Difference between revisions of "DPL94 derivatives"
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| + | These are the first and second partial derivatives of the equations of the [[DPL94]] [[:Category:Relaxation dispersion analysis|relaxation dispersion]] model. | ||
| + | |||
| + | == Equation == | ||
| + | <math> | ||
| + | \mathrm{R}_{1\rho}= \mathrm{R}_1\cos^2\theta + \left( \mathrm{R}_{1\rho}{´} + \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + \omega_\textrm{e}^2} \right) \sin^2\theta | ||
| + | </math> | ||
| + | |||
== Jacobian == | == Jacobian == | ||
=== sympy === | === sympy === | ||
| Line 47: | Line 54: | ||
------------------------------------------------------------------------------ | ------------------------------------------------------------------------------ | ||
""" % (d_f_d_R1, d_f_d_theta, d_f_d_R1rho_p, d_f_d_phi_ex, d_f_d_kex, d_f_d_we) ) | """ % (d_f_d_R1, d_f_d_theta, d_f_d_R1rho_p, d_f_d_phi_ex, d_f_d_kex, d_f_d_we) ) | ||
| + | |||
| + | #### Method 2 | ||
| + | # http://docs.sympy.org/0.7.2/modules/matrices/matrices.html | ||
| + | |||
| + | # The vectorial function. | ||
| + | X = Matrix([R1 * cos(theta)**2 + (R1rho_p + ( (phi_ex * kex) / (kex**2 + we**2) ) ) * sin(theta)**2]) | ||
| + | # What to derive for. | ||
| + | Y = Matrix([R1, theta, R1rho_p, phi_ex, kex, we]) | ||
| + | |||
| + | # Make the Jacobian | ||
| + | Jacobian = X.jacobian(Y) | ||
| + | |||
| + | jac_string = str(Jacobian) | ||
| + | jac_string_arr = jac_string.replace("Matrix", "array") | ||
| + | |||
| + | print("""Form the Jacobian matrix by: | ||
| + | ------------------------------------------------------------------------------ | ||
| + | from numpy import array, cos, sin, pi, transpose | ||
| + | |||
| + | R1 = 1.1 | ||
| + | theta = pi / 4 | ||
| + | R1rho_p = 10. | ||
| + | phi_ex = 1100. | ||
| + | kex = 2200. | ||
| + | we = 3300. | ||
| + | |||
| + | jacobian_matrix_2 = %s | ||
| + | |||
| + | print jacobian_matrix_2 | ||
| + | ------------------------------------------------------------------------------ | ||
| + | """ % (jac_string_arr) ) | ||
</source> | </source> | ||
| + | |||
=== output === | === output === | ||
| − | output | + | output is |
<source lang="python"> | <source lang="python"> | ||
from numpy import array, cos, sin, pi, transpose | from numpy import array, cos, sin, pi, transpose | ||
| Line 69: | Line 108: | ||
print jacobian_matrix | print jacobian_matrix | ||
| + | |||
| + | R1 = 1.1 | ||
| + | theta = pi / 4 | ||
| + | R1rho_p = 10. | ||
| + | phi_ex = 1100. | ||
| + | kex = 2200. | ||
| + | we = 3300. | ||
| + | |||
| + | jacobian_matrix_2 = array([[cos(theta)**2, -2*R1*sin(theta)*cos(theta) + 2*(R1rho_p + kex*phi_ex/(kex**2 + we**2))*sin(theta)*cos(theta), | ||
| + | sin(theta)**2, kex*sin(theta)**2/(kex**2 + we**2), (-2*kex**2*phi_ex/(kex**2 + we**2)**2 + phi_ex/(kex**2 + we**2))*sin(theta)**2, -2*kex*phi_ex*we*sin(theta)**2/(kex**2 + we**2)**2]]) | ||
| + | |||
| + | print jacobian_matrix_2 | ||
</source> | </source> | ||
| + | |||
| + | == Hessian == | ||
| + | === sympy === | ||
| + | <source lang="python"> | ||
| + | from sympy import * | ||
| + | |||
| + | # In contrast to other Computer Algebra Systems, in SymPy you have to declare symbolic variables explicitly: | ||
| + | R1 = Symbol('R1') | ||
