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Файл:Алгоритмизация в инженерных задачах. Учебное пособие
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D[2][3] ɩɨɥɭɱɚɟɬ ɧɨɜɨɟ ɡɧɚɱɟɧɢɟ. ɋɥɟɞɭɸɳɢɣ ɲɚɝ, ɤɨɝɞɚ ɭɫɥɨɜɢɟ ɬɚɤɠɟ ɢɫɬɢɧ-
ɧɨ ɩɪɢɜɧɨɫɢɬ ɢɡɦɟɧɟɧɢɹ ɜ ɷɥɟɦɟɧɬ, ɪɚɫɩɨɥɨɠɟɧɧɵɣ ɧɚ ɩɟɪɟɫɟɱɟɧɢɢ ɜɬɨɪɨɣ
ɫɬɪɨɤɢ ɢ ɬɪɟɬɶɟɝɨ ɫɬɨɥɛɰɚ.
ɉɪɢɦɟɪ ɪɟɚɥɢɡɚɰɢɢ ɚɥɝɨɪɢɬɦɚ [1]:
#include <vcl.h>
#pragma hdrstop
#include <iostream>
using namespace std;
const int maxV = 1000;
int i, j, n;
int GR[maxV][maxV];
//ɚɥɝɨɪɢɬɦ Ɏɥɨɣɞɚ-ɍɨɪɲɟɥɥɚ
void FU(int D[][maxV], int V)
{
int k;
for (i = 0; i<V; i++) D[i][i] = 0;
for (k = 0; k<V; k++)
for (i = 0; i<V; i++)
for (j = 0; j<V; j++)
if (D[i][k] && D[k][j] && i! = j)
if (D[i][k]+D[k][j]<D[i][j] || D[i][j] = = 0)
D[i][j] = D[i][k]+D[k][j];
for (i = 0; i<V; i++)
{
for (j = 0; j<V; j++) cout<<D[i][j]<<"\t";
cout<<endl;
}
}
//---------------------------------------------------------------------------
#pragma argsused
int main(int argc, char* argv[])
{
setlocale(LC_ALL, "Rus");
cout<<"Ʉɨɥɢɱɟɫɬɜɨ ɜɟɪɲɢɧ ɜ ɝɪɚɮɟ > "; cin>>n;
cout<<"ȼɜɟɞɢɬɟ ɦɚɬɪɢɰɭ ɜɟɫɨɜ ɪɟɛɟɪ:\n";
for (i = 0; i<n; i++)
for (j = 0; j<n; j++)
{
cout<<"GR["<<i+1<<"]["<<j+1<<"] > ";
cin>>GR[i][j];
}
41
cout<<"Ɇɚɬɪɢɰɚ ɤɪɚɬɱɚɣɲɢɯ ɩɭɬɟɣ:"<<endl;
FU(GR, n);
system("pause>>void");
}
1.10. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 1.1.
ɋɨɫɬɚɜɢɬɶ ɚɥɝɨɪɢɬɦ ɧɚɯɨɠɞɟɧɢɹ ɬɚɤɨɝɨ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ
ɪɟɡɚɧɢɹ V ɩɪɢ ɫɜɟɪɥɟɧɢɢ ɨɬɜɟɪɫɬɢɹ, ɩɪɢ ɤɨɬɨɪɨɦ ɲɟɪɨɯɨɜɚɬɨɫɬɶ ɨɛɪɚɛɨɬɚɧɧɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɩɨ ɩɚɪɚɦɟɬɪɭ RZ – ɜɵɫɨɬɚ ɦɢɤɪɨɧɟɪɨɜɧɨɫɬɟɣ ɩɪɨɮɢɥɹ ɩɨ ɞɟɫɹɬɢ
ɬɨɱɤɚɦ ɛɭɞɟɬ ɦɟɧɶɲɟ 20 ɦɤɦ. ɗɦɩɢɪɢɱɟɫɤɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɩɚɪɚɦɟɬɪɚ ɲɟɪɨɯɨɜɚɬɨɫɬɢ RZ
ɨɬ ɪɟɠɢɦɚ ɨɛɪɚɛɨɬɤɢ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ
0,17 0,46
DS
Rz = 48,7
o
,
0,04
V
ɝɞɟ D – ɞɢɚɦɟɬɪ ɫɜɟɪɥɚ, ɦɦ: D = 12 ɦɦ; S
– ɩɨɞɚɱɚ, ɦɦ/ɨɛ: Sɨ = 0,1 ɦɦ/ɨɛ;
ɨ
V – ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ, ɦ/ɦɢɧ.
ɇɚɱɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ V
ɪɨɫɬɢ ɪɟɡɚɧɢɹ h
= 0,1 ɦ/ɦɢɧ.
V
= 12 ɦ/ɦɢɧ, ɲɚɝ ɢɡɦɟɧɟɧɢɹ ɫɤɨ-
0
Ɋɟɲɟɧɢɟ. Ɂɚɞɚɱɚ ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɚ ɫ ɩɨɦɨɳɶɸ ɰɢɤɥɚ ɫ ɩɨɫɬɭɫɥɨɜɢɟɦ
(ɩɪɢɦɟɧɟɧɢɟ ɨɩɟɪɚɬɨɪɨɜ do/while) ɢɥɢ ɰɢɤɥɚ ɫ ɩɪɟɞɭɫɥɨɜɢɟɦ (while/do). Ⱥɥɝɨɪɢɬɦɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɩɪɟɞɫɬɚɜɥɟɧɵ ɧɚ ɪɢɫ. 10.
