수치해석 MATLAB으로 exponential 값을 근사하라.

clear all

close all

b = 2; %구간 설정

m = 40; %구간개수 설정

degree = 5; %전개할 테일러함수의 항 개수

degrees = 5; %마지막으로 출력할 테일러함수의 항 개수 (degree보다 크다)

a = 0; %f(x-a) 의 a 값

delta_x = 2*b/m; %구간 사이의 거리

func = exp(a); %테일러함수 0째 항

eps_t = 0; %오차값 초기화

f_true = 0; %실제값 초기화

j = 1; %테일러함수 1항으로부터 시작.

while degree <=degrees

fprintf ('\ndegree : %d\n',degree);

fprintf ('----x값\t테일러함수값\t--실제함수값\t\t오차\n');

x=-b; %x 는 -b로부터

while x<b+delta_x %b 까지 계산한다.

f_true = exp(x); %실제값은 e^x

y=funct(a,j,x,func,degree); %각 x값마다 계산한다.

eps_t=(f_true-y)/f_true; %오차값 계산

fprintf('%2.4f\t%2.7f\t%2.7f\t%2.7f\n', x,y,f_true,eps_t);

%x값, 테일러함수, 실제값, 오차 출력

x = x+delta_x; %x를 증분 delta_x만큼 증가.

end

degree=degree+1; %다음항 계산분까지 출력 증가.

end

funct.m 파일 (funct 함수를 이용하기 위해.)

function y=funct(a,j,x,func,degree)

y=func;

while j<=degree

y=y+exp(a)/factorial(j).*(x-a).^j;

j=j+1;

end

end

결과

degree:1

x값	테일러값	실제함수값	오차
-2.0000	-1.0000000	0.1353353	8.3890561
-1.9000	-0.9000000	0.1495686	7.0173050
-1.8000	-0.8000000	0.1652989	5.8397180
-1.7000	-0.7000000	0.1826835	4.8317632
-1.6000	-0.6000000	0.2018965	3.9718195
-1.5000	-0.5000000	0.2231302	3.2408445
-1.4000	-0.4000000	0.2465970	2.6220800
-1.3000	-0.3000000	0.2725318	2.1007890
-1.2000	-0.2000000	0.3011942	1.6640234
-1.1000	-0.1000000	0.3328711	1.3004166
-1.0000	0.0000000	0.3678794	1.0000000
-0.9000	0.1000000	0.4065697	0.7540397
-0.8000	0.2000000	0.4493290	0.5548918
-0.7000	0.3000000	0.4965853	0.3958742
-0.6000	0.4000000	0.5488116	0.2711525
-0.5000	0.5000000	0.6065307	0.1756394
-0.4000	0.6000000	0.6703200	0.1049052
-0.3000	0.7000000	0.7408182	0.0550988
-0.2000	0.8000000	0.8187308	0.0228778
-0.1000	0.9000000	0.9048374	0.0053462
0.0000	1.0000000	1.0000000	0.0000000
0.1000	1.1000000	1.1051709	0.0046788
0.2000	1.2000000	1.2214028	0.0175231
0.3000	1.3000000	1.3498588	0.0369363
0.4000	1.4000000	1.4918247	0.0615519
0.5000	1.5000000	1.6487213	0.0902040
0.6000	1.6000000	1.8221188	0.1219014
0.7000	1.7000000	2.0137527	0.1558050
0.8000	1.8000000	2.2255409	0.1912079
0.9000	1.9000000	2.4596031	0.2275176
1.0000	2.0000000	2.7182818	0.2642411
1.1000	2.1000000	3.0041660	0.3009707
1.2000	2.2000000	3.3201169	0.3373727
1.3000	2.3000000	3.6692967	0.3731769
1.4000	2.4000000	4.0552000	0.4081673
1.5000	2.5000000	4.4816891	0.4421746
1.6000	2.6000000	4.9530324	0.4750691
1.7000	2.7000000	5.4739474	0.5067545
1.8000	2.8000000	6.0496475	0.5371631
1.9000	2.9000000	6.6858944	0.5662510
2.0000	3.0000000	7.3890561	0.5939942

