Regresii rezolvate ca Functii Lineare .pdf
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Transcript of Regresii rezolvate ca Functii Lineare .pdf
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7/25/2019 Regresii rezolvate ca Functii Lineare .pdf
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Functia de gradul 1, de 1 variabila, notata f(x,a,b). Deducerea regresiei, Matriceal 1( )
1( ) f x a b( ) a b x Regresia care trebuie determinata
2( ) x 1 2 3 4 5 6 7 8 9 10( ) Valorile masurate ale variabilei independente, x
3( ) y 2.1 3.6 4.5 4 .9 5.4 5 .9 6.6 7 .9 8.1 9.5( ) Valorile masurate ale variabilei dependente, y
4( ) Expunere teorie generala : Intre (4) - (9). Se formeaza matricea X, astfel :
5( ) A a b( ) a b( ) AT
vectorul necunoscutelor=
6( ) X A y= Ecuatia Matriceala echivalenta cu ecuatia (1)
7( ) XT
X A XTyT=
X
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
9
10
X
0 1
0
1
2
3
4
5
6
7
8
9
1 1
1 2
1 3
1 4
1 5
1 6
1 7
1 8
1 9
1 10
8( ) XT
X 1
XT
X A XTX 1
XTy
T=
9( ) A XT
X 1
XTy
T=
Calculez XT
X 1 XTyT 1.8270.732
10( ) f x( ) 1.827 0.732 x 11( ) E f xT yT Apoi calculez f xT yT E E Erori= m
0
9
i
f xT i yT i
10
m 3 10 3
0 1 2 3 4 5 6 7 8 9 10
2
4
6
8
10
2
4
6
8
10
Grafic Erori
f xT
i yT
i
i i
f xT
0
0
1
2
3
4
5
6
7
8
9
2.559
3.291
4.023
4.755
5.487
6.219
6.951
7.683
8.415
9.147
yT
0
0
1
2
3
4
5
6
7
8
9
2.1
3.6
4.5
4.9
5.4
5.9
6.6
7.9
8.1
9.5
E0
0
1
2
3
4
5
6
7
8
9
0.459
-0.309
-0.477
-0.145
0.087
0.319
0.351
-0.217
0.315
-0.353
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2( )Functie (regresie) Multilineara determinata prin Calcul Matriceal
f x y z u a b c d e( ) a b x c y d z e u
x 1.11 1.25 1.49 1.61 1.82 2.05 2.27 2.44 2.61 2.89( ) ||
||
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Valori Masurate. x,y,z,u,V, unde V = vectorul variabilei dependente
y 1.25 1.39 1.85 1.93 2.35 2.72 3.08 3.74 3.93 4.26( )A a b c d e( ) a b c d e( )
z 2.36 2.64 2.87 2.99 3.65 4.44 4.86 5.69 5.88 6.33( )
u 1.51 1.96 2.31 2.45 2.84 3.14 3.59 4.75 5.02 5.66( ) A a b c d e( )T
a
b
c
d
e
Vectorul necunoscutelor
V 14.2 15.5 18.4 20.4 22.3 22.9 26.7 35.2 44.4 53.9( )
Se formeaza Matricea Xq [10, 5], astfel :
|
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||
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Xq AT
VT
=
XqT
Xq AT XqT VT=
XqT
Xq 1
XqT
Xq AT XqTXq 1
XqT
VT
=
Xq
1
1
1
1
1
1
1
1
1
1
1.11
1.25
1.49
1.61
1.82
2.05
2.27
2.44
2.61
2.89
1.25
1.39
1.85
1.93
2.35
2.72
3.08
3.74
3.93
4.26
2.36
2.64
2.87
2.99
3.65
4.44
4.86
5.69
5.88
6.33
1.51
1.96
