14 - CALCULO-NUMERICO.sagews

%auto
typeset_mode(True, display=False)

ZEROS DE FUNÇÕES

Método de Bisseção

import scipy.optimize as scop

f =  sin(-x)*x^3 + 10

f

x^3*sin(-x) + 10

plot(f,(x,-10,10))

%time
scop.bisect(f, -8, 10)

-6.322757827602629 CPU time: 0.01 s, Wall time: 0.01 s

Método da Falsa Posição

Método Interativo Linear

Método de Newton – Raphson

import scipy.optimize as scop

f = x^2 - 2*x

df = f.derivative(x,1)

%time
scop.newton(f, 3, df)

$\displaystyle 2.0$

CPU time: 0.00 s, Wall time: 0.01 s

• Outra forma do método de Newton

x = PolynomialRing(RealField(), 'x').gen()

f = x^2 - 2*x

# para n = 10 iteracoes
# inicio com  x0 = 8
%time

f.newton_raphson(10,8)

[$\displaystyle 4.57142857142857$, $\displaystyle 2.92571428571429$, $\displaystyle 2.22250105977109$, $\displaystyle 2.02024813034049$, $\displaystyle 2.00020092503485$, $\displaystyle 2.00000002018138$, $\displaystyle 2.00000000000000$, $\displaystyle 2.00000000000000$, $\displaystyle 2.00000000000000$, $\displaystyle 2.00000000000000$]

CPU time: 0.00 s, Wall time: 0.00 s

Método da Secante

import scipy.optimize as scop

f = x^2-2*x

# se a derivada não é passada como parametro então o método de da secante é usando

%time

scop.newton(f, 8)

2.0

<string>:1: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...) See http://trac.sagemath.org/5930 for details.

CPU time: 0.01 s, Wall time: 0.01 s

Método de Resolução de Sisemas de Equações

Sistemas Lineares

Métodos Diretos

Método de eliminação de Gauss sem Pivotamento

Método de eliminação de Gauss com Pivotamento total (Linhas e colunas)

Fatoração LU

Métodos Interativos

Método Iterativo de Gauss – Jacobi

Método Iterativo de Gauss – Seidel

Interpolação

Interpolação Polinomial: Forma de Lagrange para o polinômio interpolador

Interpolação Polinomial: Forma de Newton para o polinômio interpolado

Interpolação Polinomial: Forma de Newton-Gregory para o polinômio interpolador

Funções Spline (linear) em interpolação

Integração Numérica

Regra dos Trapézios

import scipy.integrate as sci

f(x) = x^2

intervalo = srange(0,5,0.1)

f.integrate(x,0,10).n()

$333.333333333333$

sci.trapz(map(f,intervalo),intervalo)

$333.323333449004$

Regra de Simpson

sci.simps(map(f,intervalo),intervalo)

$39.2165000000000$

Fórmula de Newton-Cotes

coef, error = sci.newton_cotes(3)

interv = linspace(0,5,10)

Error in lines 1-1 Traceback (most recent call last): File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 957, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'linspace' is not defined

coef*[map(f,intervalo)]

Error in lines 1-1 Traceback (most recent call last): File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 957, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> ValueError: operands could not be broadcast together with shapes (4,) (1,50)

(intervalo)

[$0.000000000000000$, $0.100000000000000$, $0.200000000000000$, $0.300000000000000$, $0.400000000000000$, $0.500000000000000$, $0.600000000000000$, $0.700000000000000$, $0.800000000000000$, $0.900000000000000$, $1.00000000000000$, $1.10000000000000$, $1.20000000000000$, $1.30000000000000$, $1.40000000000000$, $1.50000000000000$, $1.60000000000000$, $1.70000000000000$, $1.80000000000000$, $1.90000000000000$, $2.00000000000000$, $2.10000000000000$, $2.20000000000000$, $2.30000000000000$, $2.40000000000000$, $2.50000000000000$, $2.60000000000000$, $2.70000000000000$, $2.80000000000000$, $2.90000000000000$, $3.00000000000000$, $3.10000000000000$, $3.20000000000000$, $3.30000000000000$, $3.40000000000000$, $3.50000000000000$, $3.60000000000000$, $3.70000000000000$, $3.80000000000000$, $3.90000000000000$, $4.00000000000000$, $4.10000000000000$, $4.20000000000000$, $4.30000000000000$, $4.40000000000000$, $4.50000000000000$, $4.60000000000000$, $4.70000000000000$, $4.80000000000000$, $4.90000000000000$, $5.00000000000000$, $5.10000000000000$, $5.20000000000000$, $5.30000000000000$, $5.40000000000000$, $5.50000000000000$, $5.60000000000000$, $5.70000000000000$, $5.80000000000000$, $5.90000000000000$, $5.99999999999999$, $6.09999999999999$, $6.19999999999999$, $6.29999999999999$, $6.39999999999999$, $6.49999999999999$, $6.59999999999999$, $6.69999999999999$, $6.79999999999999$, $6.89999999999999$, $6.99999999999999$, $7.09999999999999$, $7.19999999999999$, $7.29999999999999$, $7.39999999999999$, $7.49999999999999$, $7.59999999999999$, $7.69999999999999$, $7.79999999999999$, $7.89999999999999$, $7.99999999999999$, $8.09999999999999$, $8.19999999999999$, $8.29999999999999$, $8.39999999999999$, $8.49999999999999$, $8.59999999999999$, $8.69999999999999$, $8.79999999999998$, $8.89999999999998$, $8.99999999999998$, $9.09999999999998$, $9.19999999999998$, $9.29999999999998$, $9.39999999999998$, $9.49999999999998$, $9.59999999999998$, $9.69999999999998$, $9.79999999999998$, $9.89999999999998$]

