Biseccion 454r7

Emplee el método de bisección para encontrar la raíz de la siguiente ecuación en el intervalo [-1.5 , -2.5], siendo 𝐟(𝐱) = 𝐱 − 𝟒𝐜𝐨𝐬𝐱 + 𝒆𝒙 , e iterar hasta que el error estimado sea menor o igual al 1%. SOLUCIÓN: Sea 𝐟(𝐱) = 𝐱 − 𝟒𝐜𝐨𝐬𝐱 + 𝒆𝒙 , 𝒙∗ ∈ [-1.5 , -2.5]



𝑎0 + 𝑏0 2

=

−1.5− 2.5 2

= −2.000000000

𝑋0 = −2.000000000

𝑎1 = 𝑋0 = −2.000000000

f(𝑎0 )f( 𝑋0 )= f(-1.5)f(-2.000000000) > 0 ⇒ -



𝑏1 = 𝑏0 = −2.500000000

-

Sea i=1 [𝑎1 , 𝑏1 ] = [−2.000000000 , −2.500000000]

 𝑋1 = 

ξs ≤ 1%

Sea i=0 [𝑎0 , 𝑏0 ] = [−1.5 , −2.5]

 𝑋0 = 

con un

𝑎1 + 𝑏1 2

=

−2.000000000− 2.500000000 2

= −2.250000000

𝑋1 = −2.250000000

𝑎2 = 𝑎1 = −2.000000000

f(𝑎1 )f( 𝑋1 )= f(-2.000000000)f(-2.250000000) < 0 ⇒ -

+

𝑏2 = 𝑋1 = −2.250000000



Sea i=2 [𝑎2 , 𝑏2 ] = [−2.000000000 , −2.250000000]

 𝑋2 = 

𝑎2 + 𝑏2 2

=

−2.000000000− 2.250000000 2

= −2.125000000

𝑋2 = −2.125000000

𝑎3 = 𝑎2 = −2.000000000

f(𝑎2 )f( 𝑋2 )= f(-2.000000000)f(-2.125000000) < 0 ⇒ -



+

Sea i=3 [𝑎3 , 𝑏3 ] = [−2.000000000 , −2.125000000]

 𝑋3 = 

𝑎3 + 𝑏3 2

=

−2.000000000− 2.125000000 2

= −2.062500000

𝑋3 = −2.062500000

𝑎4 = 𝑋3 = −2.062500000

f(𝑎3 )f( 𝑋3 )= f(-2.000000000)f(-2.062500000) > 0 ⇒ -



𝑏4 = 𝑏3 = −2.125000000

-

Sea i=4 [a4 , b4 ] = [−2.062500000 , −2.125000000]

 𝑋4 = 

𝑏3 = 𝑋2 = −2.125000000

𝑎4 + 𝑏4 2

=

−2.062500000− 2.125000000 2

= −2.093750000

𝑋4 = −2.093750000

𝑎5 = 𝑎4 = −2.062500000

f(𝑎4 )f( 𝑋4 )= f(-2.062500000)f(-2.093750000) < 0 ⇒ -

+

𝑏5 = 𝑋4 = −2.093750000



Sea i=5 [a5 , b5 ] = [−2.062500000 , −2.093750000]

 𝑋5 = 

𝑎5 + 𝑏5 2

=

−2.062500000− 2.093750000 2

= −2.078125000

𝑋5 = −2.078125000

𝑎6 = 𝑋5 = −2.078125000

f(𝑎5 )f( 𝑋5 )= f(-2.062500000)f(-2.078125000) > 0 ⇒ -



-

Sea i=6 [a6 , b6 ] = [−2.078125000 , −2.093750000]

 𝑋6 = 

𝑏6 = 𝑏5 = −2.093750000

𝑎6 + 𝑏6 2

=

−2.078125000− 2.093750000 2

= −2.085937500

𝑋6 = −2.085937500

𝑎7 = 𝑎6 = −2.078125000

f(𝑎6 )f( 𝑋6 )= f(-2.078125000)f(-2.085937500) < 0 ⇒ -

n 0 1 2 3 4 5 6

Entonces la raíz es :

𝑏7 = 𝑋6 = −2.085937500

+

𝑋𝑖 −2.000000000 −2.250000000 −2.125000000 −2.062500000 −2.093750000 −2.078125000 −2.085937500

X5 = -2.078125000

|𝑋𝑖+1 − 𝑋𝑖 | 0.250000000 0.125000000 0.062500000 0.031250000 0.015625000 ≤ 0.01 0.007812500