Function to create model-free spectral density hessians.
The spectral density hessians
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Data structure: self.data.d2jw
Dimension: 4D, (number of NMR frequencies, 5 spectral density frequencies, model-free parameters, model-free parameters)
Type: Numeric 4D matrix, Float64
Dependencies: None
Required by: self.data.d2ri
Formulae
~~~~~~~~
Parameter transformations
~~~~~~~~~~~~~~~~~~~~~~~~~
ae = c.te
af = c.tf
as = c.ts
therefore:
tm.ae
te' = ---------
ae + c.tm
tm.af
tf' = ---------
af + c.tm
tm.as
ts' = ---------
as + c.tm
Original: Model-free parameter - Model-free parameter
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2J(w)
------ = 0
dS2**2
d2J(w) 2 (ae + c.tm)**2 - (w.tm.ae)**2
------- = - - . c . tm**2 . ----------------------------------
dS2.dae 5 ((ae + c.tm)**2 + (w.tm.ae)**2)**2
d2J(w) 4 (ae + c.tm)**3 + 3.c.tm**3.ae.w**2.(ae + c.tm) - (1/ae).(w.tm.ae)**4
------ = - - . c . tm**2 . (1 - S2) . --------------------------------------------------------------------
dae**2 5 ((ae + c.tm)**2 + (w.tm.ae)**2)**3
Original: Other parameters
~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2J(w) d2J(w) d2J(w)
-------- = 0 , -------- = 0 , ------ = 0
dS2.dRex dS2.dcsa dS2.dr
d2J(w) d2J(w) d2J(w)
-------- = 0 , -------- = 0 , ------ = 0
dae.dRex dae.dcsa dae.dr
d2J(w) d2J(w) d2J(w)
------- = 0 , --------- = 0 , ------- = 0
dRex**2 dRex.dcsa dRex.dr
d2J(w) d2J(w)
------- = 0 , ------- = 0
dcsa**2 dcsa.dr
d2J(w)
------ = 0
dr**2
Extended: Model-free parameter - Model-free parameter
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2J(w)
------- = 0
dS2f**2
d2J(w) 2*tm / 1 (as + c.tm) . as \
--------- = ---- | ------------- - ----------------------------- |
dS2f.dS2s 5 \ 1 + (w.tm)**2 (as + c.tm)**2 + (w.tm.as)**2 /
d2J(w) 2 (af + c.tm)**2 - (w.tm.af)**2
-------- = - - . c . tm**2 . ----------------------------------
dS2f.daf 5 ((af + c.tm)**2 + (w.tm.af)**2)**2
d2J(w) 2 (as + c.tm)**2 - (w.tm.as)**2
-------- = - . c . tm**2 . (1 - S2s) . ----------------------------------
dS2f.das 5 ((as + c.tm)**2 + (w.tm.as)**2)**2
d2J(w) d2J(w)
------- = 0 , ------------- = 0
dS2s**2 dS2s.daf
d2J(w) 2 (as + c.tm)**2 - (w.tm.as)**2
-------- = - - . c . tm**2 . S2f . ----------------------------------
dS2s.das 5 ((as + c.tm)**2 + (w.tm.as)**2)**2
d2J(w) 4 (af + c.tm)**3 + 3.c.tm**3.af.w**2.(af + c.tm) - (1/af).(w.tm.af)**4
------ = - - . c . tm**2 . (1 - S2f) . --------------------------------------------------------------------
daf**2 5 ((af + c.tm)**2 + (w.tm.af)**2)**3
d2J(w)
------- = 0
daf.das
d2J(w) 4 (as + c.tm)**3 + 3.c.tm**3.as.w**2.(as + c.tm) - (1/as).(w.tm.as)**4
------ = - - . c . tm**2 . S2f . (1 - S2s) . --------------------------------------------------------------------
das**2 5 ((as + c.tm)**2 + (w.tm.as)**2)**3
Extended: Other parameters
~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2J(w) d2J(w) d2J(w)
--------- = 0 , --------- = 0 , ------- = 0
dS2f.dRex dS2f.dcsa dS2f.dr
d2J(w) d2J(w) d2J(w)
--------- = 0 , --------- = 0 , ------- = 0
dS2s.dRex dS2s.dcsa dS2s.dr
d2J(w) d2J(w) d2J(w)
-------- = 0 , -------- = 0 , ------ = 0
daf.dRex daf.dcsa daf.dr
d2J(w) d2J(w) d2J(w)
-------- = 0 , -------- = 0 , ------ = 0
das.dRex das.dcsa das.dr
d2J(w) d2J(w) d2J(w)
------- = 0 , --------- = 0 , ------- = 0
dRex**2 dRex.dcsa dRex.dr
d2J(w) d2J(w)
------- = 0 , ------- = 0
dcsa**2 dcsa.dr
d2J(w)
------ = 0
dr**2
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