Lista 1-2023-1

Lista 1 1. Problemas: 1, 2, 3, 5, 6, 7, 11, 12, 13, 14, 16 da seção 2.1. 2. Problemas: 1, 2, 3, 4, 6, 7, 8, 12, 15, 17, 18, 19, 20 da seção 2.2. 3. Problemas: 1, 2, 5, 6, 7, 8, 18, 19, 20 da seção 2.3. 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[[a2]] = f((p2) = [0,5,0,75] p3 = a3 + b3 / 2 = 0,5 + 0,15 / 2 = 0,625 2. a) f(-2) = 3(-2 + 1)(-2•1/2)(-2 - 1) = -45 / 2 f(1,5) = 3(1,5 + 1)(1,5 - 1/2)(1,5 - 3) = 3,75 p1= -2 + 1,5 / 2 = -0,25 p2= -2 - 0,25 / 2 = -1,125 p3= -1,125 - 0,25 / 2 = -0,6875 d) 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 / 2 = a4 + b4 = 9 / 16 p5: a5 + b5 = 19 / 32 p6: a6 + b6 / 2 = 37 / 64 p7: a7 + b7 / 2 = 75 / 128 = 0,58 e) Meça o peso para p1, p2, p3, p4, p5, p6, p7 => p8 p = 3,00 e) Meça o peso p1 - p7 p7 = 3,4 4. a) Função metade 10^-5 => p12 = 1,5 b) p17 = 0,25 c) p17 = -0,29 d) p14= 0,29 p14 = 1,24 5. a) p17 = 0,6 b) p19 = 0,25 c) p17 = -2,19 d) p14 = 0,29 7. a) y f(x) y = f(x) y = x d) p76 = 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| = \frac{1}{m} < 10^-3 n > 1000 1. a) x = (3 + x + 2x^2)^{\frac{1}{4}} x^{\frac{1}{4}} = 3x - 2x\sqrt{2} f(x) = 0 b) x = \frac{x + 3 - x^4}{2} \Rightarrow 2x\sqrt{2} = x + 3 - x^4 \Rightarrow f(x) = 0 c) x = \left(\frac{x + 3}{2x^2 + 4}\right)^{\frac{1}{2}} \Rightarrow x\sqrt{2}(x\sqrt{2}) = x + 3 \Rightarrow f(x) = 0 d) x = \frac{3x^{\frac{1}{4}} + 2x^{\frac{2}{3}} + 3}{4x^{\frac{3}{4}} + 4x - 1} \Rightarrow 4\sqrt{4x} + 4x\sqrt{2} - x = 3x\sqrt{4} + 3x\sqrt{2} + 3 \Rightarrow f(x) = 0 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 e D converge 6 - p0=1 p4=1,324 7 - |p^m-p| \le kn |p_1-p0| \frac{z}{3} |\frac{1}{4} 8 - p0=1 p12=0,64 k=0,55 12 - \log n \max (b-a, b-p_0) n > \log 10^9 - \log \left( \max (p_0-a, b-p_0) \right) 15 - p5=1,68 17 - g(x) = \sqrt{2x-1} 18 a) p-g = g(p) - g(p_j) = g(|g_j|/|p-g|)< g_k(|p-g|) 4p=g b) p= \frac{g}{2} ( -1+ \sqrt{5} ) 18 - a) p = \frac{p + g}{p_1} b) x_0 \sqrt{2} < x_0^2 + 2 < O < x_0 < \sqrt{2} c) g'(x) = x + \frac{1+A}{2 - 2x} \text{limites iguais} g'(\sqrt{A}) = \sqrt{A} 20 a) \sqrt{0} < x_0 |\sqrt{A}| > x_0 = x_0^2 - 2x_0 \sqrt{A} + 1 l) \lim_{x\to \infty}x=-\sqrt{4} \equação 2.3 (1,2,5,6,7,8,18,19,20) \ 1.- \ x_k = x_{k-1} - \frac{f(x_{k-1})}{\phi'(x_{k-1})} p2 = 2,6 \ 2.- \ x_k = x_{k-1} - \frac{f(x_{k-1})}{\phi_1(x_{k-1})} p(0):0 \não \pode \calcular p2 = -0,86 \ 5 - a) x_k=x_{k-1} - \frac{f(x_{k+1})}{\phi'(x_{k+1})} p0 = 2 p5 = 2,68 b) p0 = 3 p3 = -2,8 c) p0 = 0 p4 = 0,7 d) p0 = 0 p3 = 0,96 6 - a) \ p0 = 1 p8 = 1,83 b) \ p0=1,5 p4 = 1,38 c) po = 2 p4 = 2,37 d) po = 1 p4 = 1,4 e) po = 1 p4 = 0,9 f) po = 0 p4 = 0,58 7. a) p4 = 2,69 e) p7 = -2,87 c) p6 = 0,74 d) p5 = 0,96 8. a) p7 = 1,8 c) p6 = 1,39 c) p6 = 2,37 d) p8 = 1,4 e) p6 = 0,91 f) p6 = 0,6 18. po = 5π po = 10π metodo uneton po = O o que difere do inunion 19 - y = x^2 y = 2 . x x1 = -2,5 = p(6,25) x -0,0001 = x f(o) 