Daftar Isi Kata Pengantar ................................................................................ Daftar Isi ........................................................................................... Daftar Padan Kata ............................................................................
iii v xi
1. Metode Numerik Secara Umum ...............................
1
1.1 1.2 1.3
Metode Analitik versus Metode Numerik .............................. Metode Numerik dalam Bidang Rekayasa ............................. Apakah Metode Numerik Hanya untuk Persoalan Matematika yang Rumit Saja ? ............................................................... Peranan Komputer dalam Metode Numerik ........................... Mengapa Kita Harus Mempelajari Metode Numerik ? ............ Tahap-tahap Memecahkan Persoalan Secara Numerik ........... Peran Ahli Informatika dalam Metode Numerik .................... Perbedaan Metode Numerik dengan Analisis Numerik ............ Materi Apa yang Terdapat di dalam Buku Ini ? .......................
7 8 9 10 11 11 12
2. Deret Taylor Dan Analisis Galat .........................
15
1.4 1.5 1.6 1.7 1.8 1.9
2.1 2.2 2.3
2.4
Daftar Isi
4 6
Deret Taylor ....................................................................... Analisis Galat ..................................................................... Sumber Utama Galat Numerik .............................................
16 21 23
2.3.1 2.3.2 2.3.3
Galat Pemotongan ..................................................................... Galat Pembulatan ...................................................................... Galat Total .................................................................................
23 26 27
Orde Penghampiran ............................................................
28
v
2.5
2.6 2.7 2.8
Bilangan Titik-Kambang ......................................................
29
2.5.1 2.5.2 2.5.3 2.5.4
Bilangan Titik-Kambang Ternormalisasi ................................ Epsilon Mesin ............................................................................ Pembulatan pada Bilangan Titik-Kambang ............................ Aritmetika Bilangan Titik-Kambang .......................................
30 32 35 38
2.5.4.1 2.5.4.2
Operasi Penambahan dan Pengurangan ......................... Operasi Perkalian dan Pembagian ..................................
38 44
Perambatan Galat ............................................................... Kondisi Buruk .................................................................... Bilangan Kondisi .................................................................
45 46 49
3. Solusi Persamaan Nirlanjar 3.1 3.2 3.3
3.4
3.5 3.6
3.7
3.8
vi
.......................................
55
Rumusan Masalah .............................................................. Metode Pencarian Akar ....................................................... Meode Tertutup ..................................................................
57 57 58
3.3.1 3.3.2
Metode Bagidua ........................................................................ Metode Regula Falsi .................................................................
61 67
Metode Terbuka ..................................................................
72
3.4.1 3.4.2 3.4.3 3.4.4
Metode Lelaran Titik-Tetap ..................................................... Metode Newton-Raphson ......................................................... Orde Konvergensi Metode Terbuka ......................................... Metode Secant ..........................................................................
72 84 91 93
Akar Ganda ........................................................................ Akar-akar Polinom ..............................................................
97 100
3.6.1 3.6.2 3.6.3
Metode Horner untuk Evaluasi Polinom ................................. Pencarian Akar-akar Polinom .................................................. Lokasi Akar Polinom ................................................................
100 101 106
Sistem Persamaan Nirlanjar .................................................
107
3.7.1 3.7.2
Metode Lelaran Titik-Tetap ..................................................... Metode Newton-Raphson .........................................................
107 110
Contoh Soal Terapan ..........................................................
112
Metode Numerik
4. Solusi Sistem Persamaan Lanjar
...........................
117
Bentuk Umum Sistem Persamaan Lanjar .............................. Metode Elimin asi Gauss ......................................................
119 120
4.2.1 4.2.2 4.2.3
Tata-ancang Pivoting ................................................................ Penskalaan ................................................................................. Kemungkinan Solusi SPL .........................................................
125 131 132
Metode Eleminasi Gauss-Jordan ........................................... Metode Matriks Balikan ...................................................... Metode Dekomposisi LU .....................................................
135 139 140
4.5.1 4.5.2
Pemfaktoran dengan Metode LU Gauss .................................. Metode Reduksi Crout ..............................................................
