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Re: Numerically computing partial derivatives

  • To: mathgroup at smc.vnet.net
  • Subject: [mg47962] Re: Numerically computing partial derivatives
  • From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
  • Date: Tue, 4 May 2004 01:08:44 -0400 (EDT)
  • References: <c72d7k$jk$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

I think you should be able to make a lot of progress analytically. I attach
an extract from one of my papers in which I compute the first derivative of
a quantity that is closely related to yours. The main tricks to use are log
det = trace log, the Baker-Hausdorff identity for expanding logs of products
of matrices, trace(commutator)=0, etc. I presume these sorts of tricks can
be used to compute the Hessian as well.

Select from (******** to *********) and paste into Mathematica.

Steve Luttrell

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  StyleBox["Differentiating",
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  " log det"
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  "We differentiate the logarithm of the determinant of a \
matrix-valued quantity. We use this in order to differentiate ",
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          {\(Step\ 1\), \(\[Delta]\ log[
                det(\[Pi]\ \[Sigma]\^\[Prime])]\), \(\(=\)\(\(-log[
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\[Delta]\[Sigma]\^\[Prime]\)\^\(-1\)))]\) +
                log[det(\[Pi]\ \(\[Sigma]\^\[Prime]\)\^\(-1\))]\)\)},
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                        1 + \(\[Sigma]\^\[Prime]\) \(\[Delta]\[Sigma]\
\^\[Prime]\)\^\(-1\)))]\) +
                tr[log(\(\[Sigma]\^\[Prime]\)\^\(-1\))]\)\)},
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\[Prime]\)\^\(-1\))]\)\)},
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We justify the various stages of this manipulation as follows:\
\>", "Text"],

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  "Step 1. Matrix invert ",
  Cell[BoxData[
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  ", which introduces a minus sign outside the logarithm function. In \
order to calculate the derivative, write the difference that results \
from changing ",
  Cell[BoxData[
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  " infinitesimally."
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  "Step 2. Use the identity ",
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      \(TraditionalForm\`log[det(X)] = tr[log(X)]\)]],
  "."
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  "Step 3. Use the identity ",
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      \(TraditionalForm\`log(X\ Y) =
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              Hausdorff\ identity)\)\)]],
  " to obtain ",
  Cell[BoxData[
      \(TraditionalForm\`tr[log(X\ Y)] = tr[log(X)] + tr[log(Y)]\)]],
  " which causes a pair of terms to cancel, leaving only the \
infinitesimal part. Note that the trace of any commutator is zero."
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  "Step 4. Expand the logarithm using ",
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