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Metode numerice pentru rezolvarea ecuat ¸iilor algebrice Octavian Cira 25 Octombrie 2005

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Page 1: Metode numerice pentru rezolvarea ecuat¸iilor algebricecira.adi.ro/cv/MetodeNumerice.pdf · Metode numerice pentru rezolvarea ecuat¸iilor algebrice Octavian Cira 25 Octombrie 2005

Metode numericepentru

rezolvarea ecuatiiloralgebrice

Octavian Cira

25 Octombrie 2005

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Cuprins

Lista figurilor. . . . . . . . . . . . . . . . . . . . . . . . . . . IX

Lista tabelelor . . . . . . . . . . . . . . . . . . . . . . . . . . XII

Prefata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XIII

Capitolul 1. Introducere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Preliminarii. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Polinoamele cu coeficienti complecsi . . . . . . . . . . . . . . 2

1.3. Polinoamele cu coeficienti reali . . . . . . . . . . . . . . . . . 15

Capitolul 2. Marginile radacinilor . . . . . . . . . . . . . . . . . . . . . . 22

2.1. Marginile radacinilor polinoamelor . . . . . . . . . . . . . . . 22

2.2. Polinoamelor cu coeficienti complecsi . . . . . . . . . . . . . . 22

2.3. Polinoamelor cu coeficienti reali. . . . . . . . . . . . . . . . . 24

Capitolul 3. Metode clasice . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1. Metoda Ruffini . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2. Metoda Lagrange . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3. Metoda Graffe . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4. Metoda Bernoulli. . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5. Metoda bisectiei . . . . . . . . . . . . . . . . . . . . . . . . . 34

Capitolul 4. Metode de separare . . . . . . . . . . . . . . . . . . . . . . . 36

4.1. Separarea radacinilor reale la polinoame cu coeficienti reali . . 36

4.2. Separarea radacinilor la polinoame cu coeficienti complecsi . . 40

4.3. Convergenta metodei Lehmer-Schur . . . . . . . . . . . . . . 47

4.4. Evaluarea erorilor la metoda Lehmer-Schur . . . . . . . . . . 48

4.5. Noua metoda Lehmer-Schur . . . . . . . . . . . . . . . . . . . 48

4.6. Metoda Weyl . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.6.1. Simplificari ale testului de proximitate. . . . . . . . . . . 60

4.6.2. Testul de proximitate Turan . . . . . . . . . . . . . . . . 61

4.6.3. Testul de proximitate Kakeya . . . . . . . . . . . . . . . 62

V

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CUPRINS VI

Capitolul 5. Metode de factorizare . . . . . . . . . . . . . . . . . . . . . 65

5.1. Elemente de analiza n-dimensionala . . . . . . . . . . . . . . 65

5.2. Metoda Lin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3. Metode de factorizare de ordinul 2 . . . . . . . . . . . . . . . 72

5.3.1. Metoda Bairstow-Newton. . . . . . . . . . . . . . . . . . 72

5.3.2. Convergenta metodei Bairstow-Newton . . . . . . . . . . 75

5.3.3. Metoda Bairstow-secanta . . . . . . . . . . . . . . . . . . 78

5.3.4. Metoda Bairstow-Steffensen . . . . . . . . . . . . . . . . 81

5.3.5. Convegenta metodelor Bairstow-secanta siBairstow-Steffensen . . . . . . . . . . . . . . . . . . . . . . 84

5.3.6. Metoda Bairstow-Fridman . . . . . . . . . . . . . . . . . 85

5.3.7. Convergenta metodei Bairstow-Fridman . . . . . . . . . . 86

5.3.8. Dimensiunea fractala . . . . . . . . . . . . . . . . . . . . 92

5.4. Metode de factorizare de ordinul 3 . . . . . . . . . . . . . . . 95

5.4.1. Metoda Bairstow-Newton. . . . . . . . . . . . . . . . . . 95

5.4.2. Metoda Bairstow-secanta . . . . . . . . . . . . . . . . . . 100

5.4.3. Metoda Bairstow-Fridman . . . . . . . . . . . . . . . . . 104

Capitolul 6. Metode de tip Newton . . . . . . . . . . . . . . . . . . . . . 107

6.1. Convergenta locala . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2. Metoda Newton . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3. Metoda parabolei tangenta . . . . . . . . . . . . . . . . . . . 115