Ɋɢɫ. 10. Ⱥɥɝɨɪɢɬɦɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ 1.1:
ɚ – ɰɢɤɥ ɫ ɩɨɫɬɭɫɥɨɜɢɟɦ; ɛ – ɰɢɤɥ ɫ ɩɪɟɞɭɫɥɨɜɢɟɦ
42
Ɋɟɡɭɥɶɬɚɬɨɦ ɪɟɲɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɚɥɝɨɪɢɬɦɨɦ
ɰɢɤɥɚ ɫ ɩɨɫɬɭɫɥɨɜɢɟɦ (ɪɢɫ. 10, ɚ) ɹɜɥɹɟɬɫɹ ɧɚɯɨɠɞɟɧɢɟ ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ V, ɩɪɢ
ɤɨɬɨɪɨɣ ɲɟɪɨɯɨɜɚɬɨɫɬɶ ɩɨɜɟɪɯɧɨɫɬɢ RZ ɛɭɞɟɬ ɦɟɧɶɲɟ ɡɚɞɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ
RZmin. Ⱦɥɹ ɷɬɨɝɨ V ɢɡɦɟɧɹɸɬ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ V
(ɩɪɢ V0 ɭɫɥɨɜɢɟ
0
RZ < RZmin ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ) ɫ ɦɢɧɢɦɚɥɶɧɵɦ ɲɚɝɨɦ ɩɪɢɪɚɳɟɧɢɹ hv. ɉɪɢ ɤɚɠ-
ɞɨɦ ɧɨɜɨɦ ɡɧɚɱɟɧɢɢ ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ ɪɚɫɫɱɢɬɵɜɚɸɬ RZ ɢ ɩɪɨɜɟɪɹɸɬ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɹ RZ < RZmin. Ʉɚɤ ɬɨɥɶɤɨ ɷɬɨ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɢɬɫɹ, ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ
ɜɵɯɨɞ ɢɡ ɬɟɥɚ ɢɬɟɪɚɰɢɨɧɧɨɝɨ ɰɢɤɥɚ ɧɚ ɜɵɜɨɞ ɪɟɡɭɥɶɬɚɬɚ ɬɨɝɨ ɩɨɫɥɟɞɧɟɝɨ ɡɧɚɱɟɧɢɹ V, ɩɪɢ ɤɨɬɨɪɨɦ ɢ ɨɛɟɫɩɟɱɢɥɨɫɶ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɹ RZ < RZmin.
Ɉɬɜɟɬ: V = 561 ɦ/ɦɢɧ.
ɉɪɢɦɟɪ ɤɨɞɚ ɞɥɹ ɪɟɚɥɢɡɚɰɢɢ ɚɥɝɨɪɢɬɦɚ ɫ ɩɨɦɨɳɶɸ ɩɪɢɥɨɠɟɧɢɹ Turbo
Borland C++ Builder:
float D,S,V,hv,Rz;
hv = 0.1;
D = StrToFloat(Edit1->Text);
S = StrToFloat(Edit2->Text);
V = 12;// ɧɚɱɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɫɜɟɪɥɟɧɢɹ
do
{
Rz = 48.7*((pow(D,0.17)*pow(S,0.46))/pow(V,0.04));
V = V+hv;
}
while (Rz>20);
Edit3->Text = FloatToStr(V);
ɉɪɢɦɟɪ ɤɨɞɚ ɞɥɹ ɪɟɚɥɢɡɚɰɢɢ ɰɢɤɥɚ ɫ ɩɪɟɞɭɫɥɨɜɢɟɦ ɫ ɩɨɦɨɳɶɸ ɩɪɢɥɨɠɟɧɢɹ Turbo Borland C++ Builder:
float D,S,V,hv,Rz;
hv = 0.1;
D = StrToFloat(Edit1->Text);
S = StrToFloat(Edit2->Text);
V = 12;
Rz = 40;
while (Rz> = 20)
{
Rz = 48.7*((pow(D,0.17)*pow(S,0.46))/pow(V,0.04));
V = V+hv;
};
Edit3->Text = FloatToStr(V);
43
Ɂɚɞɚɱɚ 1.2. ɋɨɫɬɚɜɢɬɶ ɚɥɝɨɪɢɬɦ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ V ɩɪɢ
ɫɜɟɪɥɟɧɢɢ ɨɬɜɟɪɫɬɢɹ ɫɜɟɪɥɨɦ ɞɢɚɦɟɬɪɨɦ D, ɢɡɦɟɧɹɸɳɢɦɫɹ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ D
ɞɚɱɟɣ S
ɡɧɚɱɟɧɢɹ S
= 1 ɦɦ ɞɨ ɤɨɧɟɱɧɨɝɨ ɡɧɚɱɟɧɢɹ Dn = 4 ɦɦ ɫ ɲɚɝɨɦ hD = 1 ɦɦ, ɢ ɩɨ-
ɨ
, ɢɡɦɟɧɹɸɳɟɣɫɹ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ Sɨ = 0,1 ɦɦ/ɨɛ ɞɨ ɤɨɧɟɱɧɨɝɨ
ɨ
= 0,4 ɦɦ/ɨɛ ɫ ɲɚɝɨɦ hs = 0,1 ɦɦ/ɨɛ:
n
q
CDK
VV
V=
TS
my
o
,
ɝɞɟ ɋ
= 400; q = m = y = 0,5; Kv = 1 – ɷɦɩɢɪɢɱɟɫɤɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ;
v
T = 60 ɦɢɧ – ɩɟɪɢɨɞ ɫɬɨɣɤɨɫɬɢ ɫɜɟɪɥɚ.
