ ɡɚɬɪɚɬɵ ɧɚ ɷɥɟɤɬɪɢɱɟɫɤɭɸ ɢ
ɞɪɭɝɢɟ ɜɢɞɵ ɷɧɟɪɝɢɢ, ɧɚ ɨɫɧɨɜɧɵɟ ɢ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɟ ɦɚɬɟɪɢɚɥɵ, ɧɚ ɮɨɧɞ ɡɚɪɚɛɨɬɧɨɣ ɩɥɚɬɵ ɢ ɞɪ.).
2. Ɏɭɧɤɰɢɨɧɚɥɶɧɵɟ (ɬɟɯɧɢɤɨ-ɷɤɨɧɨɦɢɱɟɫɤɢɟ): ɦɚɤɫɢɦɚɥɶɧɚɹ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ; ɧɚɢɦɟɧɶɲɟɟ ɲɬɭɱɧɨɟ ɜɪɟɦɹ; ɨɫɧɨɜɧɨɟ ɢ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɟ ɜɪɟɦɹ; ɤɨɷɮɮɢɰɢɟɧɬ ɩɨɥɟɡɧɨɝɨ ɞɟɣɫɬɜɢɹ ɨɛɨɪɭɞɨɜɚɧɢɹ; ɧɚɞɟɠɧɨɫɬɶ ɪɚɛɨɬɵ ɫɢɫɬɟɦɵ
ɧɚɢɛɨɥɶɲɚɹ ɩɪɢɛɵɥɶ; ɪɟɧɬɚɛɟɥɶɧɨɫɬɶ; ɦɢɧɢɦɚɥɶɧɵɣ
ɤɚɱɟɫɬɜɟɧɧɵɟ ɩɚɪɚɦɟɬɪɵ ɩɪɨ-
ɫɩɪɨɟɤɬɢ-
65
ɨɛɨɪɭɞɨɜɚɧɢɹ ɢɥɢ ɨɬɞɟɥɶɧɵɯ ɟɟ ɷɥɟɦɟɧɬɨɜ; ɫɬɚɧɤɨɟɦɤɨɫɬɶ ɢɡɞɟɥɢɹ; ɫɬɚɛɢɥɶɧɨɫɬɶ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɨɛɪɚɛɨɬɤɢ.
3. Ɍɟɯɧɨɥɨɝɢɱɟɫɤɢɟ: ɬɨɱɧɨɫɬɶ ɢɡɝɨɬɨɜɥɟɧɢɹ ɢɡɞɟɥɢɹ, ɩɨɤɚɡɚɬɟɥɢ ɤɚɱɟɫɬɜɚ
ɩɨɜɟɪɯɧɨɫɬɢ ɢɡɞɟɥɢɹ (ɲɟɪɨɯɨɜɚɬɨɫɬɶ, ɜɨɥɧɢɫɬɨɫɬɶ, ɦɢɤɪɨɬɜɟɪɞɨɫɬɶ, ɨɫɬɚɬɨɱɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɢ ɞɪ.); ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɢɡɞɟɥɢɣ; ɫɬɨɣɤɨɫɬɶ ɢɧɫɬɪɭɦɟɧɬɚ.
4. ɗɤɫɩɥɭɚɬɚɰɢɨɧɧɵɟ: ɢɡɧɨɫɨɫɬɨɣɤɨɫɬɶ; ɭɫɬɚɥɨɫɬɧɚɹ ɩɪɨɱɧɨɫɬɶ; ɤɨɧɬɚɤɬɧɚɹ ɠɟɫɬɤɨɫɬɶ ɢ ɞɪɭɝɢɟ ɩɨɤɚɡɚɬɟɥɢ ɞɨɥɝɨɜɟɱɧɨɫɬɢ ɢɡɞɟɥɢɣ.
5. ɉɪɨɱɢɟ: ɩɫɢɯɨɥɨɝɢɱɟɫɤɢɟ
; ɷɫɬɟɬɢɱɟɫɤɢɟ, ɷɪɝɨɧɨɦɢɱɟɫɤɢɟ.
ɇɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɵ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɨɩɬɢɦɢɡɚɰɢɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɷɤɨɧɨɦɢɱɟɫɤɢɟ ɢ ɬɟɯɧɢɤɨ-ɷɤɨɧɨɦɢɱɟɫɤɢɟ ɤɪɢɬɟɪɢɢ ɨɩɬɢɦɚɥɶɧɨɫɬɢ. ɗɬɨ ɫɜɹɡɚɧɨ ɫ ɬɟɦ, ɱɬɨ ɜ ɨɫɧɨɜɟ ɪɚɡɪɚɛɨɬɤɢ ɥɸɛɨɝɨ Ɍɉ ɢɥɢ ɪɟɲɟɧɢɹ ɛɨɥɟɟ ɱɚɫɬɧɨɣ ɡɚɞɚɱɢ, ɧɚɩɪɢɦɟɪ ɪɚɫɱɟɬɚ ɪɟɠɢɦɨɜ ɪɟɡɚɧɢɹ, ɥɟɠɚɬ ɞɜɚ ɩɪɢɧɰɢɩɚ: ɬɟɯɧɢɱɟɫɤɢɣ ɢ ɷɤɨɧɨɦɢɱɟɫɤɢɣ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɟɪɜɵɦ ɩɪɢɧɰɢɩɨɦ
ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ ɞɨɥɠɟɧ ɝɚɪɚɧɬɢɪɨɜɚɬɶ ɜɵɩɨɥɧɟɧɢɟ ɜɫɟɯ ɬɪɟɛɨɜɚɧɢɣ ɧɚ
ɢɡɝɨɬɨɜɥɟɧɢɟ ɢɡɞɟɥɢɹ. ȼɬɨɪɨɣ ɩɪɢɧɰɢɩ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɫɥɨɜɢɹɦɢ, ɤɨɬɨɪɵɟ ɨɛɟɫɩɟɱɢɜɚɸɬ ɦɢɧɢɦɚɥɶɧɵɟ ɡɚɬɪɚɬɵ ɬɪɭɞɚ ɢ ɧɚɢɦɟɧɶɲɢɟ ɢɡɞɟɪɠɤɢ ɩɪɨɢɡɜɨɞɫɬɜɚ.
ɉɟɪɜɵɣ ɩɪɢɧɰɢɩ ɧɚɢɛɨɥɟɟ ɩɨɥɧɨ ɨɬɪɚɠɚɟɬɫɹ ɦɢɧɢɦɚɥɶɧɨɣ ɫɟɛɟɫɬɨɢɦɨɫɬɶɸ ɢɡ
ɝɪɭɩɩɵ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɤɪɢɬɟɪɢɟɜ, ɚ ɜɬɨɪɨɣ – ɦɚɤɫɢɦɚɥɶɧɨɣ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶɸ ɢɡ ɝɪɭɩɩɵ ɬɟɯɧɢɤɨ-ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɤɪɢɬɟɪɢɟɜ.
)x(f
ȼɫɟ ɡɚɞɚɱɢ ɨɩɬɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɢ
ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɩɨ ɪɚɡɥɢɱɧɵɦ
ɤɪɢɬɟɪɢɹɦ [3, 14, 17–20, 28]:
ɚ) ɩɨ ɧɚɥɢɱɢɸ ɨɝɪɚɧɢɱɟɧɢɣ
– ɡɚɞɚɱɢ ɫ ɨɝɪɚɧɢɱɟɧɢɹɦɢ (ɭɫɥɨɜɧɚɹ ɨɩɬɢɦɢɡɚɰɢɹ);
min)x(f → ,
=
,0)x(h
≥
,0)x(g
<<
.2xx1x
– ɡɚɞɚɱɢ ɛɟɡ ɨɝɪɚɧɢɱɟɧɢɣ (ɛɟɡɭɫɥɨɜɧɚɹ ɨɩɬɢɦɢɡɚɰɢɹ);
min)x(f∈→
;Enx
ɛ) ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ ɜɟɤɬɨɪɚ ɯ:
– ɡɚɞɚɱɢ ɫ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ (ɨɞɧɨɦɟɪɧɵɣ ɜɟɤɬɨɪ ɯ);
– ɡɚɞɚɱɢ ɫ ɧɟɫɤɨɥɶɤɢɦɢ ɩɟɪɟɦɟɧɧɵɦɢ ɯ;
ɜ) ɩɨ ɤɨɥɢɱɟɫɬɜɭ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ:
– ɨɞɧɨɤɪɢɬɟɪɢɚɥɶɧɵɟ ɡɚɞɚɱɢ;
– ɦɧɨɝɨɤɪɢɬɟɪɢɚɥɶɧɵɟ ɡɚɞɚɱɢ.
