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MathGroup Archive 2004

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Re: A question about derivatives of quadratic products

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49114] Re: A question about derivatives of quadratic products
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 1 Jul 2004 05:26:32 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <cbu35t$5m1$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <cbu35t$5m1$1 at smc.vnet.net>, yangjq at soton.ac.uk (afh) wrote:

> I met a question about derivatives as following:
> D = Norm[w, 2]^2/2+sum(&#945;i(yi(w.x+b+sqrt(w.M.w))-1)) (which i=1 to n)
> w and x are two-dimension vectors, yi is a number either -1 or 1, b is
> a number, M is a 2*2 matrix, &#945;i is a lagrange coefficient.

A little hard to decode. I assume you are working with

  w = {w1, w2}; x = {x1, x2};  m = {{m11, m12}, {m21, m22}};

and, along with b, all are independent of i. Then your sum can be written
 
  d = 1/2 Norm[w, 2]^2 - Sum[lambda[i], {i, 1, n}] +    
  (b + w . x + Sqrt[w . m . w]) Sum[lambda[i] y[i] , {i, 1, n}]

and simplifies slightly for real w,
    
  d = Simplify[d, w \[Element] Reals]

> I want to get the value of w when D's derivative on w is equal to
> zero.

I assume that you are computing derivatives with respect to w1 and w2 
separately and then want to solve for w1 and w2? I.e.,

   Solve[{D[d,w1]==0,D[d,w2]==0}, w]

The solution of these equations is likely to be more complicated than 
you expect, and probably not of much use to you ...

> But many errors happen, and it seems mathematica does not know what it
> should do next. 

I think that _you_ need to tell Mathematica what to do ...

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)         
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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