degree:2

----x값	테일러함수값	--실제함수값	오차
-2.0000	1.0000000	0.1353353	-6.3890561
-1.9000	0.9050000	0.1495686	-5.0507345
-1.8000	0.8200000	0.1652989	-3.9607109
-1.7000	0.7450000	0.1826835	-3.0780908
-1.6000	0.6800000	0.2018965	-2.3680620
-1.5000	0.6250000	0.2231302	-1.8010557
-1.4000	0.5800000	0.2465970	-1.3520160
-1.3000	0.5450000	0.2725318	-0.9997667
-1.2000	0.5200000	0.3011942	-0.7264608
-1.1000	0.5050000	0.3328711	-0.5171038
-1.0000	0.5000000	0.3678794	-0.3591409
-0.9000	0.5050000	0.4065697	-0.2420996
-0.8000	0.5200000	0.4493290	-0.1572813
-0.7000	0.5450000	0.4965853	-0.0974952
-0.6000	0.5800000	0.5488116	-0.0568289
-0.5000	0.6250000	0.6065307	-0.0304508
-0.4000	0.6800000	0.6703200	-0.0144408
-0.3000	0.7450000	0.7408182	-0.0056448
-0.2000	0.8200000	0.8187308	-0.0015503
-0.1000	0.9050000	0.9048374	-0.0001797
0.0000	1.0000000	1.0000000	0.0000000
0.1000	1.1050000	1.1051709	0.0001547
0.2000	1.2200000	1.2214028	0.0011485
0.3000	1.3450000	1.3498588	0.0035995
0.4000	1.4800000	1.4918247	0.0079263
0.5000	1.6250000	1.6487213	0.0143877
0.6000	1.7800000	1.8221188	0.0231153
0.7000	1.9450000	2.0137527	0.0341416
0.8000	2.1200000	2.2255409	0.0474226
0.9000	2.3050000	2.4596031	0.0628569
1.0000	2.5000000	2.7182818	0.0803014
1.1000	2.7050000	3.0041660	0.0995837
1.2000	2.9200000	3.3201169	0.1205129
1.3000	3.1450000	3.6692967	0.1428875
1.4000	3.3800000	4.0552000	0.1665023
1.5000	3.6250000	4.4816891	0.1911532
1.6000	3.8800000	4.9530324	0.2166415
1.7000	4.1450000	5.4739474	0.2427768
1.8000	4.4200000	6.0496475	0.2693789
1.9000	4.7050000	6.6858944	0.2962796
2.0000	5.0000000	7.3890561	0.3233236

degree : 3

x값	테일러함수값	실제함수값	오차
-2.0000	-0.3333333	0.1353353	3.4630187
-1.9000	-0.2381667	0.1495686	2.5923572
-1.8000	-0.1520000	0.1652989	1.9195464
-1.7000	-0.0738333	0.1826835	1.4041598
-1.6000	-0.0026667	0.2018965	1.0132081
-1.5000	0.0625000	0.2231302	0.7198944
-1.4000	0.1226667	0.2465970	0.5025621
-1.3000	0.1788333	0.2725318	0.3438074
-1.2000	0.2320000	0.3011942	0.2297329
-1.1000	0.2831667	0.3328711	0.1493203
-1.0000	0.3333333	0.3678794	0.0939061
-0.9000	0.3835000	0.4065697	0.0567422
-0.8000	0.4346667	0.4493290	0.0326315
-0.7000	0.4878333	0.4965853	0.0176243
-0.6000	0.5440000	0.5488116	0.0087674
-0.5000	0.6041667	0.6065307	0.0038976
-0.4000	0.6693333	0.6703200	0.0014720
-0.3000	0.7405000	0.7408182	0.0004296
-0.2000	0.8186667	0.8187308	0.0000783
-0.1000	0.9048333	0.9048374	0.0000045
0.0000	1.0000000	1.0000000	0.0000000
0.1000	1.1051667	1.1051709	0.0000038
0.2000	1.2213333	1.2214028	0.0000568
0.3000	1.3495000	1.3498588	0.0002658
0.4000	1.4906667	1.4918247	0.0007763
0.5000	1.6458333	1.6487213	0.0017516
0.6000	1.8160000	1.8221188	0.0033581
0.7000	2.0021667	2.0137527	0.0057535
0.8000	2.2053333	2.2255409	0.0090799
0.9000	2.4265000	2.4596031	0.0134587
1.0000	2.6666667	2.7182818	0.0189882
1.1000	2.9268333	3.0041660	0.0257418
1.2000	3.2080000	3.3201169	0.0337690
1.3000	3.5111667	3.6692967	0.0430955
1.4000	3.8373333	4.0552000	0.0537253
1.5000	4.1875000	4.4816891	0.0656425
1.6000	4.5626667	4.9530324	0.0788135
1.7000	4.9638333	5.4739474	0.0931894
1.8000	5.3920000	6.0496475	0.1087084
1.9000	5.8481667	6.6858944	0.1252978
2.0000	6.3333333	7.3890561	0.1428765

