이공학도를 위한 수치해석

차례
■ 이 책의 목적 ························································································ 3
■ 이 책의 구성 ························································································ 5
■ 이 책의 특징 ························································································ 6
■ 감사의 글 ···························································································· 7

제1편 수치해석 (Numerical Methods)
1장 수치해석이란? ·················································································· 20
1.1 수치해석과 컴퓨터 / 20 1.2 수치해석의 응용과 전망 / 21
1.3 왜 Python인가? / 22 1.4 다른 언어와 Python 과의 비교 / 23
1.5 알고리즘 (Algorithms) / 25 1.6 알고리즘의 내용과 설계 / 27
2장 수치해석과 오차 ··············································································· 29
2.1 수치해석과 근삿값 / 29 2.2 모델링 오차 (modeling errors) / 30
2.3 반올림 오차 (round-off errors) / 30 2.4 절단 오차 (truncation erros) / 32
2.5 오차에 대한 수학적 해설 / 33 2.6 오차 계산 / 34
2.7 요약 / 37 2.8 과제 / 37

제2편 Python
3장 설치하기 ························································································· 42
3.1 Python은? / 42 3.2 윈도우용 Python 설치하기 / 42
4장 Python 문법 ··················································································· 46
4.1 Python 문법 / 46 4.2 리스트(lists) / 47
4.3 프로그램 작성시 유의점 / 49 4.4 연산기호 / 49
4.5 조건 (if/else) / 50 4.6 반복 (for/range) / 52
4.7 함수 (functions) / 55 4.8 과제 / 57
5장 Numpy ·························································································· 58
5.1 Numpy / 58 5.2 벡터와 행렬 / 60
5.3 arange/linspace 함수 / 62 5.4 ones/zeros 함수 / 64
5.5 행렬의 덧셈과 뺄셈 / 64 5.6 행렬과 상수의 곱셈과 나눗셈 / 65
5.7 내적과 행렬곱 / 66 5.8 조회와 추출 (indexing and slicing) / 69
5.9 벡터화 기법 / 73 5.10 요약 / 75
5.11 과제 / 75
6장 Matplotlib ······················································································ 77
6.1 함수를 그리는 법 / 77 6.2 데이터를 사용하여 그리기 / 80
6.3 요약 / 81 6.4 과제 / 81

제3편 방정식의 근 구하기
7장 방정식의 근 구하기 ········································································ 84
7.1 근의 존재구간 / 84 7.2 중간값 정리 / 85
7.3 평균값 정리 / 86 7.4 근의 존재 여부 / 87
7.5 요약 / 90 7.6 과제 / 91
8장 이분법 (Bisection methods) ··························································· 92
8.1 이분법: 해설 / 92 8.2 이분법: 코딩 / 93
8.3 요약 / 96 8.4 과제 / 96
9장 고정점 방법 (Fixed point methods) ················································ 98
9.1 고정점 방법: 해설 / 98 9.2 고정점 방법: 코딩 / 100
9.3 요약 / 102 9.4 과제 / 102
10장 Newton-Raphson 방법 ······························································ 103
10.1 Newton-Raphson 방법:해설 / 103 10.2 Newton-Raphson 방법: 코딩 / 105
10.3 요약 / 107 10.4 과제 / 108
11장 완화법 (Relaxation Methods) ····················································· 109
11.1 완화법: 해설 / 109 11.2 완화법: 코딩 / 112
11.3 요약 / 114 11.4 과제 / 114
12장 비선형 연립방정식 ····································································· 116
12.1 비선형 연립방정식: 해설 / 116 12.2 비선형 연립방정식: 코딩 / 119
12.3 요약 / 121 12.4 과제 / 121
13장 할선법 (Secant methods) ··························································· 123
13.1 할선법: 해설 / 123 13.2 할선법: 코딩 / 125
13.3 요약 / 127 13.4 과제 / 127
14장 근 구하기 알고리즘의 비교 ························································· 128
14.1 알고리즘의 문제 / 128 14.2 알고리즘의 비교 / 129
14.3 요약 / 132
15장 모듈화 프로그래밍의 응용 I ························································ 133
15.1 모듈화 프로그래밍 / 133 15.2 모듈화 프로그래밍의 응용 : 코딩 / 133

