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Ɋɟɲɟɧɢɟ. ɉɪɟɞɜɚɪɢɬɟɥɶɧɨ ɩɨɫɬɪɨɢɦ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ N(d
ɜɢɥɶɧɨ ɜɵɛɪɚɬɶ ɢɫɫɥɟɞɭɟɦɵɣ ɢɧɬɟɪɜɚɥ (ɪɢɫ. 15).
), ɱɬɨɛɵ ɩɪɚ-
1
Ɋɢɫ. 15. Ɂɚɜɢɫɢɦɨɫɬɶ N(d
)
1
ɋɨɝɥɚɫɧɨ ɝɪɚɮɢɤɭ ɡɚɜɢɫɢɦɨɫɬɢ N(d
65], ɝɞɟ N
– ɥɟɜɚɹ ɝɪɚɧɢɰɚ ɨɬɪɟɡɤɚ; Nɉ – ɩɪɚɜɚɹ ɝɪɚɧɢɰɚ, ɤɝɫ. ȼɛɥɢɡɢ ɝɪɚɧɢɰ
Ʌ
ɷɬɨɝɨ ɨɬɪɟɡɤɚ ɮɭɧɤɰɢɹ ɡɧɚɱɟɧɢɟ d
ɛɥɢɡɤɨ ɤ ɡɧɚɱɟɧɢɸ 3.
1
) ɢɫɫɥɟɞɭɟɦ ɨɬɪɟɡɨɤ [NɅ, Nɉ] = [45,
1
ɇɟɨɛɯɨɞɢɦɨ ɢɫɫɥɟɞɨɜɚɬɶ ɮɭɧɤɰɢɸ
N40
[]
σπ
p
ɢɥɢ
03
=−
.
03N18.0 =−
,
154,034518,0)N(fy
ɥɥ
−=−⋅==
ɩɩ
421,036518,0)N(fy
=−⋅==
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɥɨɜɢɟ
ɪɟɡɨɤ [N
, Nɩ] ɩɨɩɨɥɚɦ:
ɥ
Ɋɚɫɫɱɢɬɵɜɚɟɦ
ɉɪɨɜɟɪɹɟɦ ɭɫɥɨɜɢɟ
NN
+
N
=
0
0
ɩɥ
=
2
. Ɍɚɤ ɤɚɤ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɩɪɨɞɨɥ-
ε≤)N(f
0
ɜɵɩɨɥɧɹɟɬɫɹ. Ⱦɚɥɟɟ ɞɟɥɢɦ ɨɬ-
0)N(f)N(f
<⋅
ɩɥ
6545
+
2
=−⋅==
.
ɇ55
=
.
15,035518,0)N(fy
ɠɚɟɦ ɪɚɫɱɟɬɵ.
ɂɫɫɥɟɞɭɟɦ ɩɨɥɭɱɢɜɲɢɟɫɹ ɞɟɥɟɧɢɟɦ ɨɬɪɟɡɤɢ ɢ ɜɵɛɢɪɚɟɦ ɞɥɹ ɞɚɥɶɧɟɣɲɢɯ
ɢɬɟɪɚɰɢɣ ɬɨɬ, ɧɚ ɤɨɧɰɚɯ ɤɨɬɨɪɨɝɨ ɢɫɫɥɟɞɭɟɦɚɹ ɮɭɧɤɰɢɹ ɦɟɧɹɟɬ ɫɜɨɣ ɡɧɚɤ.
;
154,0)N(fy
−==
ɥɥ
;
15,0)N(fy
==
00
,
421,0)N(fy
==
ɩɩ
ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
ɢ ɩɪɨɞɨɥɠɚɟɦ ɢɫɫɥɟɞɨɜɚɬɶ ɨɬɪɟɡɨɤ [Nɥ, N0].
0)N(f)N(f
<⋅
0ɥ
.