2.31
2.45
2.84
3.14
3.59
4.75
5.02
5.66
Prelucrarea ecuatiei MatricealeA
TXq
TXq
1Xq
T V
T=
XqT
Xq 1
XqT
VT
6.594
13.99519.901
2.283
20.733
Fq x y z u( ) 6.594 13.995 x 19.901 y 2.283 z 20.773 u E Fq xT
yT
zT
uT
VT E Erori=
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3( )
Se1
100
9
i
Ei
Se 0.131
VT
0
0
1
2
3
4
5
6
7
8
9
14.2
15.5
18.4
20.4
22.3
22.9
26.7
35.2
44.4
53.9
Fq xT
yT
zT
uT
0
0
1
2
3
4
5
6
7
8
9
10.044
17.925
18.875
21.597
22.772
23.056
27.359
38.806
42.578
52.197
E
0
0
1
2
3
4
5
6
7
8
9
-4.156
2.425
0.475
1.197
0.472
0.156
0.659
3.606
-1.822
-1.703
1
100
9
i
Fq xT
yT
zT
uT
i VT i
0.131
1 0 1 2 3 4 5 6 7 8 9 100
510
15
20
2530
35
40
45
50
55
60
0
510
15
20
2530
35
40
45
50
55
60
Fq xT
yT
zT
uT
i VT
i
i i
Regresie Sinusoidala rezolvata Matriceal ca Regresie Lineara
f x a b( ) a b sin x( )
y 1.1 1.2 1.3 1.5 1.3 1.0 0.7 0.6 0.4 0.1( )
x 1 1.1 1.3 1.6 1.9 2.1 2.3 2.5 2.8 3.1( ) U 1 1 1 1 1 1 1 1 1 1( )
XS0 UT XS 1 sin xT A a b( )
a
b
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4( )XS
0 1
0
1
2
3
4
5
6
7
8
9
1 0.841
1 0.891
1 0.964
1 1
1 0.946
1 0.863
1 0.746
1 0.598
1 0.335
1 0.042
sin xT
0
0
1
2
3
4
5
6
7
8
9
0.841
0.891
0.964
1
0.946
0.863
0.746
0.598
0.335
0.042
XS A yT
= XST
XS A XST
yT
= XST
XS 1
XST
XS A XSTXS 1
XST
yT
= A XST
XS 1
XST
yT
=
XST
XS 1
XST
yT
0.078
1.381
fs z( ) 0.07825 1.38146 z z sin x
T
z
0
0
1
2
3
4
5
6
7
8
9
0.841
0.891
0.964
1
0.946
0.863
0.746
0.598
0.335
0.042
fs z( )
0
0
1
2
3
4
5
6
7
8
9
1.084
1.153
1.253
1.303
1.229
1.114
0.952
0.749
0.385
-0.021
yT
0
0
1
2
3
4
5
6
7
8
9
1.1
1.2
1.3
1.5
1.3
1
0.7
0.6
0.4
0.1
yT fs z( )
0
0
1
2
3
4
5
6
7
8
9
0.016
0.047
0.047
0.197
0.071
-0.114
-0.252
-0.149
0.015
0.121
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i 0 9 5( )
1 0 1 2 3 4 5 6 7 8 9 10 110.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Grafic Erori
fs z( )i y
T i
i i
Se1
100
9
i
yT
i fs z( ) i
Se 2.08 10 6
stdev yT
fs z( ) 0.127 Stdev yT fs z( ) 0.134
Formula stdev stdev Vi 1
n0
n 1
i
Vi mean Vi 2
1
2
=
Formula Stdev Stdev Vi 1
n 1( )0
n 1
i
Vi mean Vi 2
=
1
10
0
9
i
yT
i fs z( )( )
i 0.00119
2
1
2
0.127
1
90
9
i
yT
i fs z( ) i 0.00119
2
1
2
0.134
Regresie Logaritmica rezolvata Matriceal ca Regresie Lineara
f x a b( ) a b log x( )y 0.01 0.02 0.13 0.25 0.31 0.36 0.37 0.41 0.49 0.51( )
x 1 1.1 1.3 1.6 1.9 2.1 2.3 2.5 2.8 3.1( ) U 1 1 1 1 1 1 1 1 1 1( )
XL0 UT XL1 log xT A a b( )
a
b
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6( )
log xT
0
0
1
2
3
4
5
6
7
8
9
0
0.041
0.114
0.204
0.279
0.322
0.362
0.398
0.447
0.491