제4편 선형연립방정식
16장 선형연립방정식 ·········································································· 142
16.1 선형연립방정식: 개요 / 142 16.2 선형연립방정식의 해법의 종류 / 144
17장 Cramer 방법 ············································································· 145
17.1 Cramer 방법: 해설 / 145 17.2 Cramer 방법: 코딩 / 147
17.3 요약 / 149 17.4 과제 / 149
18장 Gauss 소거법 (Gaussian Elimination) ·········································· 152
18.1 Gauss 소거법: 개요 / 152 18.2 Gauss 소거법: 해설 / 153
18.3 Gauss 소거법: 일반화 / 155 18.4 Gauss 소거법: 코딩 / 158
18.5 Gauss 소거법: 피보팅 문제 / 161 18.6 요약 / 162
18.7 과제 / 162
19장 LU 분해법 (LU Decomposition) ·················································· 165
19.1 LU 분해법: 해설 / 165 19.2 분해법: 선형연립방정식의 해법 / 167
19.3 Cholesky 분해법: 해설 / 170 19.4 Cholesky 분해법: 코딩 / 174
19.5 요약 / 175 19.6 과제 / 176
20장 Jacobi 방법 ················································································ 178
20.1 Jacobi 방법: 해설 / 178 20.2 Jacobi 방법: 알고리즘 / 180
20.3 Jacobi 방법: 코딩 / 183 20.4 요약 / 186
20.5 과제 / 186
21장 Gauss-Seidel 방법 ······································································ 189
21.1 Gauss-Seidel 방법: 해설 / 189 21.2 Gauss-Seidel 방법: 일반화 / 191
21.3 Gauss-Seidel 방법: 코딩 / 192 21.4 요약 / 195
21.5 과제 / 195
22장 반복법에 대한 비교와 분석 ······························································ 198
22.1 반복법 알고리즘의 특성 / 198 22.2 근사해의 수렴특성 / 198
22.3 피보팅 (pivoting)의 영향 / 200 22.4 요약 / 202
23장 SOR 방법 ······················································································ 203
23.1 SOR 방법: 해설 / 203 23.2 SOR 방법: 코딩 / 205
23.3 요약 / 209 23.4 과제 / 209

제5편 데이터와 수치해석
24장 데이터와 수치해석 ·········································································· 214
24.1 데이터와 수치해석의 필요성 / 214 24.2 데이터의 분류 / 214
24.3 데이터의 기본정보 / 217 24.4 데이터 기본정보: 코딩 / 219
24.5 데이터의 가시화 / 220 24.6 요약 / 224
24.7 과제 / 225
25장 회귀법 ·························································································· 226
25.1 회귀법과 보간법 / 226 25.2 회귀법 / 226
25.3 최소자승법: 해설 / 227 25.4 최소자승법: 일반화 / 231
25.5 최소자승법: 코딩 / 232 25.6 요약 / 233
25.7 과제 / 234
26장 다항식 회귀법 ·············································································· 235
26.1 다항식 회귀법: 해설 / 235 26.2 다항식 회귀법: 코딩 / 237
26.3 요약 / 240 26.4 과제 / 240
27장 비선형 회귀법 ·············································································· 241
27.1 비선형 회귀법: 해설 / 241 27.2 비선형 회귀법: 코딩 / 243
27.3 요약 / 246 27.4 과제 / 247

제6편 보간법 (Interpolation)
28장 선형보간법 (linear interpolation) ·················································· 250
28.1 보간법의 목적 / 250 28.2 선형보간법: 해설 / 251
28.3 선형보간법: 데이터 / 252 28.4 선형보간법: 함수 / 255
28.5 요약 / 257 28.6 과제 / 257
29장 다항식 보간법 (polynomial interpolation) ······································ 259
29.1 Newton 다항식 보간법: 해설 / 259 29.2 Newton의 다항식 보간법: 데이터 / 261
29.3 Newton 다항식 보간법: 코딩 / 264 29.4 Newton의 다항식 보간법: 함수 / 266
29.5 요약 / 268 29.6 과제 / 268
30장 Lagrange 보간법 ········································································ 270
30.1 Lagrange 보간법: 해설 / 270 30.2 Lagrange 보간법: 예제 / 272
30.3 요약 / 273