Ɋɟɲɟɧɢɟ. Ⱥɥɝɨɪɢɬɦ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɪɢɫ. 11.
Ɋɢɫ. 11. Ⱥɥɝɨɪɢɬɦɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ 1.2
Ɋɟɡɭɥɶɬɚɬɨɦ ɪɟɲɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɚɥɝɨɪɢɬɦɨɦ
(ɫɦ. ɪɢɫ. 11) ɹɜɥɹɟɬɫɹ ɮɨɪɦɢɪɨɜɚɧɢɟ ɦɚɫɫɢɜɚ ɬɚɛɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ ɜɵɯɨɞɧɨɝɨ
ɩɚɪɚɦɟɬɪɚ – ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ ɩɪɢ ɫɜɟɪɥɟɧɢɢ V ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɞɜɭɯ ɜɯɨɞɧɵɯ
ɩɚɪɚɦɟɬɪɨɜ (ɩɚɪɚɦɟɬɪɨɜ ɜɧɟɲɧɟɝɨ ɢ ɜɥɨɠɟɧɧɨɝɨ ɰɢɤɥɚ) – ɞɢɚɦɟɬɪɚ ɫɜɟɪɥɚ D ɢ
ɩɨɞɚɱɢ S.
Ɉɬɜɟɬ:
RichEdit1
163,29931640625
115,470054626465
44
94,2808990478516
230,94010925293
163,29931640625
133,33332824707
282,842712402344
200
163,29931640625
326,5986328125
230,94010925293
188,561798095703
float Cv,D,K,T,S,q,m,y,V;
Cv = 400;
q = m = y = 0.5;
K = 1;
T = 60;
for (D = 1; D < = 4; D++)
{
for (S = 0.1; S < = 0.4; S = S+0.1)
{ V = (Cv*pow(D,q)*K)/(pow(T,m)*pow(S,y));
RichEdit1->Lines->Add(V);
} ;
};
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
ɑɬɨ ɬɚɤɨɟ ɚɥɝɨɪɢɬɦ? Ʉɚɤɨɜɵ ɨɫɧɨɜɧɵɟ ɫɜɨɣɫɬɜɚ ɩɪɚɜɢɥɶɧɨ ɩɨɫɬɪɨɟɧ-
1.
ɧɨɝɨ ɚɥɝɨɪɢɬɦɚ?
Ʉɚɤɢɦ ɨɛɪɚɡɨɦ ɦɨɠɧɨ ɨɩɢɫɚɬɶ ɚɥɝɨɪɢɬɦ, ɤɚɤɢɦɢ ɫɢɦɜɨɥɚɦɢ ɨɛɨɡɧɚɱɚ-
2.
ɸɬ ɨɫɧɨɜɧɵɟ ɨɩɟɪɚɬɨɪɵ ɚɥɝɨɪɢɬɦɨɜ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ȿɋɉȾ?
ɑɬɨ ɬɚɤɨɟ ɬɢɩ ɞɚɧɧɵɯ ɜ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɢ?
3.
Ʉɚɤɢɟ ɬɢɩɵ ɞɚɧɧɵɯ ɨɬɧɨɫɹɬ ɤ ɛɚɡɨɜɵɦ, ɤɚɤɢɟ – ɤ ɩɪɨɢɡɜɨɞɧɵɦ? ȼ ɱɟɦ
4.
ɢɯ ɩɪɢɧɰɢɩɢɚɥɶɧɨɟ ɨɬɥɢɱɢɟ?
Ʉɚɤɢɦɢ ɫɢɦɜɨɥɚɦɢ ɨɛɨɡɧɚɱɚɸɬ ɨɩɟɪɚɰɢɢ ɞɥɹ ɚɪɢɮɦɟɬɢɱɟɫɤɢɯ, ɥɨɝɢɱɟ-
5.
ɫɤɢɯ, ɭɧɚɪɧɵɯ ɨɩɟɪɚɰɢɣ ɜ ɋ++?
ɑɬɨ ɬɚɤɨɟ ɮɭɧɤɰɢɹ, ɫɢɧɬɚɤɫɢɫ ɨɛɴɹɜɥɟɧɢɹ, ɜɵɡɨɜɚ ɮɭɧɤɰɢɢ ɜ ɋ++?
6.
ɋ ɩɨɦɨɳɶɸ ɤɚɤɢɯ ɨɩɟɪɚɬɨɪɨɜ ɨɩɢɫɵɜɚɸɬ ɛɚɡɨɜɵɟ ɤɨɧɫɬɪɭɤɰɢɢ ɚɥɝɨ-
7.
ɪɢɬɦɨɜ (ɥɢɧɟɣɧɭɸ, ɜɟɬɜɥɟɧɢɟ, ɰɢɤɥɢɱɟɫɤɭɸ)?
Ʉɚɤ ɨɛɴɹɜɥɹɟɬɫɹ ɫɬɪɭɤɬɭɪɚ ɞɚɧɧɵɯ, ɨɛɴɟɤɬ ɫɬɪɭɤɬɭɪɵ?
8.
ɑɬɨ ɬɚɤɨɟ ɝɪɚɮ, ɞɥɹ ɚɥɝɨɪɢɬɦɢɡɚɰɢɢ ɤɚɤɢɯ ɡɚɞɚɱ ɩɪɢɦɟɧɹɸɬɫɹ ɝɪɚɮɵ?