Ɋɚɡɥɢɱɧɵɟ ɦɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ ɩɪɢɦɟɧɹɸɬɫɹ ɞɥɹ ɪɚɡɧɵɯ ɜɢɞɨɜ ɭɫɥɨɜɢɣ
ɡɚɞɚɱɢ (ɬɚɛɥ. 8–10) [3, 14, 17–20, 28].
66
ɭ
ɭ
ɭ
Ɇɟɬɨɞɵ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɛɟɡɭɫɥɨɜɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ
ɮɭɧɤɰɢɢ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ
Ɇɟɬɨɞ ɨɩɬɢɦɢɡɚɰɢɢ Ɋɟɲɚɟɦɵɟ ɡɚɞɚɱɢ
1. Ʉɥɚɫɫɢɱɟɫɤɢɣ ɦɟɬɨɞ
2. Ɇɟɬɨɞɵ ɢɫɤɥɸɱɟɧɢɹ ɢɧɬɟɪɜɚɥɨɜ
(ɨɞɧɨɦɟɪɧɵɟ ɦɟɬɨɞɵ ɩɨɢɫɤɚ):
1)
ɦɟɬɨɞ ɩɟɪɟɛɨɪɚ;
2)
ɦɟɬɨɞ ɞɢɯɨɬɨɦɢɢ (ɞɟɥɟɧɢɟ ɨɬɪɟɡɤɚ
ɩɨɩɨɥɚɦ);
3)
ɦɟɬɨɞ Ɏɢɛɨɧɚɱɱɢ;
4)
ɦɟɬɨɞ ɡɨɥɨɬɨɝɨ ɫɟɱɟɧɢɹ
3. Ɇɟɬɨɞɵ ɩɨɥɢɧɨɦɢɚɥɶɧɨɣ
ɚɩɩɪɨɤɫɢɦɚɰɢɢ:
5)
ɩɪɢɦɟɧɟɧɢɟ ɤɜɚɞɪɚɬɢɱɧɨɣ
ɚɩɩɪɨɤɫɢɦɚɰɢɢ;
6)
ɦɟɬɨɞ ɉɚ
ɷɥɥɚ
Ɇɟɬɨɞɵ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɭɫɥɨɜɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ
Ɇɟɬɨɞ ɨɩɬɢɦɢɡɚɰɢɢ Ɋɟɲɚɟɦɵɟ ɡɚɞɚɱɢ
Ɍɚɛɥɢɰɚ 8
Ⱦɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɟ ɡɚɞɚɱɢ, ɡɚɞɚɧɧɵɟ
ɞɢɮɮɟɪɟɧɰɢɪ
ɟɦɨɣ ɮɭɧɤɰɢɟɣ
Ⱦɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɟ
ɭɧɢɦɨɞɚɥɶɧɵɟ ɮɭɧɤɰɢɢ,
ɤɚɤ ɧɟɩɪɟɪɵɜɧɵɟ, ɬɚɤ ɢ ɪɚɡɪɵɜɧɵɟ
Ⱦɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɟ
ɭɧɢɦɨɞɚɥɶɧɵɟ ɮɭɧɤɰɢɢ,
ɧɟɩɪɟɪɵɜɧɵɟ
Ɍɚɛɥɢɰɚ 9
1. Ɇɟɬɨɞɵ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɐɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɥɢɧɟɣɧɚ
2. Ɇɟɬɨɞɵ ɲɬɪɚɮɧɵɯ ɮɭɧɤɰɢɣ:
1)
ɦɟɬɨɞ ɦɧɨɠɢɬɟɥɟɣ Ʌɚɝɪɚɧɠɚ;
2)
ɦɟɬɨɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɣ
ɛɟɡɭɫɥɨɜɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ;
3)
ɦɟɬɨɞ ɛɚɪɶɟɪɧɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ
3. Ɇɟɬɨɞɵ ɩɨɢɫɤɚ:
1)
ɦɟɬɨɞ Ȼɨɤɫɚ;
2)
ɦɟɬɨɞ ɏɨɥɬɨɧɚ
ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ, ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ
Ɂɚɞɚɱɢ ɧɟɥɢɧɟɣɧɨɝɨ
ɭɫɥɨɜɧɨɣ ɡɚɞɚɱɢ ɨɩɬɢɦɢɡɚɰɢɢ
ɜ ɛɟɡɭɫɥɨɜɧɭɸ
Ɂɚɞɚɱɢ ɧɟɥɢɧɟɣɧɨɝɨ
ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ
Ɇɟɬɨɞɵ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɛɟɡɭɫɥɨɜɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɢ
ɦɧɨɝɢɯ ɩɟɪɟɦɟɧɧɵɯ
Ɇɟɬɨɞ ɨɩɬɢɦɢɡɚɰɢɢ Ɋɟɲɚɟɦɵɟ ɡɚɞɚɱɢ
1. Ʉɥɚɫɫɢɱɟɫɤɢɣ ɦɟɬɨɞ
Ⱦɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɟ ɡɚɞɚɱɢ, ɡɚɞɚɧɧɵɟ
ɞɢɮɮɟɪɟɧɰɢɪ
ɟɦɨɣ ɮɭɧɤɰɢɟɣ
67
Ɍɚɛɥɢɰɚ 10
Ɉɤɨɧɱɚɧɢɟ ɬɚɛɥɢɰɵ 10
Ɇɟɬɨɞ ɨɩɬɢɦɢɡɚɰɢɢ Ɋɟɲɚɟɦɵɟ ɡɚɞɚɱɢ
2. Ɇɟɬɨɞɵ ɩɪɹɦɨɝɨ ɩɨɢɫɤɚ:
1)
ɦɟɬɨɞɵ, ɨɫɧɨɜɚɧɧɵɟ
ɧɚ ɜɵɱɢɫɥɟɧɢɢ ɬɨɥɶɤɨ ɰɟɥɟɜɨɣ
ɮɭɧɤɰɢɢ (ɦɟɬɨɞ ɩɨɢɫɤɚ ɩɨ ɫɢɦɩɥɟɤɫɭ,
ɦɟɬɨɞ ɩɨɢɫɤɚ ɏɭɤɚ-Ⱦɠɢɜɫɚ, ɦɟɬɨɞ
ɫɨɩɪɹɠɟɧɧɵɯ ɧɚɩɪɚɜɥɟɧɢɣ ɉɚɭɷɥɥɚ,
ɤɚɤ ɧɟɩɪɟɪɵɜɧɵɟ, ɬɚɤ ɢ ɪɚɡɪɵɜɧɵɟ
Ⱦɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɟ
ɭɧɢɦɨɞɚɥɶɧɵɟ ɮɭɧɤɰɢɢ,
ɦɟɬɨɞ ɇɟɥɞɟɪɚ-Ɇɢɞɚ, ɦɟɬɨɞ
ɫɥɭɱɚɣɧɨɝɨ ɩɨɢɫɤɚ);
2) ɝɪɚɞɢɟɧɬɧɵɟ ɦɟɬɨɞɵ, ɜ ɤɨɬɨɪɵɯ
ɩɪɢɦɟɧɹɸɬɫɹ ɬɨɱɧɵɟ ɡɧɚɱɟɧɢɹ
ɩɪɨɢɡɜɨɞɧɵɯ (ɦɟɬɨɞ Ʉɨɲɢ,
Ⱦɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɟ ɭɧɢɦɨɞɚɥɶɧɵɟ
ɮɭɧɤɰɢɢ, ɧɟɩɪɟɪɵɜɧɵɟ
ɦɟɬɨɞ ɇɶɸɬɨɧɚ);
3) ɦɟɬɨɞɵ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ,
ɝɞɟ ɩɪɢɦɟɧɹɟɬɫɹ ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ
(ɦɟɬɨɞ Ɇɚɪɤɜɚɪɞɬɚ)
Ⱦɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɟ ɭɧɢɦɨɞɚɥɶɧɵɟ
ɮɭɧɤɰɢɢ, ɧɟɩɪɟɪɵɜɧɵɟ
Ƚɪɚɞɢɟɧɬɧɵɟ ɦɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ ɨɬɧɨɫɹɬɫɹ ɤ ɱɢɫɥɟɧɧɵɦ ɦɟɬɨɞɚɦ ɩɨɢɫɤɨɜɨɝɨ ɬɢɩɚ. Ɉɧɢ ɭɧɢɜɟɪɫɚɥɶɧɵ, ɯɨɪɨɲɨ ɩɪɢɫɩɨɫɨɛɥɟɧɵ ɞɥɹ ɪɚɛɨɬɵ ɫ ɫɨɜɪɟɦɟɧɧɵɦɢ ɰɢɮɪɨɜɵɦɢ ɜɵɱɢɫɥɢɬɟɥɶɧɵɦɢ ɦɚɲɢɧɚɦɢ ɢ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ
ɨɱɟɧɶ ɷɮɮɟɤɬɢɜɧɵ ɩɪɢ ɩɨɢɫɤɟ ɷɤɫɬɪɟɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɧɟɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ ɫ
ɨɝɪɚɧɢɱɟɧɢɹɦɢ ɢ ɛɟɡ ɧɢɯ, ɚ ɬɚɤɠɟ ɬɨɝɞɚ, ɤɨɝɞɚ ɚɧɚɥɢɬɢɱɟɫɤɢɣ ɜɢɞ ɮɭɧɤɰɢɢ ɜɨɨɛɳɟ ɧɟɢɡɜɟɫɬɟɧ. ȼɫɥɟɞɫɬɜɢɟ
ɷɬɨɝɨ ɝɪɚɞɢɟɧɬɧɵɟ, ɢɥɢ ɩɨɢɫɤɨɜɵɟ, ɦɟɬɨɞɵ ɲɢ-
ɪɨɤɨ ɩɪɢɦɟɧɹɸɬɫɹ ɧɚ ɩɪɚɤɬɢɤɟ.