degree : 4

x값	테일러함수값	실제함수값	오차
-2.0000	0.3333333	0.1353353	-1.4630187
-1.9000	0.3048375	0.1495686	-1.0381113
-1.8000	0.2854000	0.1652989	-0.7265694
-1.7000	0.2741708	0.1826835	-0.5007967
-1.6000	0.2704000	0.2018965	-0.3393000
-1.5000	0.2734375	0.2231302	-0.2254619
-1.4000	0.2827333	0.2465970	-0.1465402
-1.3000	0.2978375	0.2725318	-0.0928541
-1.2000	0.3184000	0.3011942	-0.0571252
-1.1000	0.3441708	0.3328711	-0.0339463
-1.0000	0.3750000	0.3678794	-0.0193557
-0.9000	0.4108375	0.4065697	-0.0104972
-0.8000	0.4517333	0.4493290	-0.0053510
-0.7000	0.4978375	0.4965853	-0.0025216
-0.6000	0.5494000	0.5488116	-0.0010721
-0.5000	0.6067708	0.6065307	-0.0003960
-0.4000	0.6704000	0.6703200	-0.0001193
-0.3000	0.7408375	0.7408182	-0.0000260
-0.2000	0.8187333	0.8187308	-0.0000032
-0.1000	0.9048375	0.9048374	-0.0000001
0.0000	1.0000000	1.0000000	0.0000000
0.1000	1.1051708	1.1051709	0.0000001
0.2000	1.2214000	1.2214028	0.0000023
0.3000	1.3498375	1.3498588	0.0000158
0.4000	1.4917333	1.4918247	0.0000612
0.5000	1.6484375	1.6487213	0.0001721
0.6000	1.8214000	1.8221188	0.0003945
0.7000	2.0121708	2.0137527	0.0007855
0.8000	2.2224000	2.2255409	0.0014113
0.9000	2.4538375	2.4596031	0.0023441
1.0000	2.7083333	2.7182818	0.0036598
1.1000	2.9878375	3.0041660	0.0054353
1.2000	3.2944000	3.3201169	0.0077458
1.3000	3.6301708	3.6692967	0.0106630
1.4000	3.9974000	4.0552000	0.0142533
1.5000	4.3984375	4.4816891	0.0185759
1.6000	4.8357333	4.9530324	0.0236823
1.7000	5.3118375	5.4739474	0.0296148
1.8000	5.8294000	6.0496475	0.0364067
1.9000	6.3911708	6.6858944	0.0440814
2.0000	7.0000000	7.3890561	0.0526530

degree : 5

x값	테일러함수값	실제함수값	오차
-2.0000	0.0666667	0.1353353	0.5073963
-1.9000	0.0984959	0.1495686	0.3414667
-1.8000	0.1279360	0.1652989	0.2260323
-1.7000	0.1558494	0.1826835	0.1468885
-1.6000	0.1830187	0.2018965	0.0935026
-1.5000	0.2101563	0.2231302	0.0581450
-1.4000	0.2379147	0.2465970	0.0352085
-1.3000	0.2668964	0.2725318	0.0206779
-1.2000	0.2976640	0.3011942	0.0117207
-1.1000	0.3307499	0.3328711	0.0063723
-1.0000	0.3666667	0.3678794	0.0032967
-0.9000	0.4059168	0.4065697	0.0016059
-0.8000	0.4490027	0.4493290	0.0007262
-0.7000	0.4964369	0.4965853	0.0002988
-0.6000	0.5487520	0.5488116	0.0001087
-0.5000	0.6065104	0.6065307	0.0000334
-0.4000	0.6703147	0.6703200	0.0000080
-0.3000	0.7408173	0.7408182	0.0000013
-0.2000	0.8187307	0.8187308	0.0000001
-0.1000	0.9048374	0.9048374	0.0000000
0.0000	1.0000000	1.0000000	0.0000000
0.1000	1.1051709	1.1051709	0.0000000
0.2000	1.2214027	1.2214028	0.0000001
0.3000	1.3498578	1.3498588	0.0000008
0.4000	1.4918187	1.4918247	0.0000040
0.5000	1.6486979	1.6487213	0.0000142
0.6000	1.8220480	1.8221188	0.0000389
0.7000	2.0135714	2.0137527	0.0000900
0.8000	2.2251307	2.2255409	0.0001843
0.9000	2.4587583	2.4596031	0.0003435
1.0000	2.7166667	2.7182818	0.0005942
1.1000	3.0012584	3.0041660	0.0009679
1.2000	3.3151360	3.3201169	0.0015002
1.3000	3.6611119	3.6692967	0.0022306
1.4000	4.0422187	4.0552000	0.0032011
1.5000	4.4617188	4.4816891	0.0044560
1.6000	4.9231147	4.9530324	0.0060403
1.7000	5.4301589	5.4739474	0.0079994
1.8000	5.9868640	6.0496475	0.0103780
1.9000	6.5975124	6.6858944	0.0132192
2.0000	7.2666667	7.3890561	0.0165636