| + | theta = Symbol('theta') | ||
| + | R1rho_p = Symbol('R1rho_p') | ||
| + | phi_ex = Symbol('phi_ex') | ||
| + | kex = Symbol('kex') | ||
| + | we = Symbol('we') | ||
| + | |||
| + | # Define function | ||
| + | f = R1 * cos(theta)**2 + (R1rho_p + ( (phi_ex * kex) / (kex**2 + we**2) ) ) * sin(theta)**2 | ||
| + | |||
| + | print("Now calculate the Hessian.\n") | ||
| + | |||
| + | # Define symbols to derive for. | ||
| + | syms = [R1, theta, R1rho_p, phi_ex, kex, we] | ||
| + | |||
| + | # Do the hessian. | ||
| + | hess = hessian(f, syms) | ||
| + | |||
| + | hess_string = str(hess) | ||
| + | hess_string_arr = hess_string.replace("Matrix", "array") | ||
| + | |||
| + | print("""Form the Hessian matrix by: | ||
| + | ------------------------------------------------------------------------------ | ||
| + | from numpy import array, cos, sin, pi | ||
| + | |||
| + | R1 = 1.1 | ||
| + | theta = pi / 4 | ||
| + | R1rho_p = 10. | ||
| + | phi_ex = 1100. | ||
| + | kex = 2200. | ||
| + | we = 3300. | ||
| + | |||
| + | hessian_matrix = %s | ||
| + | |||
| + | print hessian_matrix | ||
| + | ------------------------------------------------------------------------------ | ||
| + | """ % (hess_string_arr) ) | ||
| + | </source> | ||
| + | |||
| + | === output === | ||
| + | output is | ||
| + | <source lang="python"> | ||
| + | from numpy import array, cos, sin, pi | ||
| + | |||
| + | R1 = 1.1 | ||
| + | theta = pi / 4 | ||
| + | R1rho_p = 10. | ||
| + | phi_ex = 1100. | ||
| + | kex = 2200. | ||
| + | we = 3300. | ||
| + | |||
| + | hessian_matrix = array([[0, -2*sin(theta)*cos(theta), 0, 0, 0, 0], | ||
| + | [-2*sin(theta)*cos(theta), 2*R1*sin(theta)**2 - 2*R1*cos(theta)**2 - 2*(R1rho_p + kex*phi_ex/(kex**2 + we**2))*sin(theta)**2 + 2*(R1rho_p + kex*phi_ex/(kex**2 + we**2))*cos(theta)**2, 2*sin(theta)*cos(theta), | ||
| + | 2*kex*sin(theta)*cos(theta)/(kex**2 + we**2), 2*(-2*kex**2*phi_ex/(kex**2 + we**2)**2 + phi_ex/(kex**2 + we**2))*sin(theta)*cos(theta), -4*kex*phi_ex*we*sin(theta)*cos(theta)/(kex**2 + we**2)**2], | ||
| + | [0, 2*sin(theta)*cos(theta), 0, 0, 0, 0], [0, 2*kex*sin(theta)*cos(theta)/(kex**2 + we**2), 0, 0, -2*kex**2*sin(theta)**2/(kex**2 + we**2)**2 + sin(theta)**2/(kex**2 + we**2), -2*kex*we*sin(theta)**2/(kex**2 + we**2)**2], | ||
| + | [0, 2*(-2*kex**2*phi_ex/(kex**2 + we**2)**2 + phi_ex/(kex**2 + we**2))*sin(theta)*cos(theta), 0, -2*kex**2*sin(theta)**2/(kex**2 + we**2)**2 + sin(theta)**2/(kex**2 + we**2), | ||
| + | (8*kex**3*phi_ex/(kex**2 + we**2)**3 - 6*kex*phi_ex/(kex**2 + we**2)**2)*sin(theta)**2, (8*kex**2*phi_ex*we/(kex**2 + we**2)**3 - 2*phi_ex*we/(kex**2 + we**2)**2)*sin(theta)**2], | ||
| + | [0, -4*kex*phi_ex*we*sin(theta)*cos(theta)/(kex**2 + we**2)**2, 0, -2*kex*we*sin(theta)**2/(kex**2 + we**2)**2, | ||
| + | (8*kex**2*phi_ex*we/(kex**2 + we**2)**3 - 2*phi_ex*we/(kex**2 + we**2)**2)*sin(theta)**2, 8*kex*phi_ex*we**2*sin(theta)**2/(kex**2 + we**2)**3 - 2*kex*phi_ex*sin(theta)**2/(kex**2 + we**2)**2]]) | ||
| + | |||
| + | print hessian_matrix | ||
| + | |||
| + | </source> | ||
| + | |||
| + | == See also == | ||
| + | [[Category:Relaxation_dispersion analysis]] | ||
Latest revision as of 08:16, 13 September 2014
These are the first and second partial derivatives of the equations of the DPL94 relaxation dispersion model.