142 147
Determinan ......................................................................... Kondisi Buruk ..................................................................... Bilangan Kondisi Matriks .................................................... Metode Lelaran Untuk Menyelesaikan SPL ..........................
150 154 160 165
4.9.1 4.9.2
Metode Lelaran Jacobi .............................................................. Metode Lelaran Gauss-Seidel ..................................................
166 167
4.10 Contoh Soal Terapan ..........................................................
171
4.1 4.2
4.3 4.4 4.5
4.6 4.7 4.8 4.9
5
Interpolasi dan Regresi ................................................ 179 5.1
5.2 5.3 5.4 5.5
Persoalan Interpolasi Polinom ..............................................
182
5.1.1 5.1.2 5.1.3
Interpolasi Lanjar ...................................................................... Interpolasi Kuadratik ................................................................ Interpolasi Kubik .......................................................................
183 185 186
Polinom Lagrange ............................................................... Polinom Newton ................................................................ Keunikan Polinom Interpolasi .............................................. Galat Intepolasi Polinom ......................................................
188 193 201 202
5.5.1 5.5.2 5.5.3
Daftar Isi
Batas Antara Galat Interpolasi Untuk Titik-titik yang Berjarak Sama .......................................................................... Taksiran Galat Interpolasi Newton .......................................... Taksiran Galat Interpolasi Lagrage ..........................................
207 211 212
vii
5.6
Polinom Newton-Gregory ....................................................
212
5.6.1
213
Polinom Newton-Gregory Maju ............................................... 5.6.1.1 5.6.1.2 5.6.1.3
Tabel Sesilih Maju ........................................................ Penurunan Rumus Polinom Newton-Gregory Ma ju ......... Menghitung Batas Galat Interpolasi Polinom Newton-Gregory Maju ................................................... Taksiran Galat Interpolasi Newton-Gregory Maju .......... Manfaat Tabel Selisih Maju ...........................................
213 214
Polinom Interpolasi Newton-Gregory Mundur ........................
224
Ekstrapolasi......................................................................... Interpolasi Dwimatra ........................................................... Contoh Soal Terapan Interpolasi ........................................... Regresi ...............................................................................
226 226 229 231
5.10.1 Regresi Lanjar ........................................................................... 5.10.2 Pelanjaran ..................................................................................
233 236
5.11 Contoh Penerapan Regresi dalam Bidang Rekayasa ...............
241
5.6.1.4 5.6.1.5
5.6.2
5.7 5.8 5.9 5.10
6
Integrasi Numerik 6.1 6.2 6.3
............................................................
249
Terapan Integral dalam Bidang Sains dan Rekayasa ............... Persoalan Integrasi Numerik ............................................... Metode Pias ........................................................................
251 253 253
6.3.1 6.3.2 6.3.3 6.3.4
254 257 258 260
Kaidah Segiempat ..................................................................... Kaidah Trapesium ..................................................................... Kaidah Titik Tengah ................................................................ Galat Metode Pias .................................................................... 6.3.4.1 6.3.4.2
6.4
6.5 6.6
viii
218 220 221
Galat Kaidah Trapesium ................................................ Galat Kaidah Titik Tengah .............................................
261 264
Metode Newton-Cotes .........................................................
265
6.4.1 6.4.2 6.4.3 6.4.4 6.4.5
Kaidah Trapesium ..................................................................... Kaidah Simpson 1/3 ................................................................... Kaidah Simpson 3/8 ................................................................... Metode Integrasi Numerik Untuk h yang Berbeda-beda............. Bentuk Umum Metode Newton-Cotes......................................
266 268 276 279 280
Singularitas ......................................................................... Penggunaan Ekstrapolasi untuk Integrasi ..............................
282 286
6.6.1 6.6.2
286 291
Ekstrapolasi Richardson ........................................................... Metode Romberg .......................................................................
Metode Numerik
6.6.3
6.7 6.8 6.9
7
296
Integral Ganda ................................................................... Kuadratus Gauss ................................................................ Contoh Soal Terapan ...........................................................
300 303 311
Turunan Numerik ............................................................. 317 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10
8
Ekstrapolasi Aitken ...................................................................