ɋɭɳɧɨɫɬɶ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɦɟɬɨɞɨɜ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɡɧɚɱɟɧɢɣ
ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ, ɞɚɸɳɢɯ ɧɚɢɛɨɥɶɲɢɟ ɢɡɦɟɧɟɧɢɹ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ.
Ɉɛɵɱɧɨ ɞɥɹ ɷɬɨɝɨ ɞɜɢɝɚɸɬɫɹ ɜɞɨɥɶ ɝɪɚɞɢɟɧɬɚ, ɨɪɬɨɝɨɧɚɥɶɧɨɝɨ ɤ ɤɨɧɬɭɪɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜ ɬɨɱɤɟ.
Ɋɚɡɥɢɱɧɵɟ ɩɨɢɫɤɨɜɵɟ ɦɟɬɨɞɵ ɜ ɨɫɧɨɜɧɨɦ ɨɬɥɢɱɚɸɬɫɹ ɨɞɢɧ ɨɬ ɞɪɭɝɨɝɨ
ɫɩɨɫɨɛɨɦ ɨɩɪɟɞɟɥɟɧɢɹ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ ɤ
ɨɩɬɢɦɭɦɭ, ɪɚɡɦɟɪɨɦ ɲɚɝɚ ɢ
ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɶɸ ɩɨɢɫɤɚ ɜɞɨɥɶ ɧɚɣɞɟɧɧɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ, ɤɪɢɬɟɪɢɹɦɢ ɨɤɨɧɱɚɧɢɹ ɩɨɢɫɤɚ, ɩɪɨɫɬɨɬɨɣ ɚɥɝɨɪɢɬɦɢɡɚɰɢɢ ɢ ɩɪɢɦɟɧɢɦɨɫɬɶɸ ɞɥɹ ɪɚɡɥɢɱɧɵɯ
ɗȼɆ. Ɍɟɯɧɢɤɚ ɩɨɢɫɤɚ ɷɤɫɬɪɟɦɭɦɚ ɨɫɧɨɜɚɧɚ ɧɚ ɪɚɫɱɟɬɚɯ, ɤɨɬɨɪɵɟ ɩɨɡɜɨɥɹɸɬ
ɨɩɪɟɞɟɥɢɬɶ ɧɚɩɪɚɜɥɟɧɢɟ ɧɚɢɛɨɥɟɟ ɛɵɫɬɪɨɝɨ ɢɡɦɟɧɟɧɢɹ ɨɩɬɢɦɢɡɢɪɭɟɦɨɝɨ ɤɪɢɬɟɪɢɹ.