Equation
[math] \mathrm{R}_{1\rho}= \mathrm{R}_1\cos^2\theta + \left( \mathrm{R}_{1\rho}{´} + \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + \omega_\textrm{e}^2} \right) \sin^2\theta [/math]
Jacobian
sympy
from sympy import *
# In contrast to other Computer Algebra Systems, in SymPy you have to declare symbolic variables explicitly:
R1 = Symbol('R1')
theta = Symbol('theta')
R1rho_p = Symbol('R1rho_p')
phi_ex = Symbol('phi_ex')
kex = Symbol('kex')
we = Symbol('we')
# Define function
f = R1 * cos(theta)**2 + (R1rho_p + ( (phi_ex * kex) / (kex**2 + we**2) ) ) * sin(theta)**2
print("Now calculate the Jacobian. The partial derivative matrix.\n")
print("Jacobian is m rows with function derivatives and n columns of parameters.")
d_f_d_R1 = diff(f, R1)
d_f_d_theta = diff(f, theta)
d_f_d_R1rho_p = diff(f, R1rho_p)
d_f_d_phi_ex = diff(f, phi_ex)
d_f_d_kex = diff(f, kex)
d_f_d_we = diff(f, we)
print("""Form the Jacobian matrix by:
------------------------------------------------------------------------------
from numpy import array, cos, sin, pi, transpose
R1 = 1.1
theta = pi / 4
R1rho_p = 10.
phi_ex = 1100.
kex = 2200.
we = 3300.
d_f_d_R1 = %s
d_f_d_theta = %s
d_f_d_R1rho_p = %s
d_f_d_phi_ex = %s
d_f_d_kex = %s
d_f_d_we = %s
jacobian_matrix = transpose(array( [d_f_d_R1 , d_f_d_theta, d_f_d_R1rho_p, d_f_d_phi_ex, d_f_d_kex, d_f_d_we] ) )
print jacobian_matrix
------------------------------------------------------------------------------
""" % (d_f_d_R1, d_f_d_theta, d_f_d_R1rho_p, d_f_d_phi_ex, d_f_d_kex, d_f_d_we) )
#### Method 2
# http://docs.sympy.org/0.7.2/modules/matrices/matrices.html
# The vectorial function.
X = Matrix([R1 * cos(theta)**2 + (R1rho_p + ( (phi_ex * kex) / (kex**2 + we**2) ) ) * sin(theta)**2])
# What to derive for.
Y = Matrix([R1, theta, R1rho_p, phi_ex, kex, we])
# Make the Jacobian
Jacobian = X.jacobian(Y)
jac_string = str(Jacobian)
jac_string_arr = jac_string.replace("Matrix", "array")
print("""Form the Jacobian matrix by:
------------------------------------------------------------------------------
from numpy import array, cos, sin, pi, transpose
R1 = 1.1
theta = pi / 4
R1rho_p = 10.
phi_ex = 1100.
kex = 2200.
we = 3300.
jacobian_matrix_2 = %s
print jacobian_matrix_2
------------------------------------------------------------------------------
""" % (jac_string_arr) )
output
output is
from numpy import array, cos, sin, pi, transpose
R1 = 1.1
theta = pi / 4
R1rho_p = 10.
phi_ex = 1100.
kex = 2200.
we = 3300.
d_f_d_R1 = cos(theta)**2
d_f_d_theta = -2*R1*sin(theta)*cos(theta) + 2*(R1rho_p + kex*phi_ex/(kex**2 + we**2))*sin(theta)*cos(theta)
d_f_d_R1rho_p = sin(theta)**2
d_f_d_phi_ex = kex*sin(theta)**2/(kex**2 + we**2)
d_f_d_kex = (-2*kex**2*phi_ex/(kex**2 + we**2)**2 + phi_ex/(kex**2 + we**2))*sin(theta)**2
d_f_d_we = -2*kex*phi_ex*we*sin(theta)**2/(kex**2 + we**2)**2
jacobian_matrix = transpose(array( [d_f_d_R1 , d_f_d_theta, d_f_d_R1rho_p, d_f_d_phi_ex, d_f_d_kex, d_f_d_we] ) )
print jacobian_matrix
R1 = 1.1
theta = pi / 4
R1rho_p = 10.
