Lista 2-2023-1

Lista 2 1. Problemas: 3, 4, 5, 6, 9, 11, 13 da seção 6.1. 2. Problemas: 1, 2, 3, 4, 9, 10, 30, 31 da seção 6.2. 3. Problemas: 1, 2, 3, 4, 7, 8, 9, 10 da seção 6.5. 4. Problemas: 1, 2, 3, 4, 5, 6, 7, 8, 15, 16 da seção 7.3. Referências [1] BURDEN, Richard L.; FAIRES, J D.; BURDEN, Annette M. Análise Numérica - Tradução da 10ª edição norte-americana. Cengage Learning Brasil, 2016. Seção 2.1 1. f(x) = √x - cos x f(p2) = √(0,15) - cos 0,15 = 0,13 > 0 f[az] + f([pz) ∈ [0,5,0,75] p3 = (az + b3)/2 = 0,5 + 0,15 = 0,625 2. a) f(-2) = 3(-2 + 1)(-2 - 1/2)(-2 - 1)= 45/2 f(1,5) = 3(1,5 + n)(1,5 - 1/2)(1,5 - 1) p1 = (-2 + 1,5)/2 = -0,25 p2 = (-2 - 0,25)/2 = -1,125 p3 = (-1,125 - 0,25)/2 = -0,6875 b) f(-1,25) = -2,95 f(2,5) = 3,75 p1 = (-1,25 + 2,5)/2 = 0,625 p2 = (0,625 + 2,5)/2 = 1,56 p3 = (0,625 + 1,56)/2 = 1,09 3. a) (b-a)/2^n => f(0) = -6 f(1) = 2 p1 = a1 + b1 = 1/2 p2 = a2 + b2 = 3/4 p3 = a3 + b3 = 5/8 p4 = (1/2) + (5/8) = (a4 + b4)/2 = 9/16 p5 = a5 + b5 = 19/32 p6 = a6 + b6 = 37/64 p7 = (a7 + b7)/2 = 75/128 = 0,58 b) Faça os pares para p1, p2, p3, p4, p5, p6, p7 p7 a p9 p = 3,00 c) Faça os pares p1 - p7 p7 = 3,4 6. a) Função mista 10^-5 b) p17 = 1,5 c) p17 = 0,25 c) p17 = -2,29 p17 = -0,79 d) p14 = 0,129 p14 = 1,24 5- a) p17 = 0,6 b) p19 = 0,25 c) p17 = -2,19 d) => p14 = 0,29 7- a) y y = f(x) x = x b) p18 = 1,89 11- a) 2 b) -2 c) -1 d) 1 12- 3√ = p14 = 1,73 13- 3√ = p14 ≠ 2,92 14- n > 12 p12 = 1,37 16- |p - pm| = 1/m < 10^-3 n > 1000 1- a) x = (3 + x + 2x²) ^4 x 4 = 3x - 2x² f(x) = 0 b) x = x + 3 - x 4 3 2 ⇒ 2x z = x + 3 - x 4 ⇒ f(x)=0 c) x = (x + 3) 1/2 ⇒ x z = (x z) = x + 3 ⇒ f(x) = 0 x z² x+z z d) x = 3x 4 + 2x z 3 4∂4x 4 + 4x z - x = 3x 4 + 3x z² + 3 ⇒ f(x) = 0 9x z³ + 4x - 1 2- a) p4 = 1,10 b) p4 = 0,98 c) p4 = 1,12 d) p4 = 1,12 e) p4 - p3 é menor 3- A B e D método converge e C não converge 4- Método A B não converge e C D converge po=1 pu=1,324 |pm - p| <= kn |p1 - po| = z/3 (1/3) po=1 p12=0,64 k=0,55 c) p0 = 2 p4 = 2,37 d) p0 = 1 p4 = 1,4 e) p0 = 1 p4 = 0,9 f) p0 = 0 p4 = 0,58 7. a) pf= 2,69 b) p7 = -2,87 c) p6 = 0,74 d) p5 = 0,96 8. a) p7 = 1,8 b) p9 = 1,39 c) p6 = 2,37 d) p8 = 1,4 e) p6 = 0,91 f) p6 = 0,6 18. p0 = 5π p0 = 10π método newton p0 = 0 o que difere do anterior 19. - y = x^2 dy = 2x x1 = -2,5 : p(6,25) x16 + 0,0001 = f(0) 20. - y = 1/x d := -x = x x1 = 0 Lição 6,1 3. a) x1 = 1; x2 = -0,98; x3 = 2,9 b) x1 = 1,1; x2 = -1,1; x3 = 2,9 4. a) x1 = 0,7 x2 = 1,1 x3 = 2,9 b) x1 = -0,8 x2 = 0,7 x3 = 3 5. a) x1 = 1,18 x2 = 1,8 x3 = 0,8 b) x1 = -1 x2 = 0 x3 = 1 c) x1 = 1,5 x2 = 2 x3 = -1,2 x4 = 3 d) c) 6. a) x1 = -4 x2 = -8 x3 = -6 b) x1 = 22/9 x2 = -4/9 x3 = 8/12 7. c) x1 = 13,5 x2 = 8 x3 = 0,2 x4 = 5 8. d) x1 = -1 x2 = -2 x3 = 0 x4 = 1 9. a) x1 := -1/3 b) L = 1/3 --> x1 = x2 + 1,5 c) x = 7 + 1/3 --> x1 = 3 2(1 + 3L) ------- x6 = -3 2(13L) 11. E = a11 x1 + a12 x2 ... a1n xn = b1 Em: am x1 + am2 x2 + amn xn = bn λaf1 x1 + λaf2 x2 + λafn xn = λbf 13. a) x1 = 0,98 x2 = -0,98 x3 = 2,9 b) x1 = 1 x2 = -1 x3 = 2,9 Lição 6,2 1. a) 7 b) 2 e 3 c) 7 d) 1 e 2 2. a) p, c e d = 7 3. a) 1 e 2 b) 1 e 3 c) 1 e 2 d) 1 e 2 4. a) 2 e 3 b) 1 e 3 c) 1 e 3 d) 1 e 2 10 a) x1=1.0 x2=0.9 b) x1=1.2 x2=0.49 x3=0.98 c) x1=-8.25 x2=8 x3=-9.04 x4=0.05 d) x1=1.33 x2=4.66 x3=-4.04 x4=-1.66 9 - a) x1=30 x2=0.99 b) x1=10 x2=10.5 x3=0.14 c) x1=0.7 x2=0.01 x3=-0.03 x4=0.05 d) x1=1.3 x2=4.6 x3=-4.04 x4=-1.66 30 - a) x1=1.0 x2=1 b) x1=10 x2=1 c) x1=0.09 x2=-0.06 x3=-0.03 x4=0.04 d) x1=1.33 x2=4.68 x3=-4.06 x4=-1.65 31- a não permite pois c=6 Lição 6.5 1- a) x1=-3 x2=3 x3=1 b) x1=1/2 x2=-5/2 x3=7/2 2 - a) x1=11/20 x2=3/10 x3=2/5 b) x1=17/6 x2=-5/1 x3=24 3 - a) P= [1 0 0] [0 0 1] [0 1 0] b) P= [1 0 0 0] [0 0 1 0] [0 1 0 0] [0 0 0 1] c) P= [1 0 0 0] [0 1 0 0] [0 1 0 0] [0 0 1 0] 4 - a) P= [0 1 0] [1 0 0] [0 0 1] c) P= [1 0 0 0] [0 0 1 0] [0 0 0 1] [0 1 0 0] 7 - a) x1=1 x2=2 x3=-1 b) x1=1 x2=1 x3=1 c) x1=1.5 x2=2 x3=-1.19 x4=3 a) x1=2.9 x2=0.07 x3=5.6 x4=0.5 8 - a) x1=12 x2=14 x3=17 b) x1=495/541 x2=840/241 x3=321/141 x4=52/141 c) x1=-29/47 x2=58/17 x3=32/141 x4=52/141 d) x1=-0.7 x2=-0.18 x3=0.5 x4=0.5 9-a) PtLU = \left[ \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1/2 & 1 \end{array} \right] \left[ \begin{array}{ccc} 1 & 1 & -1 \\ 0 & 2 & 3 \\ 0 & 0 & 5/2 \end{array} \right] b) PtLU = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right] \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{ccc} 1 & 7 & 1 \\ 0 & -5 & 6 \\ 0 & 0 & 4 \end{array} \right] c) PtLU = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{cccc} 1 & -2 & 3 & 0 \\ 0 & 5 & -2 & 1 \\ 