51
+
Ɋɚɫɫɱɢɬɵɜɚɟɦ
Ɋɚɫɫɱɢɬɵɜɚɟɦ
ɉɪɨɜɟɪɹɟɦ ɭɫɥɨɜɢɟ
N* = N
= 50ɇ ɤɨɪɧɟɦ ɭɪɚɜɧɟɧɢɹ.
1
NN
=
N
1
0ɥ
2
1
1
5545
+
=
2
. Ɍɚɤ ɤɚɤ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɩɪɢɧɢɦɚɟɦ
ε≤)N(f
.
ɇ50
=
.
035018,0)N(fy
=−⋅==
Ȼɥɨɤ-ɫɯɟɦɚ ɚɥɝɨɪɢɬɦɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɩɪɟɞɫɬɚɜɥɟɧɚ ɧɚ ɪɢɫ. 16.
Ɋɢɫ. 16. Ȼɥɨɤ-ɫɯɟɦɚ ɚɥɝɨɪɢɬɦɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ 2.1
ɉɪɢɦɟɪ ɤɨɞɚ ɩɪɢɥɨɠɟɧɢɹ:
float e,N1,N2,y1,y2,N,n,y;
e = 0.1;
N1 = StrToFloat(Edit1->Text);
N2 = StrToFloat(Edit2->Text);
y1 = pow((0.18*N1),0.5)-3;
y2 = pow((0.18*N2),0.5)-3;
n = (N2+N1)/2;
y = pow((0.18*n),0.5)-3;
if ((y1*y2)>0) {
Edit4->Text = "ȼɜɟɞɢɬɟ ɤɨɪɪɟɤɬɧɵɟ ɡɧɚɱɟɧɢɹ ɧɚɱɚɥɶɧɨɝɨ ɨɬɪɟɡɤɚ"; }
52
else
{
while ((fabs(y))> = e)
{
if ((y1*y2)<0)
{ N2 = n;
n = (N2+N1)/2;}
else
{N1 = n;
n = (N2+N1)/2;}
y = pow((0.18*n),0.5)-3;
}
Edit3->Text = FloatToStr(n);
}
Ɂɚɞɚɱɚ 2.2. Ɋɚɫɫɱɢɬɚɬɶ ɨɫɟɜɭɸ ɩɨɞɚɱɭ, ɧɟɨɛɯɨɞɢɦɭɸ ɞɥɹ ɨɛɟɫɩɟɱɟɧɢɹ ɫɢ-
ɥɵ ɪɟɡɚɧɢɹ Pz = 120ɇ, ɩɪɢ ɩɪɨɞɨɥɶɧɨɦ ɧɚɪɭɠɧɨɦ ɬɨɱɟɧɢɢ ɫɬɚɥɢ 45. Ɂɚɜɢɫɢɦɨɫɬɶ ɫɢɥɵ ɨɬ ɩɚɪɚɦɟɬɪɨɜ ɪɟɠɢɦɚ ɨɛɪɚɛɨɬɤɢ
nyx
,
VStCpPz ⋅⋅⋅=
t = 2 ɦɦ, V = 100 ɦ/ɦɢɧ, Cp = 300, x = 1, y = 0,75, n = –0,15, ɧɟɨɛɯɨɞɢɦɚɹ
ɬɨɱɧɨɫɬɶ İ = 0,1.
ɉɪɢɦɟɧɢɬɶ ɦɟɬɨɞ ɤɚɫɚɬɟɥɶɧɵɯ (ɇɶɸɬɨɧɚ).
Ɋɟɲɟɧɢɟ. ɂɫɫɥɟɞɭɟɦɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɜɢɞ
12,075,0
−
.
0120100S2300)S(f
=−⋅⋅⋅=
ɇɟɨɛɯɨɞɢɦɨ ɩɪɢɦɟɧɢɬɶ ɦɟɬɨɞ ɤɚɫɚɬɟɥɶɧɵɯ, ɞɥɹ ɱɟɝɨ ɨɩɪɟɞɟɥɢɦ ɩɪɨɢɡ-
ɜɨɞɧɭɸ ɮɭɧɤɰɢɢ:
15,025,0
′
ɉɪɢɧɢɦɚɟɦ ɧɚɱɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɩɨɞɚɱɢ S
100S230075,0)S(f
⋅⋅⋅⋅=
= 0,2.
o
−−
.
Ɋɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɨɣ ɜ ɧɭɥɟɜɨɣ ɬɨɱɤɟ:
12,075,0
′
ɉɪɨɜɟɪɹɟɦ ɭɫɥɨɜɢɟ
ɭɪɚɜɧɟɧɢɹ, ɬ. ɟ. S* = S
−
−−
ε≤)S(f
, ɟɫɥɢ ɨɧɨ ɢɫɬɢɧɧɨ, ɬɨ S0 ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ
0
. Ɍɚɤ ɤɚɤ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ ( 1,0066,30 ≥− ), ɬɨ
0
−=−⋅⋅⋅=
15,025,0
=⋅⋅⋅⋅=
;
066,301201002,02300)2,0(f
.
252,3371002,0230075,0)2,0(f
ɩɟɪɟɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɨɩɟɪɚɰɢɢ. Ⱦɥɹ ɷɬɨɝɨ ɨɩɪɟɞɟɥɹɟɦ ɲɚɝ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
53
)S(f
h
0
=
0
′
)S(f
0
66,30
−
=
252,337
.
09,0
−=
Ⱦɚɥɟɟ ɪɚɫɫɱɢɬɵɜɚɟɦ ɫɥɟɞɭɸɳɟɟ ɡɧɚɱɟɧɢɟ S1 ɢ ɩɪɨɢɡɜɨɞɢɦ ɫɥɟɞɭɸɳɭɸ
ɢɬɟɪɚɰɢɸ:
.
29,0)09,0(2,0hSS
01
′
=−−=−=
12,075,0
−
15,025,0
−−
164,112010029,02300)29,0(f
−=−⋅⋅⋅=
336,30710029,0230075,0)29,0(f
=⋅⋅⋅⋅=
ɍɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ ( 1,0164,1 ≥− ), ɬɨ ɩɟɪɟɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɨɩɟ-
ɪɚɰɢɢ. Ⱦɥɹ ɷɬɨɝɨ ɨɩɪɟɞɟɥɹɟɦ ɲɚɝ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
)S(f
−
h
1
1
=
′
)S(f
1
112
164,1
=
336,307
−
.
004,0
−=
.
294,0)004,0(29,0hSS
=−−=−=
12,075,0
=−⋅⋅⋅=
;
064,012010024,02300)294,0(f
ɍɫɥɨɜɢɟ 1,0064,0 ≤ , ɢɫɬɢɧɧɨ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ S2 ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɭɪɚɜ-
ɧɟɧɢɹ, ɬ. ɟ. S* = 0,294.
Ȼɥɨɤ-ɫɯɟɦɚ ɚɥɝɨɪɢɬɦɚ ɩɪɨɝɪɚɦɦɧɨɣ ɪɟɚɥɢɡɚɰɢɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɩɪɟɞ-
ɫɬɚɜɥɟɧɚ ɧɚ ɪɢɫ. 17.
ɉɪɢɦɟɪ ɤɨɞɚ ɩɪɢɥɨɠɟɧɢɹ:
float e,x0,x1,f,f1;
e = 0.1;
x0 = StrToFloat(Edit1->Text);
f = 300*2*(pow(x0,0.75))*(pow(100,-0.15))-120;
while ((fabs(f)>e) )
{
f1 = 0.75*300*2*(pow(x0,-0.25))*(pow(100,-0.15));
f = 300*2*(pow(x0,0.75))*(pow(100,-0.15))-120;
x1 = x0-f/f1;
x0 = x1;
RichEdit1->Lines->Add(x0);
} ;
Edit2->Text = FloatToStr(x0);
;
.