phi_ex = 1100.
kex = 2200.
we = 3300.
jacobian_matrix_2 = array([[cos(theta)**2, -2*R1*sin(theta)*cos(theta) + 2*(R1rho_p + kex*phi_ex/(kex**2 + we**2))*sin(theta)*cos(theta),
sin(theta)**2, kex*sin(theta)**2/(kex**2 + we**2), (-2*kex**2*phi_ex/(kex**2 + we**2)**2 + phi_ex/(kex**2 + we**2))*sin(theta)**2, -2*kex*phi_ex*we*sin(theta)**2/(kex**2 + we**2)**2]])
print jacobian_matrix_2
Hessian
sympy
from sympy import *
# In contrast to other Computer Algebra Systems, in SymPy you have to declare symbolic variables explicitly:
R1 = Symbol('R1')
theta = Symbol('theta')
R1rho_p = Symbol('R1rho_p')
phi_ex = Symbol('phi_ex')
kex = Symbol('kex')
we = Symbol('we')
# Define function
f = R1 * cos(theta)**2 + (R1rho_p + ( (phi_ex * kex) / (kex**2 + we**2) ) ) * sin(theta)**2
print("Now calculate the Hessian.\n")
# Define symbols to derive for.
syms = [R1, theta, R1rho_p, phi_ex, kex, we]
# Do the hessian.
hess = hessian(f, syms)
hess_string = str(hess)
hess_string_arr = hess_string.replace("Matrix", "array")
print("""Form the Hessian matrix by:
------------------------------------------------------------------------------
from numpy import array, cos, sin, pi
R1 = 1.1
theta = pi / 4
R1rho_p = 10.
phi_ex = 1100.
kex = 2200.
we = 3300.
hessian_matrix = %s
print hessian_matrix
------------------------------------------------------------------------------
""" % (hess_string_arr) )
output
output is
from numpy import array, cos, sin, pi
R1 = 1.1
theta = pi / 4
R1rho_p = 10.
phi_ex = 1100.
kex = 2200.
we = 3300.
hessian_matrix = array([[0, -2*sin(theta)*cos(theta), 0, 0, 0, 0],
[-2*sin(theta)*cos(theta), 2*R1*sin(theta)**2 - 2*R1*cos(theta)**2 - 2*(R1rho_p + kex*phi_ex/(kex**2 + we**2))*sin(theta)**2 + 2*(R1rho_p + kex*phi_ex/(kex**2 + we**2))*cos(theta)**2, 2*sin(theta)*cos(theta),
2*kex*sin(theta)*cos(theta)/(kex**2 + we**2), 2*(-2*kex**2*phi_ex/(kex**2 + we**2)**2 + phi_ex/(kex**2 + we**2))*sin(theta)*cos(theta), -4*kex*phi_ex*we*sin(theta)*cos(theta)/(kex**2 + we**2)**2],
[0, 2*sin(theta)*cos(theta), 0, 0, 0, 0], [0, 2*kex*sin(theta)*cos(theta)/(kex**2 + we**2), 0, 0, -2*kex**2*sin(theta)**2/(kex**2 + we**2)**2 + sin(theta)**2/(kex**2 + we**2), -2*kex*we*sin(theta)**2/(kex**2 + we**2)**2],
[0, 2*(-2*kex**2*phi_ex/(kex**2 + we**2)**2 + phi_ex/(kex**2 + we**2))*sin(theta)*cos(theta), 0, -2*kex**2*sin(theta)**2/(kex**2 + we**2)**2 + sin(theta)**2/(kex**2 + we**2),
(8*kex**3*phi_ex/(kex**2 + we**2)**3 - 6*kex*phi_ex/(kex**2 + we**2)**2)*sin(theta)**2, (8*kex**2*phi_ex*we/(kex**2 + we**2)**3 - 2*phi_ex*we/(kex**2 + we**2)**2)*sin(theta)**2],
[0, -4*kex*phi_ex*we*sin(theta)*cos(theta)/(kex**2 + we**2)**2, 0, -2*kex*we*sin(theta)**2/(kex**2 + we**2)**2,
(8*kex**2*phi_ex*we/(kex**2 + we**2)**3 - 2*phi_ex*we/(kex**2 + we**2)**2)*sin(theta)**2, 8*kex*phi_ex*we**2*sin(theta)**2/(kex**2 + we**2)**3 - 2*kex*phi_ex*sin(theta)**2/(kex**2 + we**2)**2]])
print hessian_matrix