6.4. Metoda Ostrowski . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5. Metoda parabolei osculatoare . . . . . . . . . . . . . . . . . . 119

6.6. Metoda parabolei . . . . . . . . . . . . . . . . . . . . . . . . 122

6.7. Metoda Laguerre . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.8. Metoda Chebyshev de ordinul 1 . . . . . . . . . . . . . . . . . 128

6.9. Metoda Chebyshev de ordinul 2 . . . . . . . . . . . . . . . . . 130

6.10. Metoda Halley . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.11. Familie de metode . . . . . . . . . . . . . . . . . . . . . . . 134

6.12. Concluzii . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Capitolul 7. Metode multipas . . . . . . . . . . . . . . . . . . . . . . . . 136

7.1. Metoda coardei. . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.2. Metoda secantei . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.3. Metoda Steffensen . . . . . . . . . . . . . . . . . . . . . . . . 144

7.4. Metoda Muller . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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VII CUPRINS

Capitolul 8. Metode simultane . . . . . . . . . . . . . . . . . . . . . . . . 149

8.1. Introducere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.2. Metoda Durand-Kerner . . . . . . . . . . . . . . . . . . . . . 152

8.3. Metoda Ehrlich-Aberth . . . . . . . . . . . . . . . . . . . . . 155

8.4. Metoda Ehrlich-Aberth cu factori de corectie Newton . . . . . 159

8.5. Metoda Borsch-Supan . . . . . . . . . . . . . . . . . . . . . . 160

8.6. Metoda Borsch-Supan cu factori de corectie Weierstrass . . . 162

8.7. Metoda Tanabe . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.8. Metoda radacinii patrate . . . . . . . . . . . . . . . . . . . . 166

8.9. Metoda Wang-Zheng. . . . . . . . . . . . . . . . . . . . . . . 167

8.10. Noua metoda Newton . . . . . . . . . . . . . . . . . . . . . 169

8.11. Familie de metode . . . . . . . . . . . . . . . . . . . . . . . 176

Capitolul 9. Teoria estimarii punctului . . . . . . . . . . . . . . . . . . . 178

9.1. Estimari ale punctului pentru metoda Newton . . . . . . . . . 178

9.2. Estimari ale punctului pentru metode cu convergenta 2 . . . . 194

9.3. Estimari ale punctului pentru metode cu convergenta 3 . . . . 197

9.4. Dezvoltari ale estimarii punctului pentru metoda Newton. . . 201

9.5. Metode mixte . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9.5.1. Metoda Lehmer-Schur-Newton . . . . . . . . . . . . . . . 209