4.2. Ƚɪɚɞɢɟɧɬɧɵɟ ɦɟɬɨɞɵ ɞɥɹ ɨɩɬɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɢ
ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ
ɉɪɟɞɫɬɚɜɢɦ ɱɟɥɨɜɟɤɚ, ɫɬɨɹɳɟɝɨ ɧɚ ɫɤɥɨɧɟ ɨɜɪɚɝɚ, ɤɨɬɨɪɨɦɭ ɧɟɨɛɯɨɞɢɦɨ
ɫɩɭɫɬɢɬɶɫɹ ɜɧɢɡ (ɧɚ ɞɧɨ). ɇɚɢɛɨɥɟɟ ɟɫɬɟɫɬɜɟɧɧɵɦ, ɤɚɠɟɬɫɹ, ɧɚɩɪɚɜɥɟɧɢɟ ɜ ɫɬɨɪɨɧɭ ɧɚɢɛɨɥɶɲɟɣ ɤɪɭɬɢɡɧɵ ɫɩɭɫɤɚ, ɬ. ɟ. ɧɚɩɪɚɜɥɟɧɢɟ (–grad f(x)). ɉɨɥɭɱɚɟɦɚɹ
68
ɩɪɢ ɷɬɨɦ ɫɬɪɚɬɟɝɢɹ, ɧɚɡɵɜɚɟɦɚɹ ɝɪɚɞɢɟɧɬɧɵɦ ɦɟɬɨɞɨɦ, ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɲɚɝɨɜ, ɤɚɠɞɵɣ ɢɡ ɤɨɬɨɪɵɯ ɫɨɞɟɪɠɢɬ ɞɜɟ ɨɩɟɪɚɰɢɢ [15, 19–20]:
ɚ) ɨɩɪɟɞɟɥɟɧɢɟ ɧɚɩɪɚɜɥɟɧɢɹ ɧɚɢɛɨɥɶɲɟɣ ɤɪɭɬɢɡɧɵ ɫɩɭɫɤɚ (ɩɨɞɴɟɦɚ);
ɛ) ɩɟɪɟɦɟɳɟɧɢɟ ɜ ɜɵɛɪɚɧɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɧɚ ɧɟɤɨɬɨɪɵɣ ɲɚɝ.
ɉɪɚɜɢɥɶɧɵɣ ɜɵɛɨɪ ɲɚɝɚ ɢɦɟɟɬ ɫɭɳɟɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ. ɑɟɦ ɲɚɝ ɦɟɧɶɲɟ,
ɬɟɦ ɬɨɱɧɟɟ ɪɟɡɭɥɶɬɚɬ, ɧɨ ɛɨɥɶɲɟ ɜɵɱɢɫɥɟɧɢɣ. Ɋɚɡɥɢɱɧɵɟ ɦɨɞɢɮɢɤɚɰɢɢ ɝɪɚɞɢɟɧɬɧɨɝɨ ɦɟɬɨɞɚ ɢ ɫɨɫɬɨɹɬ ɜ ɩɪɢɦɟɧɟɧɢɢ ɪɚɡɥɢɱɧɵɯ ɫɩɨɫɨɛɨɜ ɨɩɪɟɞɟɥɟɧɢɹ ɲɚɝɚ.
ȿɫɥɢ ɧɚ ɤɚɤɨɦ-ɥɢɛɨ ɲɚɝɟ ɡɧɚɱɟɧɢɟ F(x) ɧɟ ɭɦɟɧɶɲɢɥɨɫɶ, ɷɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɬɨɱɤɭ ɦɢɧɢɦɭɦɚ «ɩɪɨɩɭɫɬɢɥɢ», ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɟɨɛɯɨɞɢɦɨ ɜɟɪɧɭɬɶɫɹ ɤ ɩɪɟɞɵɞɭɳɟɣ ɬɨɱɤɟ ɢ ɭɦɟɧɶɲɢɬɶ ɲɚɝ, ɧɚɩɪɢɦɟɪ ɜ 2 ɪɚɡɚ.
Ƚɪɚɞɢɟɧɬɧɵɟ ɦɟɬɨɞɵ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɪɢɛɥɢɠɟɧɧɵɟ (ɢɬɟɪɚɰɢɨɧɧɵɟ)
ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɧɟɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ
ɩɪɚɤɬɢɱɟɫɤɢ ɥɸɛɭɸ ɡɚɞɚɱɭ. Ɉɞɧɚɤɨ ɩɪɢ ɷɬɨɦ ɨɩɪɟɞɟɥɹɟɬɫɹ ɥɨɤɚɥɶɧɵɣ ɷɤɫɬɪɟɦɭɦ. ȼɫɥɟɞɫɬɜɢɟ ɷɬɨɝɨ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɩɪɢɦɟɧɹɬɶ ɷɬɢ ɦɟɬɨɞɵ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ, ɜ ɤɨɬɨɪɵɯ ɤɚɠɞɵɣ ɥɨɤɚɥɶɧɵɣ ɷɤɫɬɪɟɦɭɦ ɹɜɥɹɟɬɫɹ ɢ ɝɥɨɛɚɥɶɧɵɦ.