sci.romb(bb)

Error in lines 1-1 Traceback (most recent call last): File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 957, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'bb' is not defined

import numpy as np

intv = np.linspace(0,5,20)

def fu(x):
  return x^2

fu(intv)

Error in lines 1-1 Traceback (most recent call last): File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 957, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'intv' is not defined

intervalo

[$0.000000000000000$, $0.100000000000000$, $0.200000000000000$, $0.300000000000000$, $0.400000000000000$, $0.500000000000000$, $0.600000000000000$, $0.700000000000000$, $0.800000000000000$, $0.900000000000000$, $1.00000000000000$, $1.10000000000000$, $1.20000000000000$, $1.30000000000000$, $1.40000000000000$, $1.50000000000000$, $1.60000000000000$, $1.70000000000000$, $1.80000000000000$, $1.90000000000000$, $2.00000000000000$, $2.10000000000000$, $2.20000000000000$, $2.30000000000000$, $2.40000000000000$, $2.50000000000000$, $2.60000000000000$, $2.70000000000000$, $2.80000000000000$, $2.90000000000000$, $3.00000000000000$, $3.10000000000000$, $3.20000000000000$, $3.30000000000000$, $3.40000000000000$, $3.50000000000000$, $3.60000000000000$, $3.70000000000000$, $3.80000000000000$, $3.90000000000000$, $4.00000000000000$, $4.10000000000000$, $4.20000000000000$, $4.30000000000000$, $4.40000000000000$, $4.50000000000000$, $4.60000000000000$, $4.70000000000000$, $4.80000000000000$, $4.90000000000000$, $5.00000000000000$, $5.10000000000000$, $5.20000000000000$, $5.30000000000000$, $5.40000000000000$, $5.50000000000000$, $5.60000000000000$, $5.70000000000000$, $5.80000000000000$, $5.90000000000000$, $5.99999999999999$, $6.09999999999999$, $6.19999999999999$, $6.29999999999999$, $6.39999999999999$, $6.49999999999999$, $6.59999999999999$, $6.69999999999999$, $6.79999999999999$, $6.89999999999999$, $6.99999999999999$, $7.09999999999999$, $7.19999999999999$, $7.29999999999999$, $7.39999999999999$, $7.49999999999999$, $7.59999999999999$, $7.69999999999999$, $7.79999999999999$, $7.89999999999999$, $7.99999999999999$, $8.09999999999999$, $8.19999999999999$, $8.29999999999999$, $8.39999999999999$, $8.49999999999999$, $8.59999999999999$, $8.69999999999999$, $8.79999999999998$, $8.89999999999998$, $8.99999999999998$, $9.09999999999998$, $9.19999999999998$, $9.29999999999998$, $9.39999999999998$, $9.49999999999998$, $9.59999999999998$, $9.69999999999998$, $9.79999999999998$, $9.89999999999998$]

bb = map(fu,intv)
bb

[$0.0$, $0.0692520775623$, $0.277008310249$, $0.623268698061$, $1.108033241$, $1.73130193906$, $2.49307479224$, $3.39335180055$, $4.43213296399$, $5.60941828255$, $6.92520775623$, $8.37950138504$, $9.97229916898$, $11.703601108$, $13.5734072022$, $15.5817174515$, $17.728531856$, $20.0138504155$, $22.4376731302$, $25.0$]

Estudo dos Erros

Equações Diferenciais

Soluções Numéricas de EDO Métodos de passo simples

Soluções Numéricas de Equações Diferenciais Ordinárias Métodos de passo simples: Método de Série de Taulor, Métdo de Euler , Método de Euler Modificado, Método de Runge – Kutta de 4.º ordem, Métodos de previsão – correção.

Método de Série de Taylor

Método de Euler

Método de Euler Modificado

Método de Runge – Kutta de 4.º ordem

Métodos de previsão – correção