DPL94 derivatives - Revision history

Bugman: Added an introductory sentence and see also section. This is to move this page out of the dead-end page list (http://wiki.nmr-relax.com/Special:DeadendPages).

2014-09-13T08:16:03Z

Added an introductory sentence and see also section. This is to move this page out of the dead-end page list (http://wiki.nmr-relax.com/Special:DeadendPages).

Troels Emtekær Linnet: /* output */

2014-09-01T09:44:16Z

output

Troels Emtekær Linnet at 09:43, 1 September 2014

2014-09-01T09:43:02Z

Troels Emtekær Linnet at 09:42, 1 September 2014

2014-09-01T09:42:31Z

Troels Emtekær Linnet: /* output */

2014-09-01T09:32:12Z

output

Troels Emtekær Linnet at 09:31, 1 September 2014

2014-09-01T09:31:50Z

Troels Emtekær Linnet: /* sympy */

2014-09-01T09:30:51Z

sympy

Troels Emtekær Linnet: /* sympy */

2014-09-01T09:30:28Z

sympy

Troels Emtekær Linnet: /* sympy */

2014-09-01T09:29:14Z

sympy

Troels Emtekær Linnet: /* Jacobian */

2014-09-01T09:06:16Z

Jacobian

@@ Line 1: / Line 1: @@
 == Equation ==
 <math>
@@ Line 189: / Line 191: @@
 </source>

@@ Line 177: / Line 177: @@
 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]])
+hessian_matrix = array([[0, -2*sin(theta)*cos(theta), 0, 0, 0, 0],
 print hessian_matrix
 </source>

← Older revision		Revision as of 09:43, 1 September 2014
Line 1:		Line 1:
		+	== 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 ===

← Older revision		Revision as of 09:42, 1 September 2014
Line 113:		Line 113:

	print jacobian_matrix_2		print jacobian_matrix_2
		+	</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) / (kex2 + we2) ) ) * 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, -2sin(theta)cos(theta), 0, 0, 0, 0], [-2sin(theta)cos(theta), 2R1sin(theta)*2 - 2R1cos(theta)2 - 2(R1rho_p + kexphi_ex/(kex2 + we2))sin(theta)*2 + 2(R1rho_p + kexphi_ex/(kex2 + we2))cos(theta)*2, 2sin(theta)cos(theta), 2kexsin(theta)cos(theta)/(kex2 + we2), 2(-2kex*2phi_ex/(kex2 + we2)2 + phi_ex/(kex2 + we*2))sin(theta)cos(theta), -4kexphi_exwesin(theta)cos(theta)/(kex2 + we2)*2], [0, 2sin(theta)cos(theta), 0, 0, 0, 0], [0, 2kexsin(theta)cos(theta)/(kex2 + we2), 0, 0, -2kex2sin(theta)2/(kex2 + we2)2 + sin(theta)2/(kex2 + we*2), -2kexwesin(theta)2/(kex2 + we2)2], [0, 2(-2kex*2phi_ex/(kex2 + we2)2 + phi_ex/(kex2 + we*2))sin(theta)cos(theta), 0, -2kex*2sin(theta)2/(kex2 + we2)2 + sin(theta)2/(kex2 + we*2), (8kex*3phi_ex/(kex2 + we2)*3 - 6kexphi_ex/(kex2 + we2)2)sin(theta)*2, (8kex*2phi_exwe/(kex2 + we2)3 - 2phi_exwe/(kex2 + we2)2)sin(theta)*2], [0, -4kexphi_exwesin(theta)cos(theta)/(kex2 + we2)*2, 0, -2kexwesin(theta)2/(kex2 + we2)2, (8kex2phi_exwe/(kex2 + we2)3 - 2phi_exwe/(kex2 + we2)2)sin(theta)*2, 8kexphi_exwe*2sin(theta)2/(kex2 + we2)3 - 2kexphi_exsin(theta)2/(kex2 + we2)*2]])
		+
		+	print hessian_matrix
	</source>		</source>

@@ Line 109: / Line 109: @@
 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]])
+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),
 print jacobian_matrix_2
 </source>

@@ Line 81: / Line 81: @@
 === output ===
-output
+output is
 <source lang="python">
 from numpy import array, cos, sin, pi, transpose
@@ Line 101: / Line 101: @@
 print jacobian_matrix
 </source>

@@ Line 2: / Line 2: @@
 === sympy ===
 <source lang="python">
-as
+from sympy import *
 </source>

@@ Line 1: / Line 1: @@
 == Jacobian ==
 <source lang="python">
 from sympy import *
@@ Line 26: / Line 27: @@
 print("""Form the Jacobian matrix by:
 ------------------------------------------------------------------------------
-from numpy import array, transpose
+from numpy import array, cos, sin, pi, transpose
-−
 d_f_d_R1 = %s
 d_f_d_theta = %s
@@ Line 35: / Line 43: @@
 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] ) )
 ------------------------------------------------------------------------------
 """ % (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) )
 </source>
+=== output ===
 output
 <source lang="python">
-from numpy import array, transpose
+from numpy import array, cos, sin, pi, transpose
-−
 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)
@@ Line 50: / Line 67: @@
 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] ) )
 </source>