9.5.2. Metoda Lehmer-Schur-Euler-Chebyshev . . . . . . . . . . 213

9.5.3. Metoda Lehmer-Schur-Halley. . . . . . . . . . . . . . . . 214

9.6. Estimarea punctului pentru metoda Durand-Kerner . . . . . . 214

Capitolul 10. Metode simultane de incluziune . . . . . . . . . . . . . . . 231

10.1. Elemente de algebra liniara . . . . . . . . . . . . . . . . . . 231

10.1.1. Polinom caracteristic . . . . . . . . . . . . . . . . . . . 231

10.1.2. Valori si vectori proprii . . . . . . . . . . . . . . . . . . 232

10.1.3. Discuri de incluziune. . . . . . . . . . . . . . . . . . . . 236

10.1.4. Intervalul complex aritmetic . . . . . . . . . . . . . . . 244

10.2. Teorema generala de convergenta . . . . . . . . . . . . . . . 248

10.3. Metoda Durand-Kerner. . . . . . . . . . . . . . . . . . . . . 253

10.4. Metoda Borsch-Supan . . . . . . . . . . . . . . . . . . . . . 264

10.5. Metoda Tanabe . . . . . . . . . . . . . . . . . . . . . . . . . 273

10.6. Familie de metode . . . . . . . . . . . . . . . . . . . . . . . 280

10.7. Metoda Ehrlich-Aberth. . . . . . . . . . . . . . . . . . . . . 292

10.8. Metoda Ehrlich-Aberth cu corectii cu factori Newton . . . . 302

10.9. Metoda Borsch-Supan cu corectii cu factori Weierstrass . . . 312

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CUPRINS VIII

10.10. Metoda Wang-Zheng . . . . . . . . . . . . . . . . . . . . . 324

10.11. Metode de tip Halley . . . . . . . . . . . . . . . . . . . . . 340

10.12. Conditia generala de convergenta. . . . . . . . . . . . . . . 361

Capitolul 11. Metode pentru polinoame cu zerouri multiple . . . . . . 365

11.1. Introducere . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

11.2. Metode de incluziune ce au la baza polinoamele Bell . . . . . 368

11.3. Metoda de incluziune Newton . . . . . . . . . . . . . . . . . 373

11.4. Metoda de incluziune Wang-Zheng . . . . . . . . . . . . . . 386

11.5. Metoda simultana Ehrlich-Kjurkchiev . . . . . . . . . . . . . 400

Capitolul A. Bibliografia McNamee . . . . . . . . . . . . . . . . . . . . . 404

A.1. Metodele Bernoulli si QD . . . . . . . . . . . . . . . . . . . . 404

A.2. Metoda Graeffe . . . . . . . . . . . . . . . . . . . . . . . . . 410

A.3. Metoda Lehmer . . . . . . . . . . . . . . . . . . . . . . . . . 416

A.4. Metodele Lin si Bairstow . . . . . . . . . . . . . . . . . . . . 420

A.5. Metoda Newton . . . . . . . . . . . . . . . . . . . . . . . . . 425

A.6. Metode simultane . . . . . . . . . . . . . . . . . . . . . . . . 446

A.7. Metode de incluziune . . . . . . . . . . . . . . . . . . . . . . 455

Capitolul B. Indexuri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

B.1. Index de notatii . . . . . . . . . . . . . . . . . . . . . . . . . 461

B.2. Index de subiecte . . . . . . . . . . . . . . . . . . . . . . . . 462

B.3. Index de nume . . . . . . . . . . . . . . . . . . . . . . . . . . 468

Bibliografie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

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Lista figurilor

Figura 1.1: Coroana C(0, k, K). . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Figura 1.2: Coroana C(0,m2,M2). . . . . . . . . . . . . . . . . . . . . . . . . . 6

Figura 1.3: Discurile de raza R si R′. . . . . . . . . . . . . . . . . . . . . . . . . 8

Figura 1.4: Exemple pentru programul Schur. . . . . . . . . . . . . . . . . . . . 12

Figura 1.5: Grafice pentru programul Schur(Comp).. . . . . . . . . . . . . . . . 14

Figura 1.6: V (m1M1(c)T , Sturm(c, 10−12)) = 3. . . . . . . . . . . . . . . . . . . 21

Figura 1.7: V (m2M2(c)T , Sturm(c, 10−12)) = 3. . . . . . . . . . . . . . . . . . . 21

Figura 1.8: V (Kakeya(c)T , Sturm(c, 10−12)) = 3. . . . . . . . . . . . . . . . . . 21

Figura 2.1: Coroana C(0,m, M) determinat cu algoritmul Alg1. . . . . . . . . . 24

Figura 2.2: Coroana C(0,m, M) pentru algoritmul Alg2. . . . . . . . . . . . . . 25

Figura 4.1: Coroana C(0,m, M) pentru algoritmul Alg3. . . . . . . . . . . . . . 39

Figura 4.2: Acoperirea coroanei circulare cu discuri.. . . . . . . . . . . . . . . . 41

Figura 4.3: Suprafata utila din discul de acoperire. . . . . . . . . . . . . . . . . 42

Figura 4.4: Graficul functiei 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Figura 4.5: Graficul descresterii razelor discurilor de acoperire. . . . . . . . . . . 48

Figura 4.6: Graficul erorilor absolute. . . . . . . . . . . . . . . . . . . . . . . . 49

Figura 4.7: Graficul functiei 4.5.1. . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figura 4.8: Acoperirea C(0, 1.5r, r) cu 12 discuri. . . . . . . . . . . . . . . . . . 50

Figura 4.9: Graficul erorilor absolute. . . . . . . . . . . . . . . . . . . . . . . . 54