54
Ɋɢɫ. 17. Ȼɥɨɤ-ɫɯɟɦɚ ɚɥɝɨɪɢɬɦɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ 2.2
Ɂɚɞɚɱɚ 2.3. Ɋɚɫɫɱɢɬɚɬɶ ɨɫɟɜɭɸ ɩɨɞɚɱɭ, ɧɟɨɛɯɨɞɢɦɭɸ ɞɥɹ ɨɛɟɫɩɟɱɟɧɢɹ
ɫɢɥɵ ɪɟɡɚɧɢɹ Pz = 120ɇ, ɩɪɢ ɩɪɨɞɨɥɶɧɨɦ ɧɚɪɭɠɧɨɦ ɬɨɱɟɧɢɢ ɫɬɚɥɢ 45. Ɂɚɜɢ-
nyx
ɫɢɦɨɫɬɶ ɫɢɥɵ ɨɬ ɩɚɪɚɦɟɬɪɨɜ ɪɟɠɢɦɚ ɨɛɪɚɛɨɬɤɢ
, t = 2 ɦɦ,
VStCpPz ⋅⋅⋅=
V = 100 ɦ/ɦɢɧ, Cp = 300, x = 1, y = 0,75, n = –0,15, ɧɟɨɛɯɨɞɢɦɚɹ ɬɨɱɧɨɫɬɶ
İ = 0,1. ɉɪɢɦɟɧɢɬɶ ɦɟɬɨɞ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ.
Ɋɟɲɟɧɢɟ. ɂɫɫɥɟɞɭɟɦɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɜɢɞ
12,075,0
−
.
0120100S2300)S(f
=−⋅⋅⋅=
ɇɟɨɛɯɨɞɢɦɨ ɩɪɢɦɟɧɢɬɶ ɦɟɬɨɞ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ. ɉɪɢɧɢɦɚɟɦ ɧɚɱɚɥɶɧɨɟ
ɡɧɚɱɟɧɢɟ ɩɨɞɚɱɢ S
= 0,2.
o
55
Ɋɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɧɭɥɟɜɨɣ ɬɨɱɤɟ:
12,075,0
−
−=−⋅⋅⋅=
.
066,301201002,02300)2,0(f
ɉɪɨɜɟɪɹɟɦ ɭɫɥɨɜɢɟ
ɭɪɚɜɧɟɧɢɹ, ɬ. ɟ. S* = S
ɩɟɪɟɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɨɩɟɪɚɰɢɢ.
ɉɨɞɛɟɪɟɦ ɲɚɝ ɩɪɢɪɚɳɟɧɢɹ ɚɪɝɭɦɟɧɬɚ k, ɩɪɢɦɟɧɹɹ ɡɚɜɢɫɢɦɨɫɬɢ:
. Ɍɚɤ ɤɚɤ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ ( 1,0066,30 ≥− ), ɬɨ
0
′
1
−=
k
Ɋɚɫɫɱɢɬɵɜɚɟɦ ɭɬɨɱɧɟɧɧɵɣ ɤɨɪɟɧɶ ɭɪɚɜɧɟɧɢɹ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
Ɋɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ S
ɉɪɨɜɟɪɹɟɦ ɭɫɥɨɜɢɟ
ɬɨ ɩɟɪɟɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɨɩɟɪɚɰɢɢ.
Ɋɚɫɫɱɢɬɵɜɚɟɦ ɭɬɨɱɧɟɧɧɵɣ ɤɨɪɟɧɶ ɭɪɚɜɧɟɧɢɹ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
Ɍɚɤ ɤɚɤ 1,0)293,0(f ≥ , ɩɟɪɟɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɨɩɟɪɚɰɢɢ.
′
−=
)S(f
0
001
112
294,0S3= ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
Ȼɥɨɤ-ɫɯɟɦɚ ɚɥɝɨɪɢɬɦɚ ɩɪɨɝɪɚɦɦɧɨɣ ɪɟɚɥɢɡɚɰɢɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɩɪɟɞ-
ɫɬɚɜɥɟɧ ɧɚ ɪɢɫ. 18.
ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɭɪɚɜɧɟɧɢɹ.
294,0S3=
, ɟɫɥɢ ɨɧɨ ɢɫɬɢɧɧɨ, ɬɨ S0 ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ
ε≤)S(f
0
′
′
⋅+=
0)x(fk1w =
;
15,025,0
−−
100S230075,0)S(f
⋅⋅⋅⋅=
1
1002,0230075,0
⋅⋅⋅⋅
;
−=
15.025,0
−−
=−⋅−+=⋅+= ;
:
1
12,075,0
−
ε≤)S(f
, ɬ. ɤ. ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ (
1
15,075,0
−
064,0)294,0(f =
, 1,0)294,0(f ≤ ,
164,112010029,02300)29,0(f
−=−⋅⋅⋅=
293,0)164,1()003,0(29,0)S(fkSS
=−⋅−+=⋅+= ;
243,0120100293,02300)293,0(f
−=−⋅⋅⋅=
003,0
29,0)066,30()003,0(2,0)S(fkSS
.
;
),
1,0164,1 ≥−
;
56
ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɫɦɵɫɥ ɢɬɟɪɚɰɢɨɧɧɵɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɣ?
1.
Ʉɚɤ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɧɚɱɚɥɶɧɭɸ ɬɨɱɤɭ ɩɪɢ ɩɪɢɦɟɧɟɧɢɢ ɢɬɟɪɚɰɢɨɧ-
2.
ɧɵɯ ɦɟɬɨɞɨɜ?
ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɩɨɥɨɜɢɧɧɨɝɨ ɞɟɥɟ-
3.
ɧɢɹ.
ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɤɚɫɚɬɟɥɶɧɵɯ (ɇɶɸɬɨ-
4.
ɧɚ).
ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɩɪɨɫɬɵɯ ɢɬɟɪɚɰɢɣ.
5.
Ɋɢɫ. 18. Ȼɥɨɤ-ɫɯɟɦɚ ɚɥɝɨɪɢɬɦɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
57
ȽɅȺȼȺ 3
ɑɂɋɅȿɇɇɕȿ ɆȿɌɈȾɕ Ɋȿɒȿɇɂə ɋɂɋɌȿɆ
ɅɂɇȿɃɇɕɏ ɍɊȺȼɇȿɇɂɃ
Ɇɧɨɝɢɟ ɡɚɞɚɱɢ ɩɪɚɤɬɢɤɢ ɫɜɨɞɹɬɫɹ ɤ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ. ɉɪɢ ɤɨɧɫɬɪɭɢɪɨɜɚɧɢɢ ɢɧɠɟɧɟɪɧɵɯ ɫɨɨɪɭɠɟɧɢɣ, ɨɛɪɚɛɨɬɤɟ
ɪɟɡɭɥɶɬɚɬɨɜ ɢɡɦɟɪɟɧɢɣ, ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɩɥɚɧɢɪɨɜɚɧɢɹ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɢ ɪɹɞɚ ɞɪɭɝɢɯ ɡɚɞɚɱ ɬɟɯɧɢɤɢ, ɷɤɨɧɨɦɢɤɢ, ɧɚɭɱɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɪɢɯɨɞɢɬɫɹ ɪɟɲɚɬɶ ɫɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ.
ɋɢɫɬɟɦɚ m ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ n ɧɟɢɡɜɟɫɬɧɵɦɢ (ɢɥɢ ɥɢɧɟɣɧɚɹ ɫɢɫɬɟɦɚ) ɜ ɥɢɧɟɣɧɨɣ ɚɥɝɟɛɪɟ
, ɯ2, …, xn – ɧɟɢɡɜɟɫɬɧɵɟ, ɤɨɬɨɪɵɟ ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ; a11, ɚ12, …, amn –
ɝɞɟ ɯ
1
ɤɨɷɮɮɢɰɢɟɧɬɵ ɫɢɫɬɟɦɵ; b1, b2 ,…, bm – ɫɜɨɛɨɞɧɵɟ ɱɥɟɧɵ (ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɢɡɜɟɫɬɧɵɦɢ).
Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ – ɫɨɜɨɤɭɩɧɨɫɬɶ ɡɧɚɱɟɧɢɣ ɯ
ɸɬ ɜɫɟ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ ɜ ɬɨɠɞɟɫɬɜɚ [11].
ɋɭɳɟɫɬɜɭɟɬ ɦɧɨɠɟɫɬɜɨ ɪɚɫɱɟɬɧɵɯ ɢ ɝɪɚɮɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ, ɩɨɡɜɨɥɹɸɳɢɯ
ɪɟɲɚɬɶ ɩɨɞɨɛɧɵɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɚɧɚɥɢɬɢɱɟɫɤɢ ɥɢɛɨ ɩɪɢɛɥɢɠɟɧɧɵɦɢ ɦɟɬɨɞɚɦɢ. ȼ ɭɱɟɛɧɨɦ ɩɨɫɨɛɢɢ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɦɟɬɨɞɵ, ɤɨɬɨɪɵɟ ɩɪɨɫɬɨ ɚɥɝɨɪɢɬɦɢɡɢɪɨɜɚɬɶ.
ɗɬɨɬ ɦɟɬɨɞ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɩɪɢɦɟɧɹɟɬɫɹ ɞɥɹ ɦɚɬɪɢɰ ɫ ɧɟɧɭɥɟɜɵɦ ɨɩɪɟɞɟɥɢɬɟɥɟɦ.
ɉɭɫɬɶ ɞɚɧɚ ɫɢɫɬɟɦɚ n ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ n ɧɟɢɡɜɟɫɬɧɵɦɢ, ɬɨɝɞɚ ɟɟ
ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɟ
– ɷɬɨ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɜɢɞɚ
+++=
ax ax ax b
11 1 12 2 1 1
°
............................................
®
°
+++=
ax a x ax b
11 2 2
mm mnnm
¯
3.1. Ɇɟɬɨɞ ɦɚɬɪɢɰ
... ;
nn
... ,
1
, ɯ2, …, x
ɤɨɬɨɪɵɟ ɨɛɪɚɳɚ-
n,
BXA =⋅ ,
ɝɞɟ Ⱥ – ɦɚɬɪɢɰɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɫɢɫɬɟɦɵ, ȼ ɢ ɏ – ɫɬɨɥɛɰɵ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ ɢ
ɪɟɲɟɧɢɣ ɫɢɫɬɟɦɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ:
A
§
¨
¨
=
¨
¨
¨
©
a...aa
·
n11211
¸
a...aa
¸
n22221
,
¸
............
¸
¸
a...aa
¹
nn2n1n
b
§
·
1
¨
¸
b
¨
¸
B
2
=
,
¨
¸
...
¨
¸
¨
¸
b
©
¹
n
X
x
§
·
1
¨
¸
x
¨
¸
2
=
.
¨
¸
...
¨
¸
¨
¸
x
©
¹
n
58
j
-1
ɍɦɧɨɠɢɦ ɷɬɨ ɦɚɬɪɢɱɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫɥɟɜɚ ɧɚ Ⱥ
– ɦɚɬɪɢɰɭ, ɨɛɪɚɬɧɭɸ
ɦɚɬɪɢɰɟ Ⱥ [11]:
Ɍɚɤ ɤɚɤ E)A(A
1=⋅−
, ɩɨɥɭɱɚɟɦ BAX
11
−−
.
