def test_newton(self):
"""
뉴튼 랩슨법 함수를 검증하는 method
:return:
"""
#1 변수 방정식의 해를 뉴튼 랩슨법으로 찾음
result = rf.newton(f, dfdx, 100, epsilon=1e-8)
# 유효숫자 6번쨰 자리까지 f(x) ==0임을 확인함
self.assertAlmostEqual(0.0, f(result), places=6)
# -*- coding: utf8 -*-
# [2011112047] [한소운]
"""
1변수 방정식의 근을 찾는 방법 중 Newton-Raphson method 를 사용하여
어떤 함수 g(x)의 근을 찾고자 함
아래 예는 Newton-Raphson method 를 사용하기 곤란한 경우임
"""
# "Newton's method," Wikipedia. [Online]. Available: http://en.wikipedia.org/wiki/newton's_method.
# 1변수 방정식의 근을 찾는 함수를 모아둔 root_finding 모듈을 불러들임
import root_finding as rf
help(root_finding)
def g(x):
# 근을 구하고자 하는 함수
return x ** 3 - 2 * x + 2
def dgdx(x):
# g(x) 의 x 에 대한 미분
return 3.0 * x ** 2.0 - 2.0
if "__main__" == __name__:
# 주어진 초기값에서 시작하여 g(x) = 0 인 x 를 찾고자 함
# 생각보다 시간이 많이 걸릴 수 있음
x_nr = rf.newton(g, dgdx, 0)
print('x = %g, f(%g) = %g' % (x_nr, x_nr, g(x_nr)))
def bmain():
"""
Test of the first root finding method
"""
# Initialization
x0 = np.array([3, -1], dtype='float64')
dx = np.array([1e-9, 1e-9], dtype='float64')
print('Check part A')
print('Solve the system A*x*y = 1')
print(', where exp(-x) + exp(-y) = 1 + 1/A, and A=10000.')
print('Starting point: x0 =\n', x0)
print('f(x0) =\n', systems.sys(x0))
print('The roots using Newton method:')
roots = root_finding.newton_jacobian(systems.sys, x0, systems.jacobian_sys)
print(roots)
print('f(roots) =\n', systems.sys(roots))
print('Number of calls:', globvar.ncalls)
globvar.ncalls = 0
roots = root_finding.newton(systems.sys, x0, dx)
print('Using the Newton method but without the Jacobian, the roots were')
print(roots)
print('Using the method from part A, the number of calls were',
globvar.ncalls)
print('\n\n')
# Initialization
globvar.ncalls = 0
x0 = np.array([1, 400], dtype='float64')
dx = np.array([1e-9, 1e-9], dtype='float64')
print('Check part B')
print('Solve the Rosenbrock valley function')
print('Starting point: x0 =\n', x0)
print('f(x0) =\n', systems.rosenbrock(x0))
print('The roots using Newton method:')
roots = root_finding.newton_jacobian(systems.rosenbrock, x0,
systems.jacobian_rosenbrock)
print(roots)
print('f(roots) =\n', systems.rosenbrock(roots))
print('Number of calls:', globvar.ncalls)
globvar.ncalls = 0
roots = root_finding.newton(systems.rosenbrock, x0, dx)
print('Using the Newton method but without the Jacobian, the roots were')
print(roots)
print('Using the method from part A, the number of calls were',
globvar.ncalls)
print('\n\n')
# Initialization
globvar.ncalls = 0
x0 = np.array([1, -1], dtype='float64')
dx = np.array([1e-9, 1e-9], dtype='float64')
print('Check part C')
print('Solve the Himmelblau function')
print('Starting point: x0 =\n', x0)
print('f(x0) =\n', systems.himmelblau(x0))
print('The roots using Newton method:')
roots = root_finding.newton_jacobian(systems.himmelblau, x0,
systems.jacobian_himmelblau)
print(roots)
print('f(roots) =\n', systems.himmelblau(roots))
print('Number of calls:', globvar.ncalls)
globvar.ncalls = 0
roots = root_finding.newton(systems.himmelblau, x0, dx)
print('Using the Newton method but without the Jacobian, the roots were')
print(roots)
print('Using the method from part A, the number of calls were',
globvar.ncalls)
print('\n\n')
# Initialization
globvar.ncalls = 0
x0 = np.array([1, -1], dtype='float64')
dx = np.array([1e-9, 1e-9], dtype='float64')
print('Check part D: Make some interesting example')
print('Solve the Matya function')
print('Starting point: x0 =\n', x0)
print('f(x0) =\n', systems.matya(x0))
print('The roots using Newton method:')
roots = root_finding.newton_jacobian(systems.matya, x0,
systems.jacobian_matya)
print(roots)