0 & 0 & -1 & -2 \\ 0 & 0 & 0 & 3 \end{array} \right] d) PtLU = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{cccc} 1 & -2 & 3 & 0 \\ 0 & 5 & -3 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 1 \end{array} \right] 10-a) \frac{1}{3}n^3 - 1 \sum n b) P = -\det P P = (-1)^n \det P c) n - 1 \cdot \frac{1}{3}n^3 - 1 \sum n a) A = -341 Lista 7.3 1-a) x^2: (0,14; -0,35; 0,49)^t b) x^2: (0,87; 0,81; 0,74)^t c) x^2: (-0,65; 1,65; -0,4; -2,4)^t d) x^2: (1,3; -1,6; 1,6; 2,4)^t 2-a) x^2: (1,25; 1,3; 0,2)^t b) x^2: (-1; 1; -1; -1,3)^t c) x^2: (-0,5; -0,4; -0,2; 0,4)^t d) x^2: (0,7; 1,1; 0,6; 1,3; 0,5; 1,3)^t 3-a) x^2: (0,1; -0,2; 0,6)^t b) x^2: (0,97; 0,94; 0,78)^t c) x^2: (1,05; 2,64; -0,3; -2,64)^t d) x^2: (1; -1,5; 1,8; 1,88; 2,5)^t 4-a) x^2: (1,3; -0,9; 0,06)^t b) x^2: (-1,6; 1,3; 1; 0,5)^t c) x^2: (-0,61; 0; -0,2; 0,6)^t d) x^2: (0,68; 1,5; 0,25; 1,7; 0,7; 1,87)^t 5. (a) x10)= (0.03507839, -0.2369262, 0.6578015) (b) x16)= (0.9957250, 0.9577750,0.7914500) (c) x22)= (-0.7975853, 2.794795, -0.2588888, -2.251879) (d) x(14) = (-0.7529267, 0.04078538, -0.2806091, 0.6911662) (e) x12) = (0.7870883, -1.003036, 1.866048, 1.912449, 1.985707) (f) x(T) =(0.9996805, 1.999774, 0.9996805, 1.999840, 0.9995482, 1.999840) 6. (a) x10)= (1.447642384, -0.8355647882, -0.0450226618) (b) x(25)= (-1.500322611, 1.500322611, -0.9997048580) (c) x14) = (-0.7529267,0.04078538, -0.2806091, 0.6911662) (d) x(17) = (0.9996805,1.999774, 0.9996805, 1.999840, 0.9995482, 1.999840)+ 7. (a) x1) = (0.03535107, -0.2367886, 0.6577590) (b) x(4) = (0.9957475, 0.9578738, 0.7915748) (c) ×(10) = (-0.7973091, 2.794982, -0.2589884, -2.251798) (d) x(7) = (0.7866825, -1.002719, 1.866283, 1.912562, 1.989790) 8. (a) x6) = (1.447816350, -0.8358173037, -0.0447996186)+ (b) x8) = (-1.500228624, 1.499713760, -0.9998475841) (c) x8) = (-0.7531763, 0.04101049, -0.2807047, 0.6916305) (d) x(10) = (0.9998334, 1.999858, 0.9999393, 1.999899, 0.9999142, 1.999963) 15. w= 1.012823 C 4) = (0.9957846, 0.9578935, 0.7915788) w = 1.153499 x l ) = (-0.7977651, 2.795343, -0.2588021, -2.251760) 16. w= 1.033370453 x(5)= (0.3571407017, 1.428570817, 0.357142771, 1.571421010, 0.2857118407, 1.571428256) (a) p-s = g(p) - g(pj) = g1(e1)(p-s) <= k |p-s| ep=g lp=1/2 (-1 + √5)\ 18 - a) p = p + g -------- p1