BA)XA(A
⋅=⋅⋅
1
−
⋅=
. ɉɪɚɜɚɹ ɱɚɫɬɶ ɷɬɨɝɨ ɭɪɚɜ-
ɧɟɧɢɹ ɞɚɫɬ ɫɬɨɥɛɟɰ ɪɟɲɟɧɢɣ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ. ɇɟɨɛɯɨɞɢɦɵɦ ɢ ɞɨɫɬɚɬɨɱɧɵɦ
ɭɫɥɨɜɢɟɦ ɩɪɢɦɟɧɢɦɨɫɬɢ ɦɟɬɨɞɚ ɹɜɥɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ ɧɭɥɸ ɨɩɪɟɞɟɥɢɬɟɥɹ ɦɚɬɪɢɰɵ Ⱥ (
0)Adet( ≠ ) [11].
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɧɚɯɨɠɞɟɧɢɹ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ:
ɇɚɯɨɞɹɬ ɨɩɪɟɞɟɥɢɬɟɥɶ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ.
1.
ɋɨɫɬɚɜɥɹɟɬɫɹ ɫɨɸɡɧɚɹ ɦɚɬɪɢɰɚ, ɷɥɟɦɟɧɬɚɦɢ ɤɨɬɨɪɨɣ ɹɜɥɹɸɬɫɹ ɚɥɝɟɛɪɚ-
2.
ɢɱɟɫɤɢɟ ɞɨɩɨɥɧɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ.
ɉɨɥɭɱɟɧɧɭɸ ɫɨɸɡɧɭɸ ɦɚɬɪɢɰɭ ɬɪɚɧɫɩɨɧɢɪɭɸɬ.
3.
ɋɨɸɡɧɭɸ ɦɚɬɪɢɰɭ ɞɟɥɹɬ ɧɚ ɨɩɪɟɞɟɥɢɬɟɥɶ ɢ ɩɨɥɭɱɚɸɬ ɨɛɪɚɬɧɭɸ ɦɚɬɪɢɰɭ
4.
−
Ⱥ ⋅=
1
T*1
.
A
Adet
Ⱥɥɝɟɛɪɚɢɱɟɫɤɢɦ ɞɨɩɨɥɧɟɧɢɟɦ Aij ɤ ɷɥɟɦɟɧɬɭ aij ɨɩɪɟɞɟɥɢɬɟɥɹ n-ɝɨ ɩɨɪɹɞɤɚ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨ
Aij = (–1)
i+j
·Mij.
Ɇɢɧɨɪɨɦ Mij ɤ ɷɥɟɦɟɧɬɭ aij ɨɩɪɟɞɟɥɢɬɟɥɹ n-ɝɨ ɩɨɪɹɞɤɚ ɧɚɡɵɜɚɟɬɫɹ ɨɩɪɟɞɟɥɢɬɟɥɶ (n–1)-ɝɨ ɩɨɪɹɞɤɚ, ɩɨɥɭɱɟɧɧɵɣ ɢɡ ɢɫɯɨɞɧɨɝɨ ɨɩɪɟɞɟɥɢɬɟɥɹ ɜɵɱɟɪɤɢɜɚɧɢɟɦ i-ɣ ɫɬɪɨɤɢ ɢ j-ɝɨ ɫɬɨɥɛɰɚ.
Ɉɩɪɟɞɟɥɢɬɟɥɟɦ 3-ɝɨ ɩɨɪɹɞɤɚ ɦɚɬɪɢɰɵ ɧɚɡɵɜɚɟɬɫɹ ɫɭɦɦɚ ɩɪɨɢɡɜɟɞɟɧɢɣ
ɷɥɟɦɟɧɬɨɜ ɩɟɪɜɨɣ ɫɬɪɨɤɢ ɦɚɬɪɢɰɵ ɧɚ ɢɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ ɞɨɩɨɥɧɟɧɢɹ.