Figura 4.10: Algoritmul Weyl.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Figura 5.1: Bazinul de atractie al radacinii x∗ = 5. . . . . . . . . . . . . . . . . 72

Figura 5.2: Bazinele de atractie pentru metoda B-N. . . . . . . . . . . . . . . . 78

Figura 5.3: Bazinele de atractie pentru metoda B-N ın sectiunea p = −1. . . . . 79

Figura 5.4: Bazinele de atractie pentru metoda B-s. . . . . . . . . . . . . . . . . 82

IX

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LISTA FIGURILOR X

Figura 5.5: Bazinele de atractie pentru metoda B-S. . . . . . . . . . . . . . . . 84

Figura 5.6: Bazinele de atractie pentru metoda Baistow-Fridman. . . . . . . . . 86

Figura 5.7: Descompunere ın triunghiuri.. . . . . . . . . . . . . . . . . . . . . . 93

Figura 6.1: Metoda Newton ın R1. . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figura 6.2: Bazinele de atractie pentru contractia z − P (z)/Pz) . . . . . . . . . 113

Figura 6.3: Bazinele de atractie pentru MN. . . . . . . . . . . . . . . . . . . . . 114

Figura 6.4: Bazinele de atractie pentru MTP. . . . . . . . . . . . . . . . . . . . 117

Figura 6.5: Bazinele de atractie pentru MO. . . . . . . . . . . . . . . . . . . . . 119

Figura 6.6: Bazinele de atractie pentru MOP. . . . . . . . . . . . . . . . . . . . 121

Figura 6.7: Graficul functiei |f(z)|. . . . . . . . . . . . . . . . . . . . . . . . . . 123

Figura 6.8: Metoda MP ın R1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Figura 6.9: Bazinele de atractie pentru MP. . . . . . . . . . . . . . . . . . . . . 126

Figura 6.10: Bazinele de atractie pentru MC1. . . . . . . . . . . . . . . . . . . . 129

Figura 6.11: Bazinele de atractie pentru MC2. . . . . . . . . . . . . . . . . . . . 131

Figura 6.12: Bazinele de atractie pentru MH. . . . . . . . . . . . . . . . . . . . 133

Figura 7.1: Metoda coardei. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Figura 7.2: Monotonia sirului generat de metoda coardei. . . . . . . . . . . . . . 139

Figura 7.3: Aplicarea metodei coardei. . . . . . . . . . . . . . . . . . . . . . . . 140

Figura 7.4: Metoda secantei. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figura 7.5: Bazinele de atractie pentru metoda secantei. . . . . . . . . . . . . . 144

Figura 7.6: Metoda Steffensen. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Figura 9.1: Testul α cu valoarea αS(8).. . . . . . . . . . . . . . . . . . . . . . . 193

Figura 9.2: Testul α cu valoarea αW . . . . . . . . . . . . . . . . . . . . . . . . . 194

Figura 9.3: Functia ϕ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Figura 9.4: Graficul conditiei (9.40). . . . . . . . . . . . . . . . . . . . . . . . . 222

Figura 9.5: Graficul conditiei (9.41). . . . . . . . . . . . . . . . . . . . . . . . . 222

Figura 10.1: Discuri Smith. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Figura 10.2: Discuri Braess-Hadeler. . . . . . . . . . . . . . . . . . . . . . . . . 238

Figura 10.3: Discuri Gerschgorin. . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Figura 10.4: Functile g si 1/(2φ(n)). . . . . . . . . . . . . . . . . . . . . . . . . 258

Figura 10.5: Functia β si Φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Figura 10.6: Functiile f si λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Figura 10.7: Functiile h, f si g. . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

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XI LISTA FIGURILOR

Figura 10.8: Functiile h si f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Figura 10.9: Functia τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Figura 10.10: Functiile h si φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Figura 10.11: Functile γ si φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Figura 10.12: Functile φ si γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Figura 10.13: Functile τ si τφ.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Figura 10.14: Functile φ si λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

Figura 10.15: Functile h si hλ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

Figura 10.16: Functia τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

Figura 11.1: Discul de incluziune D(a,R). . . . . . . . . . . . . . . . . . . . . . 386