x	degree : 1	degree : 2	degree : 3	degree : 4	degree : 5	TRUE
-2	-1	1	-0.3333	0.3333	0.0667	0.1353
-1.9	-0.9	0.905	-0.2382	0.3048	0.0985	0.1496
-1.8	-0.8	0.82	-0.152	0.2854	0.1279	0.1653
-1.7	-0.7	0.745	-0.0738	0.2742	0.1558	0.1827
-1.6	-0.6	0.68	-0.0027	0.2704	0.183	0.2019
-1.5	-0.5	0.625	0.0625	0.2734	0.2102	0.2231
-1.4	-0.4	0.58	0.1227	0.2827	0.2379	0.2466
-1.3	-0.3	0.545	0.1788	0.2978	0.2669	0.2725
-1.2	-0.2	0.52	0.232	0.3184	0.2977	0.3012
-1.1	-0.1	0.505	0.2832	0.3442	0.3307	0.3329
-1	0	0.5	0.3333	0.375	0.3667	0.3679
-0.9	0.1	0.505	0.3835	0.4108	0.4059	0.4066
-0.8	0.2	0.52	0.4347	0.4517	0.449	0.4493
-0.7	0.3	0.545	0.4878	0.4978	0.4964	0.4966
-0.6	0.4	0.58	0.544	0.5494	0.5488	0.5488
-0.5	0.5	0.625	0.6042	0.6068	0.6065	0.6065
-0.4	0.6	0.68	0.6693	0.6704	0.6703	0.6703
-0.3	0.7	0.745	0.7405	0.7408	0.7408	0.7408
-0.2	0.8	0.82	0.8187	0.8187	0.8187	0.8187
-0.1	0.9	0.905	0.9048	0.9048	0.9048	0.9048
0	1	1	1	1	1	1
0.1	1.1	1.105	1.1052	1.1052	1.1052	1.1052
0.2	1.2	1.22	1.2213	1.2214	1.2214	1.2214
0.3	1.3	1.345	1.3495	1.3498	1.3499	1.3499
0.4	1.4	1.48	1.4907	1.4917	1.4918	1.4918
0.5	1.5	1.625	1.6458	1.6484	1.6487	1.6487
0.6	1.6	1.78	1.816	1.8214	1.822	1.8221
0.7	1.7	1.945	2.0022	2.0122	2.0136	2.0138
0.8	1.8	2.12	2.2053	2.2224	2.2251	2.2255
0.9	1.9	2.305	2.4265	2.4538	2.4588	2.4596
1	2	2.5	2.6667	2.7083	2.7167	2.7183
1.1	2.1	2.705	2.9268	2.9878	3.0013	3.0042
1.2	2.2	2.92	3.208	3.2944	3.3151	3.3201
1.3	2.3	3.145	3.5112	3.6302	3.6611	3.6693
1.4	2.4	3.38	3.8373	3.9974	4.0422	4.0552
1.5	2.5	3.625	4.1875	4.3984	4.4617	4.4817
1.6	2.6	3.88	4.5627	4.8357	4.9231	4.953
1.7	2.7	4.145	4.9638	5.3118	5.4302	5.4739
1.8	2.8	4.42	5.392	5.8294	5.9869	6.0496
1.9	2.9	4.705	5.8482	6.3912	6.5975	6.6859
2	3	5	6.3333	7	7.2667	7.3891