Persoalan Turunan Numerik ................................................. Tiga Pendekatan dalam Menghitung Turunan Numerik ......... Penurunan Rumus Turunan dengan Deret Taylor ................... Penurunan Rumus Turunan Numerik dengan Polinom Interpolasi ... Menentukan Orde Galat ........................................................ Program Menghitung Turunan ............................................. Ringkasan Rumus-rumus Turunan .......................................... Contoh Perhitungan Turunan .................................................. Ekstrapolasi Richardson ....................................................... Terapan Turunan Numerik dalam Bidang Pengolahan Citra .......
Solusi Persamaan Diferensial Biasa
318 318 320 323 325 326 326 328 329 332
....................
341
Kelompok Persamaan Diferensial ......................................... Terapan Persamaan Diferensial ............................................ PDB Orde Satu ................................................................... Metode Euler ......................................................................
342 343 345 346
8.4.1 8.4.2
Tafsiran Geometri Metode PDB .............................................. Analisis Galat Metode Euler ....................................................
347 349
Metode Heun ( Perbaikan Metode Euler ) .............................
352
8.5.1 8.5.2 8.5.3
Tafsiran Geometri Metode Heun ............................................. Galat Metode Heun ................................................................... Perluasan Metode Heun ............................................................
353 353 357
8.6 8.7
Metode Deret Taylor ........................................................... Orde Metode PDB ...............................................................
358 361
8.8
Metode Runge -Kutta ...........................................................
364
8.8.1
365
8.1 8.2 8.3 8.4
8.5
Daftar Isi
Metode Runge-Kutta Orde Satu ..............................................
ix
8.8.2 8.8.3 8.8.4
Metode Runge-Kutta Orde Dua .............................................. Metode Runge-Kutta Orde Tiga .............................................. Metode Runge-Kutta Orde Empat ..........................................
365 368 369
8.9 Ekstrapolasi Richardson ....................................................... 8.10 Metode Banyak-Langkah .....................................................
371 371
8.10.1 8.10.2 8.10.3 8.10.4 8.10.5
Metode Adams-Bashforth-Moulton ........................................ Metode Milne-Simpson ........................................................... Metode Hamming ..................................................................... Prosedur Pendahuluan .............................................................. Keidealan Metode Predictor-Corrector .................................
372 376 376 376 377
Pemilihan Ukuran Langkah yang Optimal ............................ Sistem Persamaan Diferensial ............................................. Persamaan Diferensial Orde Lanjut ...................................... Ketidakstabilan Metode PDB ............................................... Contoh Soal Terapan ..........................................................
379 381 384 387 389
Daftar Pustaka ...................................................................................
397
8.11 8.12 8.13 8.14 8.15
Daftar Padan x
Metode Numerik
Bahasa Inggris – Bahasa Indonesia Inggris
Indonesia
approximate (v) approximation (n) accurate (adj) array (n) continuous (adj) continuous function (n) derivative (n) discrete (adj) dimension (n) design (art) (v) domain (n) effective (adj) efficient (adj) efficiency (n) engineering engineer error (n) estimate (v) estimation (n) exact (adj) exact solution (n) floating-point numbers (n) generalize (v) generalization (n) iteration (n) interval (n) subinterval (n) linier (adj) loop (n)
menghampiri hampiran teliti larik menerus, sinambung fungsi menerus turunan farik matra meripta ranah sangkil mangkus kemangkusan rekayasa rekayasawan galat menaksir, pemperkirakan taksiran sejati solusi sejati bilangan titik-kambang merampatkan rampatan lelaran selang upaselang lanjar kalang
manipulate (v) multistep (n)
mengutak-atik bahu-langkah
Daftar Isi
xi
non linier (adj) problem (n) plot (v) round-off (n) rule (n) single-precission numbers (n) double-precission numbers (n) strategy (n) substitute (v) substitution (n) significant figure (digit) (n) subscript (n) stagnant (adj) step (n) strip (n) truncation (n)
xii
nirlanjar soal, persoalan, masalah merajah pembualatan aturan bilangan berketelitian tunggal bilangan berketlitian ganda tata-ancang menyulih sulihan angka bena, angka berarti tikalas mandek langkah pias pemotongan
Metode Numerik