ɉɪɨɢɡɜɟɞɟɧɢɟ ɞɜɭɯ ɫɨɝɥɚɫɨɜɚɧɧɵɯ ɦɚɬɪɢɰ (ɭ ɤɨɬɨɪɵɯ ɱɢɫɥɨ ɫɬɨɥɛɰɨɜ
ɦɚɬɪɢɰɵ A ɪɚɜɧɨ ɱɢɫɥɭ ɫɬɪɨɤ ɦɚɬɪɢɰɵ B) A = [m×n] ɢ B = [n×k] – ɷɬɨ ɧɨɜɚɹ
ɦɚɬɪɢɰɚ C = [m×k], ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɣ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
ba...babaɫ ⋅++⋅+⋅= ,
pjipj22ij11iij
m,...,.1i =
; m,...,1
= .
3.2. Ɇɟɬɨɞ ɢɬɟɪɚɰɢɣ (ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ)
ɗɬɨɬ ɦɟɬɨɞ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɛɥɚɞɚɟɬ ɯɨɪɨɲɟɣ ɫɯɨɞɢɦɨɫɬɶɸ ɢ ɯɨɪɨɲɨ ɚɥɝɨɪɢɬɦɢɡɢɪɭɟɬɫɹ.
ɉɭɫɬɶ ɞɚɧɚ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɬɢɩɚ
°
12111
2
®
°
¯
=+++
;bxa...xaxa
1nn1
............................................
=+++
.bxa...xaxa
nnnn22n11n
59
Ɋɚɡɪɟɲɢɦ ɫɢɫɬɟɦɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɟɦɟɧɧɵɯ x
(ɩɨɥɭɱɢɦ ɧɨɪɦɚɥɶɧɵɣ
i
ɜɢɞ ɫɢɫɬɟɦɵ):
°
1211111
2
®
°
¯
α++α+α+β=
;x...xxx
nn1
............................................
α++α+α+β=
.x...xxx
nnn22n11n22
b
i
=β
i
a
ii
,
a
ij
−=α
ij
a
ii
.
Ⱦɚɥɟɟ ɫɢɫɬɟɦɭ ɪɟɲɚɸɬ ɦɟɬɨɞɨɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ. Ɂɚ ɧɭ-
0
ɥɟɜɨɟ ɡɧɚɱɟɧɢɟ ɩɟɪɟɦɟɧɧɵɯ ɩɪɢɧɢɦɚɸɬ
ɯ β= . Ⱦɚɥɟɟ ɫɬɪɨɹɬ ɩɟɪɜɨɟ ɩɪɢɛɥɢ-
i
i
ɠɟɧɢɟ ɜ ɜɢɞɟ [11]
1
i
0
xɯ ⋅α+β= .
ii
ɂɬɟɪɚɰɢɢ ɩɪɨɞɨɥɠɚɸɬ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɧɟ ɜɵɩɨɥɧɢɬɫɹ ɭɫɥɨɜɢɟ
−1n
n
xxmax
i
i
.
ε≤−
Ɇɟɬɨɞ ɢɬɟɪɚɰɢɣ (ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ) ɫɯɨɞɢɬɫɹ, ɟɫɥɢ ɜɵɩɨɥɧɹɟɬɫɹ ɨɞɧɨ ɢɡ ɭɫɥɨɜɢɣ:
ɇɨɪɦɚ ɩɪɟɨɛɪɚɡɨɜɚɧɧɨɣ ɦɚɬɪɢɰɵ ɦɟɧɶɲɟ ɟɞɢɧɢɰɵ:
1)
¦
n
1max
<α=α
.
ij
=
1j,j
2) ɂɫɯɨɞɧɚɹ ɦɚɬɪɢɰɚ ɢɦɟɟɬ ɞɢɚɝɨɧɚɥɶɧɨɟ ɩɪɟɨɛɥɚɞɚɧɢɟ
n
>
aa .
¦
ijii
≠=
ji,1j,i
3.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 3.1.
Ɋɟɲɢɬɶ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ ɨɛɪɚɬɧɵɯ ɦɚɬ-
ɪɢɰ
=−+
°
®
°
¯
;8ɯ08,0ɯ24,0ɯ4
321
=−+
;9ɯ15,0ɯ3ɯ09,0
323
=+−
.20ɯ4ɯ08,0ɯ04,0
321
60
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