XL
0 1
0
1
2
3
4
5
6
7
8
9
1 0
1 0.041
1 0.114
1 0.204
1 0.279
1 0.322
1 0.362
1 0.398
1 0.447
1 0.491
XL A yT
= XLT
XL A XLT
yT
= XLT
XL 1
XLT
XL A XLTXL 1
XLT
yT
= A XLT
XL 1
XLT
yT
= XLT
XL 1
XLT
yT
6.158 10
3
1.053
fo z( ) 0.00616 1.05259 z z log xT z
0
0
1
2
3
4
5
6
78
9
0
0.041
0.114
0.204
0.279
0.322
0.362
0.3980.447
0.491
fo z( )
0
0
1
2
3
4
5
6
78
9
-36.1610
0.05
0.126
0.221
0.3
0.345
0.387
0.4250.477
0.523
yT
0
0
1
2
3
4
5
6
78
9
0.01
0.02
0.13
0.25
0.31
0.36
0.37
0.410.49
0.51
yT
fo z( )
0
0
1
2
3
4
5
6
78
9
-33.8410
-0.03
-33.90410
0.029
0.01
0.015
-0.017
-0.0150.013
-0.013
Se1
100
9
i
yT
i fo z( ) i
Se 3.313 10 6
stdev yT
fo z( ) 0.017 Stdev yT fo z( ) 0.018 i 0 9
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1 0 1 2 3 4 5 6 7 8 9 10 110.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Grafic Erori
fo z( )i y
T i
i i
7( )
stdev Vi 1
n0
n 1
i
Vi mean Vi 2
1
2
=
Stdev Vi 1
n 1( ) 0
n 1
i
Vi mean Vi 2
=
1
100
9
i
yT
i fo z( ) i 0.00119
2
1
2
0.017
1
9
0
9
i
yT
i fo z( ) i 0.00119
2
1
2
0.018
Regresie Exponentiala cu baza [e], rezolvata ca Regresie Lineara
f x a b( ) a b ex
y 3.1 3.2 3.9 4.5 7.8 12.1 19.1 35.5 46.7 58.8( )
x 1 1.1 1.3 1.6 1.9 2.4 2.9 3.5 3.8 4.1( ) U 1 1 1 1 1 1 1 1 1 1( ) yT
0
0
1
2
3
4
5
6
7
8
9
3.1
3.2
3.9
4.5
7.8
12.1
19.1
35.5
46.7
58.8
XE0 UT XE1 ex
T
A a b( )a
b
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8( )
ex
T
0
0
1
2
3
4
5
6
7
8
9
2.718
3.004
3.669
4.953
6.686
11.023
18.174
33.115
44.701
60.34
XE
0 1
0
1
2
3
4
5
6
7
8
9
1 2.718
1 3.004
1 3.669
1 4.953
1 6.686
1 11.023
1 18.174
1 33.115
1 44.701
1 60.34
XE A yT
= XET
XE A XET
yT
= XET
XE 1
XET
XE A XETXE 1
XET
yT
= A XET
XE 1
XET
yT
= XET
XE 1
XET
yT
0.668
0.998
fe z( ) 0.668 0.998 z z ex
T z
0
0
1
2
3
4
5
6
7
8
9
2.718
3.004
3.669
4.953
6.686
11.023
18.174
33.115
44.701
60.34
fe z( )
0
0
1
2
3
4
5
6
7
8
9
3.381
3.666
4.33
5.611
7.341
11.669
18.806
33.717
45.28
60.888
yT
0
0
1
2
3
4
5
6
7
8
9
3.1
3.2
3.9
4.5
7.8
12.1
19.1
35.5
46.7
58.8
yT
fe z( )
0
0
1
2
3
4
5
6
7
8
9
-0.281
-0.466
-0.43
-1.111
0.459
0.431
0.294
1.783
1.42
-2.088
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9( )Se
1
100
9
i
yT
i fe z( ) i
Se 1.185 10 3
stdev yT
fe z( ) 1.084 Stdev yT fe z( ) 1.143 i 0 9
1 0 1 2 3 4 5 6 7 8 9 10 110
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
fe z( )i y
T i
i i
stdev Vi 1
n0
n 1
i
Vi mean Vi 2
1
2
=
Stdev Vi 1
n 1( )0
n 1
i
Vi mean Vi 2
=
1
100
9
i
yT
i fe z( ) i 0.001192
1
2
1.084
1
90
9
i
yT
i fe z( ) i 0.001192
1
2
1.143
