Métodos numéricos

Método de bisección

f(x)

i	a	b	a+b/2	f(a)	f(b)	f((a+b)/2)
0	1	1	1	-3	-3	-3
1	1	1	1	-3	-3	-3
2	1	1	1	-3	-3	-3
3	1	1	1	-3	-3	-3
4	1	1	1	-3	-3	-3
5	1	1	1	-3	-3	-3
6	1	1	1	-3	-3	-3

f(x)

f(x)=7~{\left(\frac{ x}{7}\right)}^{3}+1

\textcolor{#6fbdd7}{f_x(x)}= \frac{{ x}^{2}\cdot3}{49}

x_i = x_i - f(x_i)/f_x(x_i)

$i$	$x_i$	$f(x_i)$	$f_x(x_i)$	$\small f(x_i)/f_x(x_i)$
0	7.4	9.269877551020409	3.032219378611455	2.764937910883857
1	4.635062089116143	3.032219378611455	1.2580761661091793	2.3052834478411866
2	2.329778641274957	1.2580761661091793	0.9370107561106986	3.785754447465806
3	-1.4559758061908492	0.9370107561106986	-12.325813326020086	7.219565896231217
4	-8.675541702422066	-12.325813326020086	-3.409718218592956	-2.6748362972927264
5	-6.00070540512934	-3.409718218592956	-0.8033260141319896	-1.546638094168845
6	-4.454067310960495	-0.8033260141319896	-0.11338140894564752	-0.6613831168166495
7	-3.7926841941438454	-0.11338140894564752	-0.0038051905970828415	-0.12874290701666435

\Large x=2~\pi+ e = 9.00146713563863

\Large (x)_2= 1001.000000000110000000100110011100111001011000110011

(x_1)_{10}= 9

(x_1)_2= 1001

9/2 = 1|4

4/2 = 0|2

2/2 = 0|1

1/2 = 1|0

(x_2)_{10}= 0.001467135638631

(x_2)_2= 0.00000000011000000010011001110011100101100011001100001000101111

0.001467135638631 \times 2 = 0.002934271277262 |_{\mathbb{N}} = 0

0.002934271277262 \times 2 = 0.005868542554524 |_{\mathbb{N}} = 0

0.005868542554524 \times 2 = 0.011737085109048 |_{\mathbb{N}} = 0

0.011737085109048 \times 2 = 0.023474170218096 |_{\mathbb{N}} = 0

0.023474170218096 \times 2 = 0.046948340436192 |_{\mathbb{N}} = 0

0.046948340436192 \times 2 = 0.093896680872384 |_{\mathbb{N}} = 0

0.093896680872384 \times 2 = 0.187793361744768 |_{\mathbb{N}} = 0

0.187793361744768 \times 2 = 0.375586723489536 |_{\mathbb{N}} = 0

0.375586723489536 \times 2 = 0.751173446979072 |_{\mathbb{N}} = 0

0.751173446979072 \times 2 = 1.502346893958144 |_{\mathbb{N}} = 1

0.502346893958144 \times 2 = 1.004693787916288 |_{\mathbb{N}} = 1

0.004693787916288 \times 2 = 0.009387575832576 |_{\mathbb{N}} = 0

0.009387575832576 \times 2 = 0.018775151665152 |_{\mathbb{N}} = 0

0.018775151665152 \times 2 = 0.037550303330304 |_{\mathbb{N}} = 0

0.037550303330304 \times 2 = 0.075100606660608 |_{\mathbb{N}} = 0

0x+0y+0z=0\\0x+0y+0z=0\\0x+0y+0z=0

R1

R2

R3

\Large \begin{bmatrix} 0&0&0\\0&0&0\\0&0&0 \end{bmatrix} \begin{bmatrix} x&y&z\\a&b&c\\d&e&f \end{bmatrix} = \begin{bmatrix} 0&0&0\\0&0&0\\0&0&0 \end{bmatrix}

0x+0y+0z=0\\0x+0y+0z=0\\0x+0y+0z=0

R1+

R2+

R3+

\Large \begin{bmatrix} 0&0&0\\0&0&0\\0&0&0 \end{bmatrix} \begin{bmatrix} x&y&z\\a&b&c\\d&e&f \end{bmatrix} = \begin{bmatrix} 0&0&0\\0&0&0\\0&0&0 \end{bmatrix}

i	a	b	a+b/2	f(a)	f(b)	f((a+b)/2)
0	1	1	1	-3	-3	-3
1	1	1	1	-3	-3	-3
2	1	1	1	-3	-3	-3
3	1	1	1	-3	-3	-3
4	1	1	1	-3	-3	-3
5	1	1	1	-3	-3	-3
6	1	1	1	-3	-3	-3

i	a	b	a+b/2	f(a)	f(b)	f((a+b)/2)
0	1	1	1	-3	-3	-3
1	1	1	1	-3	-3	-3
2	1	1	1	-3	-3	-3
3	1	1	1	-3	-3	-3
4	1	1	1	-3	-3	-3
5	1	1	1	-3	-3	-3
6	1	1	1	-3	-3	-3

i	a	b	a+b/2	f(a)	f(b)	f((a+b)/2)
0	1	1	1	-3	-3	-3
1	1	1	1	-3	-3	-3
2	1	1	1	-3	-3	-3
3	1	1	1	-3	-3	-3
4	1	1	1	-3	-3	-3
5	1	1	1	-3	-3	-3
6	1	1	1	-3	-3	-3