9.
Ʉɚɤ ɮɨɪɦɢɪɭɟɬɫɹ ɦɚɬɪɢɰɚ ɫɦɟɠɧɨɫɬɢ, ɦɚɬɪɢɰɚ ɢɧɰɢɞɟɧɬɧɨɫɬɢ?
10.
45
ȽɅȺȼȺ 2
ɑɂɋɅȿɇɇɕȿ ɆȿɌɈȾɕ Ɋȿɒȿɇɂə
ɍɊȺȼɇȿɇɂɃ
ȼɨ ɦɧɨɝɢɯ ɩɪɚɤɬɢɱɟɫɤɢɯ ɫɥɭɱɚɹɯ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɩɚɪɚɦɟɬɪ ɫɢɫɬɟɦɵ, ɤɨɬɨɪɚɹ ɨɩɢɫɵɜɚɟɬɫɹ ɫɥɨɠɧɵɦ ɚɥɝɟɛɪɚɢɱɟɫɤɢɦ ɭɪɚɜɧɟɧɢɟɦ. Ⱥɧɚɥɢɬɢɱɟɫɤɢ
ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɚɥɝɟɛɪɚɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɣɬɢ ɧɟ ɜɫɟɝɞɚ ɭɞɚɟɬɫɹ, ɬɚɤ ɤɚɤ
ɨɬɫɭɬɫɬɜɭɸɬ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɜɵɫɨɤɢɯ ɫɬɟɩɟɧɟɣ ɜ
ɨɛɳɟɦ ɜɢɞɟ. Ⱦɥɹ ɬɪɚɧɫɰɟɧɞɟɧɬɧɵɯ ɭɪɚɜɧɟɧɢɣ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɦɨɠɧɨ ɧɚɣɬɢ ɜ
ɧɟɦɧɨɝɢɯ ɫɚɦɵɯ ɩɪɨɫɬɵɯ ɫɥɭɱɚɹɯ.
ȿɫɥɢ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɬɪɭɞɧɨ
ɦɟɬɨɞɵ, ɧɚɩɪɢɦɟɪ ɩɪɢɛɥɢɠɟɧɧɵɟ.
ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɞɥɹ ɪɟɲɟɧɢɹ ɧɟɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɧɚ ɉɗȼɆ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɸɬɫɹ ɫɩɟɰɢɚɥɶɧɵɟ ɦɟɬɨɞɵ, ɤɨɬɨɪɵɟ ɨɬɧɨɫɹɬɫɹ ɤ ɦɟɬɨɞɚɦ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɦɚɬɟɦɚɬɢɤɢ. ɇɚ ɢɯ ɨɫɧɨɜɟ ɫɨɡɞɚɧɨ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɚɥɝɨɪɢɬɦɨɜ ɞɥɹ ɉɗȼɆ.
ȼ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ ɞɥɹ ɪɟɲɟɧɢɹ ɧɟɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ ɩɨɦɨɳɶɸ
ɝɪɚɦɦɧɵɯ ɫɪɟɞɫɬɜ ɩɪɢɦɟɧɹɸɬ
ɢɬɟɪɚɰɢɨɧɧɵɯ ɦɟɬɨɞɨɜ ɹɜɥɹɟɬɫɹ ɦɧɨɝɨɤɪɚɬɧɨɟ ɩɨɜɬɨɪɟɧɢɟ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ
ɧɚɛɨɪɚ ɞɟɣɫɬɜɢɣ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɪɟɡɭɥɶɬɚɬɚ (ɱɬɨ ɥɟɝɤɨ ɪɟɚɥɢɡɭɟɬɫɹ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɰɢɤɥɨɜ). ɉɨɫɥɟ ɤɚɠɞɨɝɨ ɩɨɜɬɨɪɟɧɢɹ (ɢɬɟɪɚɰɢɢ) ɩɪɨɢɫɯɨɞɢɬ ɨɱɟɪɟɞɧɨɟ
ɩɪɢɛɥɢɠɟɧɢɟ ɤ ɤɨɪɧɸ ɭɪɚɜɧɟɧɢɹ.
Ɉɫɨɛɟɧɧɨɫɬɶɸ ɩɪɢɦɟɧɟɧɢɹ ɩɪɢɛɥɢɠɟɧɧɵɯ ɢɬɟɪɚɰɢɨɧɧɵɯ ɦɟɬɨɞɨɜ ɹɜɥɹɟɬɫɹ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɩɪɨɜɟɞɟɧɢɹ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ ɭɪɚɜɧɟɧɢɹ.
ȿɫɥɢ ɭɪɚɜɧɟɧɢɟ ɧɟ ɭɞɚɟɬɫɹ ɪɟɲɢɬɶ
ɥɢɬɶ, ɫɤɨɥɶɤɨ ɨɧɨ ɢɦɟɟɬ ɤɨɪɧɟɣ ɢ ɤɚɤɨɜɚ ɢɯ ɩɪɢɪɨɞɚ
ɩɥɟɤɫɧɵɯ ɢɥɢ ɜɟɳɟɫɬɜɟɧɧɵɯ, ɫɤɨɥɶɤɨ ɨɬɪɢɰɚɬɟɥɶɧɵɯ ɢɥɢ ɩɨɥɨɠɢɬɟɥɶɧɵɯ.