Regresie Exponentiala [cu baza n>0, n rezolvata Matriceal ca Regresie Lineara
f x m n( ) m nx
ln f( ) ln m( ) x ln n( )= F a b x=
y 2.4 2.8 4.5 7.9 13.5 20.8 30.7 43.1 53.1 77.2( ) w ln yT
ln yT
0
0
1
2
3
4
5
6
7
8
9
0.875
1.03
1.504
2.067
2.603
3.035
3.424
3.764
3.972
4.346
x 1 1.1 1.3 1.6 1.9 2.1 2.3 2.5 2.6 2.8( )
A a b( )a
b
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10( )U 1 1 1 1 1 1 1 1 1 1( )
xT
0
0
1
2
3
4
5
6
7
8
9
1
1.1
1.3
1.6
1.9
2.1
2.3
2.5
2.6
2.8
XN 0 UT XN 1 xT XN
0 1
0
1
2
3
4
5
6
7
8
9
1 1
1 1.1
1 1.3
1 1.6
1 1.9
1 2.1
1 2.3
1 2.5
1 2.6
1 2.8
XN A w= XNT
XN A XNT
w= XNT
XN 1
XNT
XN A XNTXN 1
XNT
w= A XNT
XN 1
XNT
w= XNT
XN 1
XNT
w 1.049
1.933
1.04947 ln m( )= m e 1.04947( )
m 0.35
1.93306 ln n( )= n e1.93306
n 6.911
fm x( ) 0.35012( ) 6.91062( )x
z xT
z
0
0
1
2
3
4
5
6
7
8
9
1
1.1
1.3
1.6
1.9
2.1
2.3
2.5
2.6
2.8
fm z( )
0
0
1
2
3
4
5
6
7
8
9
2.42
2.936
4.321
7.717
13.782
20.286
29.861
43.955
53.329
78.499
yT
0
0
1
2
3
4
5
6
7
8
9
2.4
2.8
4.5
7.9
13.5
20.8
30.7
43.1
53.1
77.2
yT
fm z( )
0
0
1
2
3
4
5
6
7
8
9
-0.02
-0.136
0.179
0.183
-0.282
0.514
0.839
-0.855
-0.229
-1.299
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11( )
Se1
100
9
i
yT
i fm z( )i
Se 0.11 stdev yT
fm z( ) 0.59 Stdev yT fm z( ) 0.622 i 0 9
1 0 1 2 3 4 5 6 7 8 9 10 110
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
Grafic Erori
fm z( )i y
T i
i i
stdev Vi 1
n0
n 1
i
Vi mean Vi 2
1
2
=
Stdev Vi 1
n 1( )0
n 1
i
Vi mean Vi 2
=
1
100
9
i
yT
i fm z( ) i 0.001192
1
2
0.6
1
9 0
9
i
yT
i fm z( )i 0.001192
1
2
0.633
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Regresie multilineara REZOLVATA CLASIC : Metoda Sumei celor mai mici patrate
y(x1,x2,x3,x4) = a + bx1 + cx2 + dx3 + hx4 12( )
Spatiul cu 5 dimensiuni :
Fma b c d h x1 x2 x3 x4 a b x1 c x2 d x3 h x4 x3t 2.36 2.64 2.87 2.99 3.65 4.44 4.86 5.69 5.88 6.33( )
x1t 1.11 1.25 1.49 1.61 1.82 2.05 2.27 2.44 2.61 2.89( ) x4t 1.51 1.96 2.31 2.45 2.84 3.14 3.59 4.75 5.02 5.66( )
x2t 1.25 1.39 1.85 1.93 2.35 2.72 3.08 3.74 3.93 4.26( ) Yt 14.2 15.5 18.4 20.4 22.3 22.9 26.7 35.2 44.4 53.9( )
x1tT
0
0
1
2
3
4
5
6
7
8
9
1.11
1.25
1.49
1.61
1.82
2.05
2.27
2.44
2.61
2.89
x2tT
0
0
1
2
3
4
5
6
7
8
9
1.25
1.39
1.85
1.93
2.35
2.72
3.08
3.74
3.93
4.26
x3tT
0
0
1
2
3
4
5
6
7
8
9
2.36
2.64
2.87
2.99
3.65
4.44
4.86
5.69
5.88
6.33
x4tT
0
0
1
2
3
4
5
6
7
8
9
1.51
1.96
2.31
2.45
2.84
3.14
3.59
4.75
5.02
5.66
Se defineste: U 1 1 1 1 1 1 1 1 1 1( ) Necunoscute a, b, c, d, h
S a b c d h( )
0
n
i
Fi yi 2
= n 5 Fi yi diferenta pentru fiecare punct= Fm xtT
1 YtT
1 Fm xtT
2 YtT
2 . . .