20 - y = 1/x d= - x= 1/x1 = 0 Lição 6,1 3 - a) x1 = 1, x2 = -0,98, x3 = 2,9 b) x1 = 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 6 - a) x1 = -4 x2 = -8 x3 = -6 b) x1 = 22 x2 = 4 x3 = 8 c) x1 = 13,5 x2 = 8 x3 = 0,2 x4 = 5 d) x1 = -1 x2 = -2 x3 = 0 x4 = 1 9 - a) L = -1/3 b) L = 1/3 + 1 x1 = x2 + 1,5 x1 = 3 c) x1 = 3 x3 = 2(1+3x/ 11. E = a11 x1 + a12 x2 ... a1n xn = b1 E = a1 fx1 + λ a12 x2 + etc 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) 2,3 c) 7 d) 1 e 2 2 - a b, 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 = 10 X2 = 9, 9 b) X1 = 12 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) Xf = 30 X2 = 0,99 b) X1 = 10 X2 = 10,5 X3 = 0,14 c) X2 = 0,2 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 = 10 X2 = 1 b) X1 = 10 X2 = 1 c) X1 = 0,09 X1 = 0,06 X3 = 0,03 X4 = 0,04 d) X1 = 1,33 X2 = 4,68 X3 = 4,06 X4 = 1,65 31 - a não possui pol. f: 6 Seção 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 = 50 X3 = 24 3 - a) P = [1 0 0 0 0 1 0 1 0] c) P = [1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1] d) P = [0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1] 4 - a) P = [0 1 0 1 0 0 0 0 1] c) P = [1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1] d) P = [0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1] 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 = 1,7 b) X1 = 495/141 X2 = 840/241 X3 = 321/141 X4 = 52/141 c) X1 = 29/47 X2 = 58/47 X3 = 32/141 X4 = 52/141 d) X1 = 0,7 X2 = 0,18 X3 = 0,5 X4 = 0,5 a) p^tLU = (\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}) (\begin{bmatrix} 1 & 0 & 0 \\ 0 & -\frac{1}{2} & 1 \\ 0 & 1 & 0 \end{bmatrix}) (\begin{bmatrix} 1 & 1 & -1 \\ 0 & 2 & 3 \\ 0 & 0 & 5/2 \end{bmatrix}) e leqness= lessness (\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}) (\begin{bmatrix} 1 & 0 & 7 \\ 0 & 0 & 1 \\ 2 & 0 & 0 \end{bmatrix}) (\begin{bmatrix} 1 & -5 & 6 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \end{bmatrix}) c) p^tLU = (\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}) (\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix}) (\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}) d) p^2LU = (\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}) (\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}) 10-a) \frac{1}{3} n^{3}-1\frac{1}{3} \leqness \inelessness b) P = -det P c) P = (-1)^k det P d) (n-1) \cdot \frac{1}{3} n^{3-1} \frac{1}{3} Mesa 7.3 1-a) x^2 = (0,14, +0,35; 0,4)^t b) x^2 = (0,87; 0,31)^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; 1,0,4; -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; 0,3)^t b) x^2 = (0,97; 0,94; 0,78)^t c) x^2 = (1,05; 2,64; -0,3; -2,6)^t d) x^2 = (1,1; -1,5; 1,8; 1,8; 2,5) 4-a) x^2 = (1,31, -0,9; 0,06)^t b) x^2 = (-1,6; 1,3; 1,0; 0,5)^t c) x^2 = (-0,6; 1,5) 0-0,2; 0,6)^t d) x^2 = (0,68; 1,5; 0,25; 1,2; 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)