제7편 적분과 미분 (Calculus)
31장 사다리꼴 적분공식 ········································································ 276
31.1 수치적분 / 276 31.2 사다리꼴 적분공식: 해설 / 277
31.3 조합 사다리꼴 적분공식: 해설 / 279 31.4 조합 사다리꼴 적분공식: 코딩 / 282
31.5 조합 사다리꼴 적분공식: 오차 / 283 31.6 요약 / 284
31.7 과제 / 285
32장 Simpson 적분공식 ······································································ 287
32.1 Simpson 적분공식: 해설 / 287 32.2 조합 Simpson 적분공식: 해설 / 289
32.3 조합 Simpson 적분공식: 코딩 / 290 32.4 조합 Simpson 적분공식: 오차 / 291
32.5 요약 / 292 32.6 과제 / 293
33장 데이터에 대한 수치적분 ······························································ 295
33.1 데이터와 수치적분 / 295
33.2 데이터에 대한 사다리꼴 수치적분: 코딩 / 295
33.3 데이터에 대한 Simpson 수치적분: 코딩 / 297
33.4 등 간격이 아닌 데이터에 대한 수치적분: 코딩 / 298
33.5 요약 / 300
33.6 과제 / 301
34장 Gauss 적분법 ············································································· 303
34.1 Gauss 적분법: 해설 / 303 34.2 Gauss 적분: 코딩 / 306
34.3 요약 / 307 34.4 과제 / 308
35장 수치미분 (numerical differentiation) ············································ 310
35.1 수치미분: 해설 / 310 35.2 수치미분: 함수에 대한 1차 미분 / 312
35.3 수치미분: 함수에 대한 2차 미분 / 314 35.4 수치미분: 코딩 / 314
35.5 수치미분: 데이터에 대한 미분 / 316 35.6 요약 / 317
35.7 과제 / 318

제8편 상미분 방정식 (Ordinary Differential Equations)
36장 Euler 방법 ················································································ 322
36.1 상미분방정식 / 322 36.2 초기값 문제 / 323
36.3 Euler 방법: 해설 / 323 36.4 Euler 방법: 코딩 / 325
36.5 Euler 방법에서의 오차 / 327 36.6 요약 / 330
36.7 과제 / 330
37장 Runge-Kutta 방법 ····································································· 332
37.1 Runge-Kutta 방법: 해설 / 332 37.2 2차 Runge-Kutta 방법: 해설 / 333
37.3 2차 Runge-Kutta 방법: 코딩 / 335 37.4 2차 Runge-Kutta 방법: 오차의 감소 /337
37.5 요약 / 338 37.6 과제 / 338
38장 4차 Runge-Kutta 방법 ································································ 340
38.1 4차 Runge-Kutta 방법: 해설 / 340 38.2 4차 Runge-Kutta 방법: 예제 / 341
38.3 4차 Runge-Kutta 방법: 코딩 / 343 38.4 4차 Runge-Kutta 방법: 오차 / 344
38.5 요약 / 345 38.6 과제 / 346
39장 예측자-수정자 방법 ···································································· 347
39.1 예측자와 수정자 / 347 39.2 Heun 방법: 해설 / 347
39.3 Heun 방법: 코딩 / 349 39.4 Heun 방법: 오차 / 351
39.5 요약 / 352 39.6 과제 / 352
40장 Adams-Bashforth 방법 ······························································ 354
40.1 다단계 방법 (multi-step methods) / 354 40.2 Adams-Bashforth 방법: 해설 / 355
40.3 Adams-Bashforth 방법: 코딩 / 358 40.4 Adams-Bashforth 방법: 오차 / 360
40.5 요약 / 361 40.6 과제 / 361
41장 Adams-Bashforth-Moulton 방법 ················································ 363
41.1 Adams-Bashforth-Moulton 방법: 해설 / 363
41.2 2단계 Adams-Bashforth-Moulton 방법 : 코딩 / 365
41.3 4단계 Adams-Bashforth-Moulton 방법: 해설 및 코딩 / 366
41.4 Adams-Bashforth-Moulton 방법: 오차 / 369
41.5 요약 / 370
41.6 과제 / 371
42장 연립미분방정식 ·········································································· 373
42.1 연립미분방정식: 해설 / 373 42.2 연립미분방정식: Euler 방법 코딩 / 375
42.3 연립미분방정식: RK4 방법 해설 / 376 42.4 연립미분방정식: RK4 방법 코딩 / 378
42.5 연립미분방정식: 오차 / 379 42.6 요약 / 381
42.7 과제 / 381
43장 고차미분방정식 (Higher-order Differential Equations) ···················382
43.1 고차미분방정식과 Euler 방법: 해설 / 382
43.2 고차미분방정식과 Euler 방법: 코딩 / 384
43.3 고차미분방정식과 4차 Runge-Kutta 방법: 해설 / 386
43.4 고차미분방정식과 RK4 방법: 코딩 / 389
43.5 요약 / 391
43.6 과제 / 391
44장 모듈화 프로그래밍의 응용 II ······················································· 393
44.1 모듈화 프로그래밍 / 393 44.2 모듈화 프로그래밍의 응용 II : 코딩 / 393

Appendices
■ 과제 해답 ······················································································400
■ 참고문헌 ······················································································ 449
■ 찾아보기 ······················································································ 450