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13( )
with :a
Sd
d0=
bS
d
d0=
cS
d
d0=
dS
d
d0=
hS
d
d0=
S a b c d h( )
0
9
j
Fm xitT
.j Yt T
j
2
=
a, b, c, d, h
|
|
|
|
|
|
|
|
|
|
|
|
|
|
aS
d
d0=
bS
d
d0=
cS
d
d0= 2( ) Sistem de 5 ecuatii cu 5 necunoscute
dS
d
d0=
hS
d
d0=
S a b c d h( )
0
9
i
a b x1tT
i c x2t
T
i
d x3tT
i
h x4tT
i
YtT
i
2
U 0 1 2 3 4 5 6 7 8 9
0 1 1 1 1 1 1 1 1 1 1
aS
d
d2
0
9
i
a b x1tT
i c x2t
T
i
d x3tT
i
h x4tT
i
YtT
i
= 0=
0
9
i
YtT
i
U YtT
= v0 U Yt T
bS
d
d2
0
9
i
a b x1tT
i c x2t
T
i
d x3tT
i
h x4tT
i
YtT
i
x1tT
i
= 0=
0
9
i
YtT
i x1t
T
i
Yt x1tT
= v1 Yt x1tT
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14( )
cS
d
d2
0
9
i
a b x1tT
i c x2t
T
i
d x3tT
i
h x4tT
i
YtT
i
x2tT
i
= 0=
0
9
i
YtT
i x2t
T
i
Yt x2tT
= v2 Yt x2tT
dS
d
d2
0
9
i
a b x1tT
i c x2t
T
i
d x3tT
i
h x4tT
i
YtT
i
x3tT
i
= 0=
0
9
i
YtT
i x3t
T
i
Yt x3tT
= v3 Yt x3tT
hS
d
d2
0
9
i
a b x1tT
i c x2t
T
i
d x3tT
i
h x4tT
i
YtT
i
x4tT
i
= 0=
0
9
i
YtT
i x4t
T
i
Yt x4tT
= v4 Yt x4tT
0
9
i
1
10
0
9
i
x1tT
i
U xtT
=
0
9
i
x2tT
i
U x2tT
=
Definesc : Elementele matricei M[5 x 5]
M1 10 M2 U x1tT
M3 U x2tT
M4 U x3tT
M5 U x4tT
M2 U x1tT
N1 x1tx1tT
N2 x1t x2t
T N3 x1t x3t
T N4 x1t x4t
T
M3 U x2tT
N2 x1tx2tT
P1 x2t x2tT
P2 x3tx2tT
P3 x4t x2tT
M4 U x3tT
P2 x2t x3tT
Q1 x3t x3tT
Q2 x4t x3tT
N3 x1t x3t
T
Q2 x3t x4tT
R1 x4t x4tT
M5 U x4t
T N4 x1t x4t
T P3 x2t x4t
T
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15( )
M
M1
M2
M3
M4
M5
M2
N1
N2
N3
N4
M3
N2
P1
P2
P3
M4
N3
P2
Q1
Q2
M5
N4
P3
Q2
R1
M
10
19.54
26.5
41.71
33.23
19.54
41.392
57.528
89.311
72.317
26.5
57.528
80.629
124.678
101.491
41.71
89.311
124.678
193.391
156.821
33.23
72.317
101.491
156.821
128.072
V1
v0
v1
v2
v3
v4
V1
273.9
601.08
845.39
1.304 103
1.071 103
M 0.192 M 1
10.205
15.201
15.671
3.796
1.834
15.201
30.073
22.887
1.39
3.399
15.671
22.887
36.151
9.657
7.966
3.796
1.39
9.657
5.647
0.939
1.834
3.399
7.966
0.939
3.727
Fma b c d h x1 x2 x3 x4 a b x1 c x2 d x3 h x4
M 1
V1
6.594
13.995
19.901
2.283
20.733
a 6.594
b 13.995
c 19.901
d 2.283
h 20.733
Se por compara
rezultatele obtinute
prin cele doua
metode.