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Lista tabelelor

Tabelul 1.1: Variatia semnului cand Q este crescator. . . . . . . . . . . . . . . . 16

Tabelul 1.2: Variatia semnului cand Q este descrescator. . . . . . . . . . . . . . 16

Tabelul 1.3: Variatia functiei V . Cazul 1. . . . . . . . . . . . . . . . . . . . . . 18

Tabelul 1.4: Variatia functiei V . Cazul 2. . . . . . . . . . . . . . . . . . . . . . 18

Tabelul 1.5: Variatia functiei V . Cazul 3. . . . . . . . . . . . . . . . . . . . . . 19

Tabelul 1.6: Variatia functiei V . Cazul 4. . . . . . . . . . . . . . . . . . . . . . 19

Tabelul 5.1: Dimensiunea fractala. . . . . . . . . . . . . . . . . . . . . . . . . . 95

Tabelul 6.1: Dimensiunea fractala pentru metode cu convergenta patratica. . . . 135

Tabelul 6.2: Dimensiunea fractala pentru metode cu convergenta cubica. . . . . 135

Tabelul 9.1: Intervale [0, rn). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Tabelul 9.2: Valorile constantei αS(n). . . . . . . . . . . . . . . . . . . . . . . . 191

Tabelul 9.3: Convergenta patratica a metodei MN. . . . . . . . . . . . . . . . . 192

Tabelul 9.4: Valorile η1(n). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Tabelul 9.5: Testul η1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Tabelul 10.1: Radacinile pozitive ale ecuatiei xn − x − 3 = 0. . . . . . . . . . . . 342

XII

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Prefata

Studiul radacinilor polinoamelor algebrice a fost unul din cele mai prolifi-ce subiecte ın istoria milenara a matematicii. Determinarea valorii pentru carepolinomul algebric se anuleaza a fost ın atentia a mii de matematicieni. JohnMichael McNamee a publicat o bibliografie pe aceasta tema ın patru editii[110], [111], [112] si [113], ultima contine contine 32 de capitole ın cadrul apeste 300 de pagini, avand aproximativ 10 000 de titluri, unde apar doar rezul-tatele publicate ın limbile de larga circulatie internationala (engleza, fracezasi germana), ın reviste si edituri de prestigiu. O lucrarea de referinta pen-tru tema prezentata ın aceasta carte este volumul VIII, Numerical Solution ofPolynomial Equations din seria Handbook of Numerical Analysis [161], tiparitasub egida Elsevier Science.

Teoria matematica a ecuatiilor algebrice este un capitol ıncheiat al alge-brei ınca de la sfarsitul secolului al XVIII-lea odata cu demonstrarea, de catreGauss (1799), a teoremei fundamentale a algebrei. A ramas deschisa problemadeterminarii practice a solutiilor ecuatiilor algebrice.

Odata cu ideea geniala a lui Isac Newton (1669)[116] de a aproxima radacina unui polinom cu punctul deintersectie al tangentei cu axa OX, s-a deschis o noua calede determinare a solutiilor ecuatiilor. Niels Abel a demon-strat (1826) ca formulele de calcul a radacinilor cu radi-cali pot fi folosite numai pentru ecuatii algebrice de gradcel mult 4. Din acest moment era clar ca determinareasolutiilor ecuatiilor algebrice de grad mai mare decat 4 seva putea face numai cu metode de aproximare.

Multe cercetari au fost efectuate pentru a determinamarginile radacinilor polinoamelor. Exista o multitudine de constante caremajoreaza si, respectiv, minoreaza marginea superioara si marginea inferioaraale radacinilor polinomului. In cele mai multe cazuri majorarea sau mino-rarea este grosiera. Prezentam, ın aceasta carte, algoritmi de determinare aunui majorant si, respectiv, minorant ce aproximeaza marginile superioara si,respectiv, inferioara a radacinilor polinomului cu o precizie data [31].

Problema are importanta ei pentru ca permite delimitarea cat mai exacta

XIII

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Capitolul 0. Prefata XIV

a coroanei circulare ın care se gasesc toate radacinile complexe si segmentelede pe axa reala care contin toate radacinile reale.