Ɍɚɤɠɟ ɧɟɨɛɯɨɞɢɦɨ ɡɧɚɬɶ ɬɨɱɧɨɫɬɶ, ɫ ɤɨɬɨɪɨɣ ɧɚɣɞɟɧɧɨɟ ɪɟɲɟɧɢɟ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɤɨɪɧɟɦ ɜ ɡɚɞɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɡɚɞɚɱɢ.
Ⱦɥɹ ɜɫɟɯ ɢɬɟɪɚɰɢɨɧɧɵɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɣ f(x
ɉɗȼɆ ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɜɚɬɶ ɧɚɱɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɤɨɪɧɟɣ x
ɧɭɸ ɬɨɱɧɨɫɬɶ
ε.
ɉɪɟɞɜɚɪɢɬɟɥɶɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɭɪɚɜɧɟɧɢɹ ɦɨɠɧɨ ɩɪɨɢɡɜɨɞɢɬɶ ɝɪɚɮɢɱɟɫɤɢɦ ɥɢɛɨ ɬɚɛɥɢɱɧɵɦ ɦɟɬɨɞɨɦ.
Ƚɪɚɮɢɱɟɫɤɢ ɢɡɨɛɪɚɠɚɸɬ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ y = f(x
ɚɪɝɭɦɟɧɬɨɜ, ɨɛɪɚɳɚɸɳɢɯ ɮɭɧɤɰɢɸ ɜ ɧɨɥɶ.
Ɍɚɛɥɢɱɧɵɦ ɦɟɬɨɞɨɦ ɨɩɪɟɞɟɥɹɸɬ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɞɥɹ ɦɚɫɫɢɜɚ ɬɨɱɟɤ ɢ
ɜɵɹɜɥɹɸɬ ɫɨɫɟɞɧɢɟ ɚɪɝɭɦɟɧɬɵ, ɩɪɢ ɤɨɬɨɪɵɯ ɮɭɧɤɰɢɹ ɦɟɧɹɟɬ ɫɜɨɣ ɡɧɚɤ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ (ɜ ɞɚɧɧɨɦ ɞɢɚɩɚɡɨɧɟ ɞɚɧɧɵɯ ɛɭɞɟɬ ɧɚɯɨɞɢɬɶɫɹ ɤɨɪɟɧɶ ɭɪɚɜɧɟɧɢɹ).
ɂɬɟɪɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɥɟɝɤɨ ɚɥɝɨɪɢɬɦɢɡɢɪɭɸɬɫɹ ɢ ɪɟɚɥɢɡɭɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɉɗȼɆ. Ʉ ɨɫɧɨɜɧɵɦ ɢɬɟɪɚɰɢɨɧɧɵɦ
− ɦɟɬɨɞ ɞɟɥɟɧɢɹ ɨɬɪɟɡɤɚ ɩɨɩɨɥɚɦ;
− ɦɟɬɨɞ ɤɚɫɚɬɟɥɶɧɵɯ;
− ɦɟɬɨɞ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ.
ɧɚɣɬɢ ɚɧɚɥɢɬɢɱɟɫɤɢ, ɬɨ ɩɪɢɦɟɧɹɸɬ ɞɪɭɝɢɟ
ɩɪɨ-
ɢɬɟɪɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ. Ƚɥɚɜɧɵɦ ɩɪɢɡɧɚɤɨɦ
ɚɧɚɥɢɬɢɱɟɫɤɢ, ɬɨ ɡɚɪɚɧɟɟ ɬɪɭɞɧɨ ɨɩɪɟɞɟ-
− ɫɤɨɥɶɤɨ ɢɡ ɧɢɯ ɤɨɦ-
) = 0 ɫ ɩɨɦɨɳɶɸ
i
ɭɪɚɜɧɟɧɢɹ ɢ ɡɚɞɚɧ-
io
) ɢ ɨɩɪɟɞɟɥɹɸɬ ɡɧɚɱɟɧɢɹ
i
ɦɟɬɨɞɚɦ ɦɨɠɧɨ ɨɬɧɟɫɬɢ [11]:
46
2.1. Ɇɟɬɨɞ ɞɟɥɟɧɢɹ ɨɬɪɟɡɤɚ ɩɨɩɨɥɚɦ
Ɇɟɬɨɞ ɩɨɥɨɜɢɧɧɨɝɨ ɞɟɥɟɧɢɹ (ɞɢɯɨɬɨɦɢɢ, ɛɢɫɟɤɰɢɢ) ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ
ɡɚɞɚɧɧɵɣ ɢɫɫɥɟɞɭɟɦɵɣ ɨɬɪɟɡɨɤ ɮɭɧɤɰɢɢ [a, b] ɞɟɥɹɬ ɩɨɩɨɥɚɦ, ɪɚɫɫɱɢɬɚɜ
xo = (a+b)/2 [11]. Ⱦɚɥɟɟ ɨɩɪɟɞɟɥɹɸɬ ɬɭ ɩɨɥɨɜɢɧɭ ɨɛɥɚɫɬɢ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ,
ɝɞɟ ɨɧɚ ɦɟɧɹɟɬ ɡɧɚɤ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ. Ⱦɥɹ ɷɬɨɝɨ ɢɡ ɞɜɭɯ ɩɨɥɭɱɟɧɧɵɯ ɨɬɪɟɡɤɨɜ [a, xo] ɢ [xo, b] ɜɵɛɢɪɚɸɬ ɞɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ ɬɨɬ, ɧɚ ɤɨɧɰɚɯ ɤɨɬɨɪɨɝɨ f(x) ɢɦɟɟɬ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɡɧɚɤɢ (ɪɢɫ
. 12). ȼɵɛɪɚɧɧɵɣ ɨɬɪɟɡɨɤ ɫɧɨɜɚ
ɩɪɨɞɨɥɠɚɸɬ ɢɫɫɥɟɞɨɜɚɬɶ, ɞɟɥɹɬ ɩɨɩɨɥɚɦ ɢ ɩɪɨɞɨɥɠɚɸɬ ɢɬɟɪɚɰɢɢ. ɉɨɜɬɨɪɟɧɢɹ
ɩɪɨɜɨɞɹɬ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ f(x) ɧɟ ɛɭɞɟɬ ɦɟɧɶɲɟ ɡɚɞɚɧɧɨɣ
ɬɨɱɧɨɫɬɢ
ε.
