---

Overview

metoda Gaussa
metody iteracyjne
zad. dom.
Arkusz6

---

Sheet 1: metoda Gaussa

4x1 + x2 + x3 = 9			Ita: dzielimy pierwsze równanie przez współczynnik przy x1 x1 + 0,25x2 + 0,25x3 = 2,25
x1 + 2x2 + x3 = 8
0,5x1 + x2 + 4x3 = 14,5			Ita: do drugiego równania mnożymy razy współczynnik przy x1 ze znakiem - (-)x1 - 0,25x2 - 0,25x3 = -2,25
			x1 + 2x2 + x3 = 8
			Ita: odejmujemy 2 - 0,25 1 - 0,25 8 - 2,25 1,75x2 + 0,75x3 = 5,75
			Ita: do trzeciego równania mnożymy razy współczynnik przy x1 ze znakiem - (-)0,5x1 - 0,125x2 - 0,125x3 = -1,125
			0,5x1 + x2 + 4x3 = 14,5
			Ita: odejmujemy 1-0,125 4-0,125 14,5-1,125 0,875x2 + 3,875x3 = 13,375
Ita: pierwsze równanie 1,75x2 + 0,75x3 = 5,75
Ita: drugie równanie 0,875x2 + 3,875x3 = 13,375
Ita: dzielenie pierwszego równania przez 1,75 x2 + 0,428x3 = 3,286
Ita: 0,428*(-0,875) 3,328 * (-0,875) (-)0,875x2 - 0,3745x3 = -2,875
0,875x2 + 3,875x3 =13,375
3,5005x3 = 10,5
x3 = 2,999

---

Sheet 2: metody iteracyjne

metoda Jacobiego
AX = P	X i P - wektory kolumnowe
Ita: kolejne przybliżenia kolejnych wektorów równania X(0) -> X(1) -> X(2)
4x1 + x2 = 1				przybliżenie startowe							4	1	0	0	0
x1 + 2x2 - x3 = 2					0						1	2	-1	0	0
x2 + 4x3 + 2x4 = 1					0						0	1	4	2	0
x3 + 6x4 + x5 = 2				X(0)	0						0	0	1	6	1
2x4 + 8x5 = 1					0						0	0	0	2	8
					0
przepisujemy równanie obliczając x1, które zostaje po lewej, reszta równań przechodzi na prawo											0.280882352941176	-0.123529411764706	-0.033823529411765	0.011764705882353	-0.001470588235294		1		0.0221
											-0.123529411764706	0.494117647058824	0.135294117647059	-0.047058823529412	0.005882352941176		2		0.9118
x1 = 1/4 (1 - x2)					0.25						0.033823529411765	-0.135294117647059	0.236764705882353	-0.082352941176471	0.010294117647059	x	1	=	-0.1544	=	X(10)
x2 = 1/2 (2 - x1 + x3)					1						-0.005882352941176	0.023529411764706	-0.041176470588235	0.188235294117647	-0.023529411764706		2		0.3529
x3 = 1/4 (1 -x2 - 2x4)				Ita: podstawiamy zerowe przybliżenie (X(0)) czyli same zera X(1)	0.25						0.001470588235294	-0.005882352941176	0.010294117647059	-0.047058823529412	0.130882352941176		1		0.0368
x4 = 1/6 (2 - x3 - x5)					0.333
x5 = 1/8 (1 - 2x4)					0.125
X(0)	X(1)	X(2)	X(3)	X(4)	X(5)	X(6)	X(7)	X(8)	X(9)	X(10)
0.0000	0.2500	0.0000	0.0000	0.0208	0.0169	0.0221	0.0216	0.0221	0.0221	0.0221
0.0000	1.0000	1.0000	0.9167	0.9323	0.9115	0.9134	0.9117	0.9118	0.9118	0.9117
0.0000	0.2500	-0.1667	-0.1354	-0.1563	-0.1563	-0.1545	-0.1548	-0.1544	-0.1544	-0.1544
0.0000	0.3333	0.2708	0.3542	0.3464	0.3533	0.3530	0.3530	0.3530	0.3529	0.3529
0.0000	0.1250	0.0417	0.0573	0.0365	0.0384	0.0367	0.0368	0.0368	0.0367	0.0368
0.2500	0.0000	-0.0208	0.0039	-0.0052	0.0005	-0.0004	0.0000	0.0000	0.0221
0.0000	0.0833	-0.0156	0.0208	-0.0020	0.0017	-0.0001	0.0000	0.0000	0.9117
0.4167	-0.0313	0.0208	0.0000	-0.0017	0.0003	-0.0004	0.0000	0.0000	-0.1544
0.0625	-0.0833	0.0078	-0.0069	0.0003	0.0000	0.0000	0.0001	0.0000	0.3529
0.0833	-0.0156	0.0208	-0.0020	0.0017	-0.0001	0.0000	0.0000	0.0000	0.0368
metoda Seidla
4x1 + x2 = 1				przybliżenie startowe
x1 + 2x2 - x3 = 2					0
x2 + 4x3 + 2x4 = 1					0
x3 + 6x4 + x5 = 2				X(0)	0
2x4 + 8x5 = 1					0
					0
x1 = 1/4 (1 - x2)					0.25
x2 = 1/2 (2 - x1 + x3)					1
x3 = 1/4 (1 -x2 - 2x4)				Ita: podstawiamy zerowe przybliżenie (X(0)) czyli same zera X(1)	0.25
x4 = 1/6 (2 - x3 - x5)					0.333
x5 = 1/8 (1 - 2x4)					0.125
X(0)	X(1)	X(2)	X(3)	X(4)	X(5)	X(6)
0.000	0.250	0.188	0.203	0.199	0.200	0.200
0.000	0.875	1.344	1.570	1.686	1.743	1.771
0.000	0.031	-0.102	-0.092	-0.125	-0.123	-0.131
0.000	0.328	0.336	0.335	0.340	0.341	0.342
0.000	0.043	0.041	0.041	0.040	0.040	0.040
	0.063	-0.016	0.004	-0.001	0.000
	-0.469	-0.227	-0.115	-0.057	-0.029
	0.133	-0.010	0.034	-0.003	0.008
	-0.008	0.001	-0.006	0.000	-0.001
	0.002	0.000	0.001	0.000	0.000