Multi matematicieni au considerat metode noi pentru aproximareasolutiilor ecuatiilor algebrice: Bernoulli [55], [81], Sturm [171], Euler si Cheby-shev [158], Laguerre [100], Weierstrass [185], Bairstow [10], Muller [115],Lehmer [102], Docev [49], Durand [50], Borsch-Supan [13], [14], Kerner [87],Ehrlich [51], Jenkins si Traub [83], Aberth [3], Halley [68], [158], [97], [11],Tanabe [174], Wang si Zheng [182] etc. Incepand cu anul 1937, Ostrowski[122] publica mai multe articole prin care demonstreaza convergenta locala ametodei Newton ın cazul functiilor de o variabila reala.

Rezultatul lui Kantorovich din 1948 [85], care de-monstreaza convergenta patratica a metodei Newton ınspatii generale Banach, este fundamental. Acest rezul-tat domina literatura de specialitate, aparand mii de arti-cole pe aceasta tema. Dupa modelul dat de Kantorovich,trei probleme se pun ın legatura cu metodele de aproxi-mare a radacinilor polinomului: convergenta, ordinul deconvergenta si estimarea erorii. Pe langa aceste directii decercetare referitor la metodele numerice pentru ecuatiilealgebrice, exita multe alte teme de interes.

Problematica stabilitatii Routh-Hurwitz a metodelornumerice [76], [155] este, ıncepand cu secolul XX, o problema intens studiatajudecand dupa impresionanta bibliografie [113, capitolul Stability Questions(criteriul Routh-Hurwitz etc.)] pe aceasta tema.

Bairstow [10] a considerat o metoda de factorizare a polinoamelor ıntermeni de ordinul 2. Metoda Bairstow s-ar putea chema Bairstow-Newtonpentru ca sistemul neliniar ce intervine ın deducerea procesului iterativ esterezolvat cu metoda Newton. In aceasta carte vom prezenta metode de tipBairstow de ordinul 2 si 3 pentru factorizarea polinomului care au fost numiteBairstow-secanta, Bairstow-Steffensen, Bairstow-Fridman [28], [29], [30], dupacum pentru deducerea procesului iterativ, ce se face prin rezolvarea unui sistemneliniar de ordinul doi sau trei, s-a folosit metoda secantei, metoda Steffensen[170] sau metoda Fridman [60].

Mult mai tarziu s-a pus problema iteratiei de start, de aici rezultandproblema bazinelor de atractie a metodelor [32], [33], [34], [45], [40]. Foarteinteresant este faptul ca bazinele de atractie pentru metode de tip Newton auo stransa legatura cu teoria Fatou-Julia a fractalilor sau mai bine zis a atracto-rilor. Aceasta observatie s-a facut ın anul 1985 cand s-a reprezentat prima datagrafic multimea punctelor, dintr-un patrat, din care metoda Newton convergepentru ecuatia z3 − 1 = 0 [54].

Algoritmul Lehmer-Schur [102] este o metoda de separare a radacinilorpolinomului cu coeficienti complecsi, care are un neajuns importat, cu cat o

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XV

radacina este separata mai tarziu, cu atat creste incertitudinea fata de pre-cizia separarii. Vom prezenta o varianta proprie a acestei metode care evitaacest neajuns [31]. Metoda Weyl [127] este o metoda de separare, fiind o ge-neralizare a algoritmului de bisectie din R, ın planul complex, unde intervalulcomplex este considerat patratul. Metoda Weyl, cunoscuta si sub denumireade constructia Quadtree [125], este una din metodele cu cea mai mica com-plexitatea a calculului, fapt ce face sa fie un algoritm ”foarte atragator”.

In 1981 Smale [167] defineste notiunea de ”zero aproximativ”, notiunecare sta la baza teoriei estimarii punctului [141]. Aceasta teorie permitedefinirea apriori a bazinelor de atractie pentru solutiile ecuatiei algebrice ınfunctie de metoda folosita.

Se remarca dezvoltarea deosebita a metodelor simultane de determinarea tuturor radacinilor ecuatiei algebrice [50], [49], [13], [87], [51], [3], [174],[182] etc. In legatura cu aceste metode s-au dezvoltat metodele de incluzi-une care folosesc o aritmetica a intervalului complex si discurile de incluziuneGerschgorin [53], unde intervalul complex este discul. O lucrare fundamen-tala pentru metodele de incluziune o datoram lui Petkovic [133], acelasi autorın anul 2002 elaboreaza o lucrare de sinteza ce cuprinde ultimele rezultatereferitoare la metodele de incluziune [137].