Ɋɢɫ. 12. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɦɟɬɨɞɚ ɞɟɥɟɧɢɹ
ɨɬɪɟɡɤɚ ɩɨɩɨɥɚɦ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ
1. Ɂɚɞɚɸɬ ɧɟɨɛɯɨɞɢɦɭɸ ɬɨɱɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɣ ε. ɍɫɬɚɧɚɜɥɢɜɚɸɬ ɝɪɚɧɢɰɵ
ɚɪɝɭɦɟɧɬɨɜ [a, b] ɞɥɹ ɮɭɧɤɰɢɢ f(x), ɬɚɤɢɟ ɱɬɨɛɵ
ɍɬɨɱɧɹɸɬ ɢɫɫɥɟɞɭɟɦɵɣ ɨɬɪɟɡɨɤ, ɪɚɫɫɱɢɬɵɜɚɸɬ xo
2.
Ɋɚɫɫɱɢɬɵɜɚɸɬ
3.
4.
ɉɪɨɜɟɪɹɸɬ ɭɫɥɨɜɢɟ
ɧɟɦ ɭɪɚɜɧɟɧɢɹ, ɬ. ɟ. ɯ* = ɯɨ
;
)ɯɨ(f
1
, ɟɫɥɢ ɨɧɨ ɢɫɬɢɧɧɨ, ɬɨ ɯɨ1 ɹɜɥɹɟɬɫɹ ɤɨɪ-
ε≤)ɯɨ(f
1
. ȿɫɥɢ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɨ ɪɟɚɥɢɡɭɸɬ ɫɥɟ-
1
ɞɭɸɳɭɸ ɢɬɟɪɚɰɢɸ;
ɂɫɫɥɟɞɭɸɬ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɧɚ ɤɨɧɰɚɯ ɨɬɪɟɡɤɨɜ [a, ɯɨ
5.
ȿɫɥɢ
= (ɯɨ
+b)/2;
1
Ⱦɚɥɟɟ ɩɪɨɜɟɪɹɸɬ ɢɫɬɢɧɧɨɫɬɶ ɭɫɥɨɜɢɹ
6.
, ɬɨ ɜɵɱɢɫɥɹɸɬ xo2 = (a+ɯɨ1)/2. ɂɧɚɱɟ ɜɵɱɢɫɥɹɸɬ xo2 =
0)ɯɨ(f)a(f
<⋅
1
47
2
0)b(f)a(f <⋅
= (a+b)/2;
1
ɢ ɬ. ɞ.
ε≤)ɯɨ(f
;
] ɢ [ɯɨ1, b].
1
2.2. Ɇɟɬɨɞ ɤɚɫɚɬɟɥɶɧɵɯ
Ɇɟɬɨɞ ɤɚɫɚɬɟɥɶɧɵɯ ɢɥɢ ɦɟɬɨɞ ɇɶɸɬɨɧɚ – ɷɬɨ ɢɬɟɪɚɰɢɨɧɧɵɣ ɱɢɫɥɟɧɧɵɣ
ɦɟɬɨɞ ɧɚɯɨɠɞɟɧɢɹ ɤɨɪɧɹ (ɧɭɥɹ) ɡɚɞɚɧɧɨɣ ɮɭɧɤɰɢɢ. Ɇɟɬɨɞ ɛɵɥ ɜɩɟɪɜɵɟ ɩɪɟɞɥɨɠɟɧ ɂɫɚɚɤɨɦ ɇɶɸɬɨɧɨɦ (1643–1727). ɉɨɢɫɤ ɪɟɲɟɧɢɹ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɭɬɟɦ
ɩɨɫɬɪɨɟɧɢɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ ɢ ɨɫɧɨɜɚɧ ɧɚ ɩɪɢɧɰɢɩɚɯ ɩɪɨɫɬɨɣ
ɢɬɟɪɚɰɢɢ. Ɍɚɤɠɟ ɦɟɬɨɞ ɇɶɸɬɨɧɚ ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɦɟɧɟɧ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɨɩɬɢɦɢɡɚɰɢɢ, ɜ ɤɨɬɨɪɵɯ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɧɭɥɶ ɩɟɪɜɨɣ
ɩɪɨɢɡɜɨɞɧɨɣ ɥɢɛɨ
ɝɪɚɞɢɟɧɬɚ ɜ ɦɧɨɝɨɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ. Ɉɧ ɨɫɧɨɜɚɧ ɧɚ ɝɪɚɮɢɱɟɫɤɨɣ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ.
Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɱɢɫɥɟɧɧɨ
ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ ɪɚɜɧɚ ɬɚɧɝɟɧɫɭ ɭɝɥɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɤɚɫɚɬɟɥɶɧɨɣ,
ɩɪɨɜɟɞɟɧɧɨɣ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ ɤ ɤɪɢɜɨɣ, ɢ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɨɫɢ ɯ
(ɪɢɫ. 13).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɦɟɟɦ
)x(f
′
tg)x(f =α=
0
0
,
h
0
ɨɬɫɸɞɚ
)x(f
0
h
=
0
.
′
)x(f
0
ɋɥɟɞɭɸɳɟɟ ɡɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
)x(f
0
ɯɯ
±=
01
.
′
)x(f
0
Ɂɧɚɤ (ɩɥɸɫ) ɩɪɢɦɟɧɹɟɬɫɹ ɞɥɹ ɥɟɜɨɫɬɨɪɨɧɧɟɣ ɫɯɨɞɢɦɨɫɬɢ, (ɦɢɧɭɫ) ɞɥɹ
ɩɪɚɜɨɫɬɨɪɨɧɧɟɣ [11].
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ
1. Ɂɚɞɚɸɬ ɧɟɨɛɯɨɞɢɦɭɸ ɬɨɱɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɣ ε. ɍɫɬɚɧɚɜɥɢɜɚɸɬ ɧɚɱɚɥɶ-
ɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɪɧɹ ɭɪɚɜɧɟɧɢɹ ɯ
Ɋɚɫɫɱɢɬɵɜɚɸɬ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɟ f(x
2.
Ɋɚɫɫɱɢɬɵɜɚɸɬ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɜ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɟ
3.
′
)x(f
;
0
4.
ɉɪɨɜɟɪɹɸɬ ɭɫɥɨɜɢɟ ε≤)ɯ(f
ɧɟɦ ɭɪɚɜɧɟɧɢɹ, ɬ. ɟ. ɯ* = ɯ
;
0
);
0
, ɟɫɥɢ ɨɧɨ ɢɫɬɢɧɧɨ, ɬɨ ɯ0 ɹɜɥɹɟɬɫɹ ɤɨɪ-
0
. ȿɫɥɢ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɨ ɩɟɪɟɯɨɞɹɬ ɤ ɫɥɟ-
0
ɞɭɸɳɟɣ ɨɩɟɪɚɰɢɢ;
48
Ɋɚɫɫɱɢɬɵɜɚɸɬ ɫɥɟɞɭɸɳɟɟ ɡɧɚɱɟɧɢɟ ɩɪɟɞɩɨɥɚɝɚɟɦɨɝɨ ɤɨɪɧɹ ɭɪɚɜɧɟɧɢɹ
5.
ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
ɯɯ
01
ɢ ɩɪɨɞɨɥɠɚɸɬ ɢɬɟɪɚɰɢɢ, ɧɚɱɢɧɚɹ ɫ ɩ. 2.
′
)x(f
0
)x(f
0
±=
Ɋɢɫ. 13. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɦɟɬɨɞɚ ɤɚɫɚɬɟɥɶɧɵɯ
2.3. Ɇɟɬɨɞ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ
ɉɭɫɬɶ ɡɚɞɚɧɚ ɮɭɧɤɰɢɹ f(ɯ), ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ f(x) = 0. Ɇɟɬɨɞ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ (ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ) ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɟɟ
ɨɛɳɢɦ, ɢ ɦɧɨɝɢɟ ɞɪɭɝɢɟ ɦɟɬɨɞɵ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɧɟɤɨɬɨɪɭɸ ɜɚɪɢɚɰɢɸ
ɦɟɬɨɞɚ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ.
ɉɪɟɞɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɮɭɧɤɰɢɢ f(x) = 0 ɜ ɜɢɞɟ ɯ = w(x). ɗɬɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ, ɧɚɩɪɢɦɟɪ, ɩɪɢɛɚɜɢɜ ɯ ɤ ɨɛɟɢɦ ɱɚɫɬɹɦ ɭɪɚɜɧɟɧɢɹ f(x) = 0 [11].
Ɇɟɬɨɞ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ ɢɦɟɟɬ
ɢɧɬɟɪɩɪɟɬɚɰɢɸ (ɪɢɫ. 14). Ɋɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ ɯ = w(x) ɛɭɞɟɬ ɚɛɫɰɢɫɫɚ ɬɨɱɤɢ
ɩɟɪɟɫɟɱɟɧɢɹ ɩɪɹɦɨɣ ɭ = ɯ ɫ ɤɪɢɜɨɣ ɭ = w(x). ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ ɢɬɟɪɚɰɢɣ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ w(x) ɜ ɬɨɱɤɟ ɯi ɧɟɨɛɯɨɞɢɦɨ ɨɬɥɨɠɢɬɶ ɩɨ ɨɫɢ ɚɛɫɰɢɫɫ. ɗɬɨ ɦɨɠɧɨ
ɫɞɟɥɚɬɶ, ɟɫɥɢ ɩɪɨɜɟɫɬɢ ɝɨɪɢɡɨɧɬɚɥɶ ɞɨ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɩɪɹɦɨɣ ɭ = ɯ ɢ ɢɡ ɬɨɱɤɢ
ɢɯ ɩɟɪɟɫɟɱɟɧɢɹ ɨɩɭɫɬɢɬɶ ɩɟɪɩɟɧɞɢɤɭɥɹɪ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ
1. Ɂɚɞɚɸɬ ɧɟɨɛɯɨɞɢɦɭɸ ɬɨɱɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɣ ε. ɍɫɬɚɧɚɜɥɢɜɚɸɬ ɧɚɱɚɥɶ-
ɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɪɧɹ ɭɪɚɜɧɟɧɢɹ ɯ
Ɋɚɫɫɱɢɬɵɜɚɸɬ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɟ f(x
2.
