Re: Numerically computing partial derivatives

• To: mathgroup at smc.vnet.net
• Subject: [mg47989] Re: Numerically computing partial derivatives
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Tue, 4 May 2004 07:03:23 -0400 (EDT)
• Organization: The University of Western Australia
• References: <c72d7k\$jk\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <c72d7k\$jk\$1 at smc.vnet.net>,
Mark Coleman <mark at markscoleman.com> wrote:

> I am working with a maximum likelihood problem in econometrics. With
> some effort I can get Mathematica v5.0 to maximize the function. In order to
> derive standard errors of the estimates, however, I need to calculate
> the Hessian of the function at the optimal solution. This requires, of
> course,  calculating the set of second derivatives of the function. Due
> to the nature of the function, however, neither the built-in D or ND
> operator seem to work (note: The function contains a term of the form
> Log[Det[I-rho*W]], where I is the nxn Identity matrix, rho is a real,
> and W is a non-symmetric (sparse) matrix of reals, or order n. In
> practice, n > 1000 at times. As a result, symbolic differentiation is
> not feasible. In addition, when I use ND [], I get nonsensical answers.

I assume you realize that

Log[Det[I-rho W]] == Sum[Log[1-rho lambda[i]],{i,n}]

where lambda[i] is the i-th eigenvalue of W. Can't you use this to
achieve symbolic differentiation?

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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