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=QE2012_AC-3_ECE580-2=
 
=QE2012_AC-3_ECE580-2=
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Put your content here . . .
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'''2. (20pts) Employ the DFP method to construct a set of Q-conjugate directions using the function'''
 +
 
 +
      <math>f = \frac{1}{2}x^TQx - x^Tb+c </math>
 +
        <math>  =\frac{1}{2}x^T
 +
\begin{bmatrix}
 +
  1 & 1 \\
 +
  1 & 2
 +
\end{bmatrix}x-x^T\begin{bmatrix}
 +
  2  \\
 +
  1
 +
\end{bmatrix} + 3.</math>
 +
 
 +
Where <span class="texhtml">''x''<sup>(0)</sup></span> is arbitrary.  
 +
 
 +
<br>
 +
 
 +
Solution:
 +
 
 +
      <math>f = \frac{1}{2}x^TQx - x^Tb+c </math>
 +
      <span class="texhtml">''U''''s''''e''</span> <span class="texhtml">''i''''n''''i''''t''''i''''a''''l''</span>  <span class="texhtml">''p''''o''''i''''n''''t''</span> <span class="texhtml">''x''<sup>(0)</sup> = [0,0]<sup>''T''</sup>''a''''n''''d''</span> <span class="texhtml">''H''<sub>0</sub> = ''I''<sub>2</sub></span>
 +
      <span class="texhtml">''I''''n''</span> <span class="texhtml">''t''''h''''i''''s''</span> <span class="texhtml">''c''''a''''s''''e'',</span>
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      <math>g^{(k)} = \begin{bmatrix}
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  1 & 1 \\
 +
  1 & 2
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\end{bmatrix} x^{(k)} - \begin{bmatrix}
 +
  2  \\
 +
  1
 +
\end{bmatrix}</math>
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 +
      <span class="texhtml">''H''''e''''n''''c''''e'',</span>
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      <math>g^{(0)} = \begin{bmatrix}
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  -2  \\
 +
  -1
 +
\end{bmatrix},</math>  <math>d^{(0)} = -H_0g^{(0)} =- \begin{bmatrix}
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  1 & 0 \\
 +
  0 & 1
 +
\end{bmatrix}\begin{bmatrix}
 +
  -2  \\
 +
  -1
 +
\end{bmatrix} = \begin{bmatrix}
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  2  \\
 +
  1
 +
\end{bmatrix}</math>
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 +
<br>
 +
 
 +
      <span class="texhtml">''B''''e''''c''''a''''u''''s''''e''</span> <span class="texhtml">''f''</span> <span class="texhtml">''i''''s''</span> <span class="texhtml">''a''</span> <span class="texhtml">''q''''u''''a''''d''''r''''a''''t''''i''''c''</span> <span class="texhtml">''f''''u''''n''''c''''t''''i''''o''''n'',</span>
 +
 
 +
      <math>\alpha_0 = argminf(x^{(0)} + \alpha d^{(0)}) = - \frac{g^{(0)^T}d^{(0)}}{d^{(0)^T}Qd^{(0)}} = - \frac{\begin{bmatrix}
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  -2 & -1
 +
\end{bmatrix}\begin{bmatrix}
 +
  2  \\
 +
  1
 +
\end{bmatrix}}{\begin{bmatrix}
 +
  2 & 1\end{bmatrix}\begin{bmatrix}
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  1 & 1 \\
 +
  1 & 2
 +
\end{bmatrix}\begin{bmatrix}
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  2  \\
 +
  1
 +
\end{bmatrix}} = \frac{1}{2}</math>
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 +
      <math>x^{(1)} = x^{(0)} + \alpha d^{(0)} = \frac{1}{2} \begin{bmatrix}
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  2  \\
 +
  1
 +
\end{bmatrix} = \begin{bmatrix}
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  1  \\
 +
  \frac{1}{2}
 +
\end{bmatrix}</math>
 +
 