Fs x1t x2t x3t x4t 6.594 13.995x1t 19.901 x2t 2.283 x3t 20.733 x4t
Fs x1tT
x2tT
x3tT
x4tT
0
0
1
2
3
4
5
6
7
8
9
9.983
17.847
18.783
21.499
22.658
22.93
27.216
38.616
42.378
51.971
YtT
0
0
1
2
3
4
5
6
7
8
9
14.2
15.5
18.4
20.4
22.3
22.9
26.7
35.2
44.4
53.9
Fs x1tT
x2tT
x3tT
x4tT
YtT
0
0
1
2
3
4
5
6
7
8
9
-4.217
2.347
0.383
1.099
0.358
0.03
0.516
3.416
-2.022
-1.929
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16( )Calculul erorilor m
0
9
i
Fs x1tT
x2tT
x3tT
x4tT
i YtT
i
10
m 2.054 10 3
0 1 2 3 4 5 6 7 8 9 10
12
24
36
48
60
12
24
36
48
60
Fs x1tT
x2tT x3t
T x4tT i Yt
T i
i i
Invers : Fiind data o relatie liniara de forma ax+by+cz+... , relatie care descrie un fenomen oarecare,
si fiind apoi posibil a se efectua masuratori asupra acestui fenomen,la momente diferite de timp : t1, t2, t3,..., rezulta in final, urmatoarele date notate (1), (2),...(6) si
marcate cu culoare verde, astfel :
Valorile notate 1,2...6 cu culoare verde,
sunt valori masurate
Valorile expresiilor algebrice
pentru [xo ; yo] determinati :x 0.51 1.51 , Relatie necesara pentru afisarea graficului
Sa se rezolve sistemul
de ecuatii simultane:
xo 1.03 yo 2.011Prelucrez sistemul de ecuatii dupa acelasi procedeu, si rezulta (xo, yo)
1( ) xo yo 3.041 e1 3 3.041 e1 0.0411( ) x y 3=
2( ) 3 xo yo 1.079 e2 1 1.079 e2 0.0792( ) 3 x y 1=
3( ) 2xo yo 4.071 e3 4.1 4.071 e3 0.0293( ) 2x y 4.1=
A
1
3
2
3
1
1
1
1
1
1
1
3
AT
A 25
5
5
14
B
3
1
4.1
5.3
1.2
6.9
4( ) 3xo yo 5.101 e4 5.3 5.101 e4 0.199
4( ) 3x y 5.3=5( ) xo yo 0.981 e5 1.2 0.981 e5 0.219
5( ) x y 1.2=6( ) xo 3yo 7.063 e6 6.9 7.063 e6 0.163
6( ) x 3y 6.9=
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17( )
AT
A 1
AT
B 1.03
2.011
xo 1.03= yo 2.011=Media erorilor = Me :
e1 e2 e3 e4 e5 e6
60.019
0.50 0.70 0.90 1.10 1.30 1.500.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
2.011
3 x
1 3x
4.1 2 x
5.3 3x
1.2 x
2.3 0.33x
1.03
x
Solutia ar trebui sa fie [1 ; 2]
Fiecare ecuatie este aprox.egala
cu valorile obtinute
din inlocuirea : xp=1 ; yp=2.
Rezultat real : xo=1.03 ; yo=2.011
xo xp 3.0% xp=
yo yp 1.1% yp=
Me Mo 1.9%=
eof
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