La ora actuala se fac eforturi importante ın directia paralelizarii algorit-milor si a folosirii calculelor ın precizii extinse pentru a rezolva ecuatii cu gradmare (> 100) si a obtine radacini cu precizii ınalte.

Nu este necesar sa avem un polinom de grad mare pentru a avea problemede determinare a radacinilor cu o acuratete mare, este suficient un polinomrau conditionat sau cu radacini multiple pentru ca sa avem dificultati de de-terminare a radacinilor.

Sunt interesante metodele mixte, ın care se combina o metoda de separarea radacinilor, ın faza initiala a algoritmului, cu o metoda rapid convergenta,ın partea a doua a algoritmului. La aceste metode este important criteriulprin care renuntam la metoda de separare si luam ın considerare metoda rapidconvergenta. Teoria estimarii punctului joaca un rol hotarator din acest punctde vedere [35], [41], [39], [36], [38].

Datorita dezvoltarii tehnologice hard si soft s-a ajus la performante re-marcabile. Determinarea tuturor radacinilor pentru ecuatiile algebrice de grad99 nu mai constituie o problema de timp CPU si acuratete de determinare. Laora actuala exista pachete de calcul stiintific (Mathcad [104], [37], [44], Mate-matica, Maple, Matlab, . . . ), sau biblioteci de programe stiintifice (IMSL [1],NAG [2], DROOTS, . . . ) care furnizeaza functii sau programe de rezolvare aecuatiilor algebrice de grad mare cu precizii de 15, 24 sau 32 de cifre zecimale.Apeland la calculul simbolic se pot furniza radacini exacte sau cu un numarmare de zecimale exacte (100–250 cifre zecimale). Pentru a vedea ultimeleprobleme referitoare la radacinile polinoamelor se poate consulta excelentul

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Capitolul 0. Prefata XVI

articol a lui Burgnano si Trigiante [18].In aceasta carte se prezinta peste 50 de metode de aproximare a ra-

dacinilor ecuatiilor algebrice, unde s-au avut ın vedere constructia metodei,convergenta, ordinul de convergenta, estimarea erorii, bazinele de atractie alemetodei, iar ın unele cazuri s-au facut referiri si la complexitatea calculului.Rezultatele teoretice sunt urmate de prezentarea algoritmilor, a programelorsi a exemplelor rezolvate. Toate exemplele si programele au fost realizate ınMathcad 2001 [37], [44]. Consideram ca aceasta carte contine multe programece pot constitui un suport de ınvatare a programarii ın Mathcad. In exem-plele date s-au preferat polinoame cu radacini numere ıntregi sau cu radacinicomplexe ce au partea reala si partea imaginara ıntreaga pentru a se evidentiausor erorile de aproximare.

Cartea se adreseaza programatorilor, cercetatorilor, cadrelor didacticedin ınvatamantul superior, studentiilor care sunt familiarizati cu analiza nu-merica. Se prezinta suportul teoretic pentru metodele numerice si algoritmiice stau la baza programelor performante de rezolvare numerica a ecuatiiloralgebrice.

Demonstratiile teoremelor si propozitiilor, ın multe cazuri, nu pot fi par-curse si verificate fara a avea la ındemana un soft (Mathcad, Maple, Matemat-ica, Scientific Word, ...) ce asigura calcul simbolic si rutine pentru rezolvareaecuatilor algebrice, pentru a verifica calcule complexe si a rezolva ecuatii al-gebrice de grad mare. Consideram ca acest fapt este o noutate importanta ınliteratura stiintifica din Romania.

Autorul detine toate documentele Mathcad versiunea 11.0 (148 de fisiereMCD si un fisier README.txt) pentru demonstratiile ın detaliu a teoremelorsi propozitiilor prezentate, acestea pot fi accesate de pe CD-ul atasat cartii.

Multumesc tuturor celor care au fost alaturi de mine pentru scriereaacestei carti.

Arad, 08 Februarie 2005 Autorul

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