Ɉɫɧɨɜɧɵɦ ɩɨɧɹɬɢɟɦ, ɩɪɢɦɟɧɹɟɦɵɦ ɜɨ ɜɫɟɯ ɝɪɚɞɢɟɧɬɧɵɯ ɦɟɬɨɞɚɯ, ɹɜɥɹɟɬɫɹ ɩɨɧɹɬɢɟ ɝɪɚɞɢɟɧɬɚ ɮɭɧɤɰɢɢ, ɤɚɤ ɧɚɩɪɚɜɥɟɧɢɹ ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɜɨɡɪɚɫɬɚɧɢɹ
ɮɭɧɤɰɢɢ. Ƚɪɚɞɢɟɧɬ ɜ
grad f(x).
Ⱦɥɹ ɥɸɛɨɣ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ ɮɭɧɤɰɢɢ ɝɪɚɞɢɟɧɬɨɦ ɜ ɬɨɱɤɟ
ɫɹ ɜɟɤɬɨɪ
Ƚɪɚɞɢɟɧɬ ɩɨɤɚɡɵɜɚɟɬ ɧɚɩɪɚɜɥɟɧɢɟ, ɜ ɤɨɬɨɪɨɦ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɮɭɧɤɰɢɢ
ɪɟɣɲɟɝɨ ɜɨɡɪɚɫɬɚɧɢɹ ɮɭɧɤɰɢɢ (ɪɢɫ. 20). ȼɫɥɟɞɫɬɜɢɟ ɷɬɨɝɨ ɞɥɹ ɩɪɨɞɜɢɠɟɧɢɹ ɤ
ɬɨɱɤɟ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɢ ɧɟɨɛɯɨɞɢɦɨ ɞɜɢɝɚɬɶɫɹ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ
ɝɪɚɞɢɟɧɬɚ. ɇɚ ɨɫɧɨɜɚɧɢɢ ɷɬɨɝɨ ɫɬɪɨɹɬ ɢɬɟɪɚɰɢɨɧɧɵɣ ɩɪɨɰɟɫɫ, ɭɬɨɱɧɹɹ ɤɨɪɧɢ
ɭɪɚɜɧɟɧɢɹ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
(ɧɚɱɚɥɶɧɨɣ), ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɣ ɩɟɪɟɯɨɞ ɜ ɧɚɩɪɚɜɥɟɧɢɢ grad
f(x), ɟɫɥɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɱɤɚ ɦɚɤɫɢɦɭɦɚ, ɢ –grad f(x) (ɚɧɬɢɝɪɚɞɢɟɧɬɚ), ɟɫɥɢ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɱɤɚ ɦɢɧɢɦɭɦɚ, ɞɨ ɬɨɱɤɢ, ɹɜɥɹɸɳɟɣɫɹ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ. ɉɪɢ
ɷɬɨɦ ɷɬɚ ɬɨɱɤɚ ɦɨɠɟɬ ɨɤɚɡɚɬɶɫɹ ɤɚɤ ɜɧɭɬɪɢ ɨɛɥɚɫɬɢ ɞɨɩɭɫɬɢɦɵɯ ɡɧɚɱɟɧɢɣ, ɬɚɤ ɢ
ɧɚ ɟɟ ɝɪɚɧɢɰɟ.
)x(f ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɶɲɟɣ, ɬ. ɟ. )x(f∇ ɨɩɪɟɞɟɥɹɟɬ ɧɚɩɪɚɜɥɟɧɢɟ ɧɚɢɫɤɨ-
ɂɬɟɪɚɰɢɢ ɩɪɟɤɪɚɳɚɸɬɫɹ, ɤɨɝɞɚ
ɉɪɨɰɟɫɫ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ, ɧɚɱɢɧɚɹ ɫ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɢ ɯ
ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɡɚɜɢɫɢɦɨɫɬɹɯ ɨɛɨɡɧɚɱɚɟɬɫɹ )x(f∇ ɥɢɛɨ
)x(f
)x(f
∂
ª
[]
T
)x(f
=∇
«
¬
+
∂
,
1x
∂
,...,
2x
∂
nn1n
=∇λ−=
ε<∇ )x(f .
ɢ ɩɨɡɜɨɥɹɸɬ ɪɟɲɢɬɶ
)x(f
∂
º
.