0
ɫɥɟɞɭɸɳɭɸ ɧɚɝɥɹɞɧɭɸ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ
ɧɚ ɨɫɶ ɚɛɫɰɢɫɫ.
;
);
0
49
k
k
k
k
ɉɪɨɜɟɪɹɸɬ ɭɫɥɨɜɢɟ
3.
ɭɪɚɜɧɟɧɢɹ, ɬ. ɟ. ɯ* = ɯ
. ȿɫɥɢ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɨ ɩɟɪɟɯɨɞɹɬ ɤ ɫɥɟɞɭɸ-
0
, ɟɫɥɢ ɨɧɨ ɢɫɬɢɧɧɨ, ɬɨ ɯ0 ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ
ε≤)ɯ(f
0
ɳɟɣ ɨɩɟɪɚɰɢɢ;
ɉɟɪɟɯɨɞɹɬ ɨɬ ɧɟɥɢɧɟɣɧɨɝɨ ɭɪɚɜɧɟɧɢɹ f(x) = 0 ɤ ɭɪɚɜɧɟɧɢɸ ɯ = w(x), ɞɥɹ
4.
ɱɟɝɨ ɭɦɧɨɠɚɟɦ ɥɟɜɭɸ ɢ ɩɪɚɜɭɸ ɱɚɫɬɢ f(x) = 0 ɧɚ ɩɪɨɢɡɜɨɥɶɧɭɸ ɤɨɧɫɬɚɧɬɭ k ɢ
ɩɪɢɛɚɜɥɹɸɬ ɤ ɨɛɟɢɦ ɱɚɫɬɹɦ ɩɨɥɭɱɢɜɲɟɝɨɫɹ ɭɪɚɜɧɟɧɢɹ ɯ, ɩɨɥɭɱɚɸɬ ɭɪɚɜɧɟɧɢɟ
ɜɢɞɚ
ɯ ⋅+=⋅+
, ɬ. ɟ. ɭɪɚɜɧɟɧɢɟ
0x)x(f
;
ɯɯ ⋅+=
)x(f
Ɋɢɫ. 14. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɦɟɬɨɞɚ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ
Ɂɧɚɱɟɧɢɟ k ɩɨɥɭɱɚɸɬ ɢɡ ɧɟɪɚɜɟɧɫɬɜɚ
5.
Ɋɚɫɫɱɢɬɵɜɚɸɬ ɭɬɨɱɧɟɧɧɵɣ ɤɨɪɟɧɶ ɭɪɚɜɧɟɧɢɹ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
6.
;
)x(fkɯɯ
⋅+=
001
7.
Ɋɚɫɫɱɢɬɵɜɚɸɬ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɟ f(x
ɉɪɨɜɟɪɹɸɬ ɭɫɥɨɜɢɟ
8.
ɭɪɚɜɧɟɧɢɹ, ɬ. ɟ. ɯ* = ɯ
ɢɥɢ
′
);
1
′
<
, ɟɫɥɢ ɨɧɨ ɢɫɬɢɧɧɨ, ɬɨ ɯ0 ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ
ε≤)ɯ(f
1
. ȿɫɥɢ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɨ ɩɟɪɟɯɨɞɹɬ ɤ ɩ. 6.
1
1)x(w0 <
′
;
1w
)x(f
⋅+=
2.4. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 2.1.
Ɉɩɪɟɞɟɥɢɬɶ ɦɟɬɨɞɨɦ ɩɨɥɨɜɢɧɧɨɝɨ ɞɟɥɟɧɢɹ ɡɧɚɱɟɧɢɟ ɜɧɟɲɧɟɝɨ ɭɫɢɥɢɹ N, ɜɨɡɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɤɪɟɩɟɠɧɵɣ ɛɨɥɬ, ɢɡɝɨɬɨɜɥɟɧɧɵɣ ɢɡ ɫɬɚɥɢ 45,
ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɜɵɞɟɪɠɚɬɶ ɷɬɨɬ ɛɨɥɬ, ɟɫɥɢ ɟɝɨ ɞɢɚɦɟɬɪ d
= 3 ɦɦ. Ɇɢɧɢɦɚɥɶɧɵɣ
1
ɞɢɚɦɟɬɪ ɛɨɥɬɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
N40
dσπ=
1
ɝɞɟ N – ɜɧɟɲɧɟɟ ɭɫɢɥɢɟ (ɭɫɢɥɢɟ ɪɚɫɬɹɠɟɧɢɹ ɛɨɥɬɚ), ɇ; [ıp] – ɞɨɩɭɫɬɢɦɨɟ
ɧɚɩɪɹɠɟɧɢɟ ɩɪɢ ɪɚɫɬɹɠɟɧɢɢ, ɤɝɫ/ɦɦ
2
([ıp] = 72 H/ɦɦ2 – ɞɥɹ ɫɬɚɥɢ 45); d1 – ɜɧɭɬ-
,
[]
p
ɪɟɧɧɢɣ ɞɢɚɦɟɬɪ ɪɟɡɶɛɵ ɛɨɥɬɚ, ɦɦ. ɋɬɟɩɟɧɶ ɬɨɱɧɨɫɬɢ İ = 0,01.
50
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