def some_complex_irrational_f(x, y, z): from pyaudi import exp, log, cos, sin, tan, sqrt, cbrt, cos, sin, tan, acos, asin, atan, cosh, sinh, tanh, acosh, asinh, atanh from pyaudi import abs as gd_abs from pyaudi import sin_and_cos, sinh_and_cosh f = (x + y + z) / 10. retval = exp(f) + log(f) + f**2 + sqrt(f) + cbrt(f) + cos(f) + sin(f) retval += tan(f) + acos(f) + asin(f) + atan(f) + cosh(f) + sinh(f) retval += tanh(f) + acosh(f) + asinh(f) + atanh(f) a = sin_and_cos(f) b = sinh_and_cosh(f) retval += a[0] + a[1] + b[0] + b[1] return retval
def some_complex_irrational_f(x,y,z): from pyaudi import exp, log, cos, sin, tan, sqrt, cbrt, cos, sin, tan, acos, asin, atan, cosh, sinh, tanh, acosh, asinh, atanh from pyaudi import abs as gd_abs from pyaudi import sin_and_cos, sinh_and_cosh f = (x+y+z) / 10. retval = exp(f) + log(f) + f**2 + sqrt(f) + cbrt(f) + cos(f) + sin(f) retval += tan(f) + acos(f) + asin(f) + atan(f) + cosh(f) + sinh(f) retval += tanh(f) + acosh(f) + asinh(f) + atanh(f) a = sin_and_cos(f) b = sinh_and_cosh(f) retval+=a[0]+a[1]+b[0]+b[1] return retval
def test_cbrt(self): from pyaudi import gdual_double as gdual from pyaudi import cbrt x = gdual(2.3, "x",3); y = gdual(1.5, "y",3); p1 = x*x*y - x*y*x*x*x + 3*y*y*y*y*x*y*x; # positive p0 p2 = x*x*y - x*y*x*x*x - 3*y*y*y*y*x*y*x; # negative coefficient self.assertTrue((cbrt(p1)*cbrt(p1)*cbrt(p1) - p1).is_zero(1e-12)) self.assertTrue((cbrt(p2)*cbrt(p2)*cbrt(p2) - p2).is_zero(1e-12))
def test_cbrt(self): from pyaudi import gdual_double as gdual from pyaudi import cbrt x = gdual(2.3, "x", 3) y = gdual(1.5, "y", 3) p1 = x * x * y - x * y * x * x * x + 3 * \ y * y * y * y * x * y * x # positive p0 p2 = x * x * y - x * y * x * x * x - 3 * y * \ y * y * y * x * y * x # negative coefficient self.assertTrue((cbrt(p1) * cbrt(p1) * cbrt(p1) - p1).is_zero(1e-12)) self.assertTrue((cbrt(p2) * cbrt(p2) * cbrt(p2) - p2).is_zero(1e-12))
from pyaudi import gdual_double as gdual from pyaudi import exp, log, cbrt # We want to compute the Taylor expansion of a function f (and thus all derivatives) at x=2, y=3 # 1 - Define the generalized dual numbers (7 is the truncation order, i.e. the maximum order of derivation we will need) x = gdual(2, "x", 7) y = gdual(3, "y", 7) # 2 - Compute your function as usual f = exp(x * x + cbrt(y) / log(x * y)) # 3 - Inspect the results (this does not require any more computations) print("Taylor polynomial: " + str(f)) # This is the Taylor expansion of f (truncated at the 7th order) print("Derivative value [1,0]: " + str(f.get_derivative( [1, 0]))) # This is the value of the derivative (d / dx) print("Derivative value [4,3]: " + str(f.get_derivative( [4, 3]))) # This is the value of the mixed derivative (d^7 / dx^4dy^3) # 4 - Using the dictionary interface (note the presence of the "d" before all variables) print("Derivative value [1,0]: " + str(f.get_derivative( {"dx": 1}))) # This is the value of the derivative (d / dx) print("Derivative value [4,3]: " + str(f.get_derivative({ "dx": 4, "dy": 3 }))) # This is the value of the mixed derivative (d^7 / dx^4dy^3)
from pyaudi import gdual_double as gdual from pyaudi import exp, log, cbrt # We want to compute the Taylor expansion of a function f (and thus all derivatives) at x=2, y=3 # 1 - Define the generalized dual numbers (7 is the truncation order, i.e. the maximum order of derivation we will need) x = gdual(2, "x", 7); y = gdual(3, "y", 7); # 2 - Compute your function as usual f = exp(x*x + cbrt(y) / log(x*y)); # 3 - Inspect the results (this does not require any more computations) print("Taylor polynomial: " + str(f)) # This is the Taylor expansion of f (truncated at the 7th order) print("Derivative value [1,0]: " + str(f.get_derivative([1,0]))) # This is the value of the derivative (d / dx) print("Derivative value [4,3]: " + str(f.get_derivative([4,3]))) # This is the value of the mixed derivative (d^7 / dx^4dy^3) # 4 - Using the dictionary interface (note the presence of the "d" before all variables) print("Derivative value [1,0]: " + str(f.get_derivative({"dx":1}))) # This is the value of the derivative (d / dx) print("Derivative value [4,3]: " + str(f.get_derivative({"dx":4, "dy":3}))) # This is the value of the mixed derivative (d^7 / dx^4dy^3)