---

Sheet 3: zad. dom.

x1 + 2x2 - x3 = 2

4x1 + x2 = 1

x2 + 4x3 + 2x4 = 1

x3 + 6x4 + x5 = 2

2x4 + 8x5 = 1

---

Sheet 4: Arkusz6

ln(5x) - x = 0
x	f(x)
0.1	-0.793147180559945
0.2	-0.2
0.3	0.105465108108165
0.4	0.293147180559945
0.5	0.416290731874155
0.6	0.49861228866811
0.7	0.552762968495368
0.8	0.586294361119891
0.9	0.604077396776274
1	0.6094379124341
1.1	0.604748092238425
1.2	0.591759469228055
1.3	0.571802176901591
1.4	0.545910149055313
1.5	0.514903020542264
1.6	0.479441541679836
1.7	0.44006616349627
1.8	0.397224577336219
1.9	0.351291798606495
2	0.302585092994045
2.1	0.251375257163478
2.2	0.19789527279837
2.3	0.142347035369204
2.4	0.084906649788
2.5	0.025728644308255
2.6	-0.035050642538464
2.7	-0.097310314555617
2.8	-0.160942670384742
2.9	-0.225851350573472
3	-0.291949798897791
3.1	-0.3591599760748
3.2	-0.42741127776022
3.3	-0.496639619093466
3.4	-0.566786655943785
3.5	-0.637799119070533
3.6	-0.709628242103836
3.7	-0.782229267915722
3.8	-0.855561020833561
3.9	-0.9295855344303
4	-1.00426772644601
4.1	-1.07957511385564
4.2	-1.15547756227658
4.3	-1.23194706486638
4.4	-1.30895754664168
4.5	-1.38648469078963
4.6	-1.46450578407085
4.7	-1.54299957884989
4.8	-1.62194616965205
4.9	-1.70132688244932
5	-1.7811241751318