»
xn
∂
¼
...2,1,0n),x(fxx
)x(f ɹɜɥɹɟɬ-
0
69
Ƚɪɚɞɢɟɧɬɧɵɟ ɦɟɬɨɞɵ ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɧɚ ɞɜɚ ɤɥɚɫɫɚ (ɝɪɭɩɩɵ). Ʉ ɩɟɪɜɨɣ
ɝɪɭɩɩɟ ɨɬɧɨɫɹɬɫɹ ɦɟɬɨɞɵ, ɜ ɤɨɬɨɪɵɯ ɜɫɟ ɢɫɫɥɟɞɭɟɦɵɟ ɬɨɱɤɢ ɩɪɢɧɚɞɥɟɠɚɬ ɞɨɩɭɫɬɢɦɨɣ ɨɛɥɚɫɬɢ. Ʉ ɬɚɤɢɦ ɦɟɬɨɞɚɦ ɨɬɧɨɫɹɬɫɹ: ɦɟɬɨɞ ɝɪɚɞɢɟɧɬɚ, ɧɚɢɫɤɨɪɟɣɲɟɝɨ
ɫɩɭɫɤɚ, Ɏɪɚɧɤɚ-ȼɭɥɮɚ. Ʉɨ ɜɬɨɪɨɣ ɝɪɭɩɩɟ ɨɬɧɨɫɹɬɫɹ ɦɟɬɨɞɵ, ɜ ɤɨɬɨɪɵɯ ɢɫɫɥɟɞɭɟɦɵɟ ɬɨɱɤɢ ɦɨɝɭɬ ɢ ɧɟ ɩɪɢɧɚɞɥɟɠɚɬɶ ɞɨɩɭɫɬɢɦɨɣ ɨɛɥɚɫɬɢ. Ɉɛɳɢɦ ɢɡ
ɬɚɤɢɯ
ɦɟɬɨɞɨɜ ɹɜɥɹɟɬɫɹ ɦɟɬɨɞ ɲɬɪɚɮɧɵɯ ɮɭɧɤɰɢɣ. ȼɫɟ ɦɟɬɨɞɵ ɲɬɪɚɮɧɵɯ ɮɭɧɤɰɢɣ
ɨɬɥɢɱɚɸɬɫɹ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɫɩɨɫɨɛɨɦ ɨɩɪɟɞɟɥɟɧɢɹ «ɲɬɪɚɮɚ».
Ɋɢɫ. 20. Ƚɟɨɦɟɬɪɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɦɟɬɨɞɚ ɝɪɚɞɢɟɧɬɧɨɝɨ ɫɩɭɫɤɚ
ɫ ɩɨɫɬɨɹɧɧɵɦ ɲɚɝɨɦ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ
ɝɪɚɞɢɟɧɬɧɨɝɨ ɫɩɭɫɤɚ
1. Ɉɩɪɟɞɟɥɹɸɬ ɧɚɱɚɥɶɧɵɣ ɜɟɤɬɨɪ ɚɪɝɭɦɟɧɬɨɜ ɮɭɧɤɰɢɢ
,
=
ɩɪɢɧɚɞɥɟɠɚɳɢɣ ɞɨɩɭɫɬɢɦɨɣ ɨɛɥɚɫɬɢ ɡɧɚɱɟɧɢɣ f(x).
2. Ɉɩɪɟɞɟɥɹɸɬ grad f(x
3. ȼɵɛɢɪɚɸɬ ɲɚɝ
4. Ɉɩɪɟɞɟɥɹɸɬ ɫɥɟɞɭɸɳɟɟ ɭɬɨɱɧɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɜɟɤɬɨɪɚ ɚɪɝɭɦɟɧɬɨɜ ɩɨ
ɡɚɜɢɫɢɦɨɫɬɢ x
(k+1)
= x
(k)
ɦɨ ɦɚɤɫɢɦɢɡɢɪɨɜɚɬɶ ɮɭɧɤɰɢɸ, «–» – ɞɥɹ ɦɢɧɢɦɢɡɚɰɢɢ.
5. Ɉɩɪɟɞɟɥɹɸɬ grad f(x
– ɟɫɥɢ
– ɟɫɥɢ ɧɟɬ, ɬɨ ɩɟɪɟɯɨɞ ɤ ɩ. 2.
ȿɫɥɢ grad f(x
(k)
) = 0, ɬɨ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɬɨɱɧɵɦ [19, 20] (ɪɢɫ. 21).
) ɢɥɢ –grad f(x0).
0
λ.
± λ grad f(x
) ɢ :
(k+1)
, ɪɟɲɟɧɢɟ ɧɚɣɞɟɧɨ;
ε<+)x(gradf
)1k(
), ɡɧɚɤ «+» ɩɪɢɦɟɧɹɟɬɫɹ, ɟɫɥɢ ɧɟɨɛɯɨɞɢ-
(k)
70
)x,…ɯ,(ɯ ɯ
n210
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