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)