print('f(roots) =\n', systems.matya(roots))
print('Number of calls:', globvar.ncalls)
globvar.ncalls = 0
roots = root_finding.newton(systems.matya, x0, dx)
print('Using the Newton method but without the Jacobian, the roots were')
print(roots)
print('Using the method from part A, the number of calls were',
globvar.ncalls)
print('\n\n')
def bmain():
"""
Test of the minimization method using the Quasi-Newton method
"""
# Initialization
globvar.ncalls = 0
x0 = np.array([-2, 2], dtype='float64')
dx = np.array([1e-9, 1e-9], dtype='float64')
alpha = 1e-4
print('Testing the quasi-Newton method with Broydens update')
print('Check part A')
print('Minimize the Rosenbrock valley function')
print('f(x, y) = (1 - x)^2 + 100 * (y - x^2)^2')
print('Starting point: x0 =\n', x0)
print('f(x0) =\n', systems.rosenbrock(x0))
print('The minimum is found as:')
mini = minimize.qnewton_minimize(systems.rosenbrock,
systems.grad_rosenbrock, x0, alpha)
print(mini)
print('f(min) =\n', systems.rosenbrock(mini))
print('Number of steps used:', globvar.ncalls)
print('We can now compare the number of steps used by different methods')
globvar.ncalls = 0
mini = minimize.newton_minimize(systems.rosenbrock,
systems.grad_rosenbrock,
systems.hessian_rosenbrock, x0, alpha)
print('Number of steps used in the Newton minimization (A) is',
globvar.ncalls)
globvar.ncalls = 0
mini = root.newton(systems.grad_rosenbrock, x0, dx)
print('Number of steps used in the Newton root-finding is', globvar.ncalls)
globvar.ncalls = 0
mini = root.newton_jacobian(systems.grad_rosenbrock, x0,
systems.hessian_rosenbrock)
print('Number of steps using Newton root-finding with the Jacobian is',
globvar.ncalls)
print('\n\n')
# Initialization
x0 = np.array([2.5, 2.0], dtype='float64')
alpha = 1e-4
globvar.ncalls = 0
print('Check part B')
print('Minimize the Himmelblau function')
print('f(x, y) = (x^2 + y -11)^2 + (x + y^2 - 7)^2')
print('Starting point: x0 =\n', x0)
print('f(x0) =\n', systems.himmelblau(x0))
print('The minimum is found as:')
mini = minimize.qnewton_minimize(systems.himmelblau,
systems.grad_himmelblau, x0, alpha)
print(mini)
print('f(min) =\n', systems.himmelblau(mini))
print('Number of steps used:', globvar.ncalls)
print('We can now compare the number of steps used by different methods')
globvar.ncalls = 0
mini = minimize.newton_minimize(systems.himmelblau,
systems.grad_himmelblau,
systems.hessian_himmelblau, x0, alpha)
print('Number of steps used in the Newton minimization (A) is',
globvar.ncalls)
globvar.ncalls = 0
mini = root.newton(systems.grad_himmelblau, x0, dx)
print('Number of steps used in the Newton root-finding is', globvar.ncalls)
globvar.ncalls = 0
mini = root.newton_jacobian(systems.grad_himmelblau, x0,
systems.hessian_himmelblau)
print('Number of steps using Newton root-finding with the Jacobian is',
globvar.ncalls)
print('\n\n')
# -*- coding: utf8 -*-
# [학번] [이름]
"""
1변수 방정식의 근을 찾는 방법 중 Newton-Raphson method 를 사용하여
어떤 함수 g(x)의 근을 찾고자 함
아래 예는 Newton-Raphson method 를 사용하기 곤란한 경우임
"""
# "Newton's method," Wikipedia. [Online]. Available: https://en.wikipedia.org/wiki/newton's_method. [Accessed: 21-Aug-2016].
# 1변수 방정식의 근을 찾는 함수를 모아둔 root_finding 모듈을 불러들임
import root_finding as rf
def g(x):
# 근을 구하고자 하는 함수
return x ** 3 - 2 * x + 2
def dgdx(x):
# g(x) 의 x 에 대한 미분
return 3.0 * x ** 2.0 - 2.0
if "__main__" == __name__:
# 주어진 초기값에서 시작하여 g(x) = 0 인 x 를 찾고자 함
# 생각 보다 시간이 많이 걸릴 수 있음
x_nr = rf.newton(g, dgdx, 0)
print('x = %g, f(%g) = %g' % (x_nr, x_nr, g(x_nr)))
def test_newton(self):
result = rf.newton(f, dfdx, 0.0001, epsilon=1e-8)
self.assertAlmostEqual(0.0, f(result), places=3)