 +
      <math>\Delta x^{(0)} = x^{(1)}- x^{(0)} = \begin{bmatrix}
 +
  1  \\
 +
  \frac{1}{2}
 +
\end{bmatrix}</math>
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      <math>g^{(1)} =\begin{bmatrix}
 +
  1 & 1 \\
 +
  1 & 2
 +
\end{bmatrix} x^{(1)} - \begin{bmatrix}
 +
  2  \\
 +
  1
 +
\end{bmatrix}= \begin{bmatrix}
 +
  -\frac{1}{2}  \\
 +
  1
 +
\end{bmatrix}</math>
 +
 
 +
      <math>\Delta g^{(0)} = g^{(1)} - g^{(0)} = \begin{bmatrix}
 +
  -\frac{3}{2}  \\
 +
  2
 +
\end{bmatrix} </math>
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 +
<br>
 +
 
 +
      <span class="texhtml">''O''''b''''s''''e''''r''''v''''e''</span> <span class="texhtml">''t''''h''''a''''t'':</span>
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      <math>\Delta x^{(0)} \Delta x^{(0)^T} = \begin{bmatrix}
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  1  \\
 +
  \frac{1}{2}
 +
\end{bmatrix} \begin{bmatrix}
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  1  & \frac{1}{2}
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\end{bmatrix} = \begin{bmatrix}
 +
  1 & \frac{1}{2}  \\
 +
  \frac{1}{2}  & \frac{1}{4}
 +
\end{bmatrix} </math>
 +
      <math> \Delta x^{(0)^T} \Delta g^{(0)} = \begin{bmatrix}
 +
  1  & \frac{1}{2}
 +
\end{bmatrix}\begin{bmatrix}
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  \frac{3}{2}  \\
 +
  2
 +
\end{bmatrix}  = \frac{5}{2}</math>
 +
      <math>H_0 \Delta g^{(0)} = \begin{bmatrix}
 +
  1 & 0 \\
 +
  0 & 1
 +
\end{bmatrix} \begin{bmatrix}
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  \frac{3}{2}  \\
 +
  2
 +
\end{bmatrix} = \begin{bmatrix}
 +
  \frac{3}{2}  \\
 +
  2
 +
\end{bmatrix},</math> <math>(H_0 \Delta g^{(0)})(H_0 \Delta g^{(0)})^T = \begin{bmatrix}
 +
  \frac{9}{4}  & 3 \\
 +
  3 & 4
 +
\end{bmatrix}</math>
 +
      <math>\Delta g^{(0)^T}H_0 \Delta g^{(0)} = \begin{bmatrix}
 +
  \frac{3}{2}  & 2
 +
\end{bmatrix} \begin{bmatrix}
 +
  1 & 0 \\
 +
  0 & 1
 +
\end{bmatrix} \begin{bmatrix}
 +
  \frac{3}{2}  \\ 2
 +
\end{bmatrix} = \frac{25}{4}</math>
 +
      <span class="texhtml">''U''''s''''i''''n''''g''</span> <span class="texhtml">''t''''h''''e''</span> <span class="texhtml">''a''''b''''o''''v''''e'',</span> <span class="texhtml">''n''''o''''w''</span> <span class="texhtml">''w''''e''</span> <span class="texhtml">''c''''o''''m''''p''''u''''t''''e''</span>
 +
      <math>H_1 = H_0 + \frac{\Delta x^{(0)} \Delta x^{(0)^T}}{\Delta x^{(0)^T} \Delta g^{(0)}} - \frac{(H_0 \Delta g^{(0)})(H_0 \Delta g^{(0)})^T}{\Delta g^{(0)^T}H_0 \Delta g^{(0)} }  = \begin{bmatrix}
 +
  1 & 0 \\
 +
  0 & 1
 +
\end{bmatrix} + \begin{bmatrix}
 +
  \frac{2}{5} & \frac{1}{5} \\
 +
  \frac{1}{5} & \frac{1}{10}
 +
\end{bmatrix} - \frac{25}{4}\begin{bmatrix}
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  \frac{9}{4} & 3 \\
 +
  3 & 4
 +
\end{bmatrix} = \begin{bmatrix}
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  \frac{26}{25} & -\frac{7}{25} \\
 +
  -\frac{7}{25} & \frac{23}{50}
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\end{bmatrix}</math>
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 +
      <span class="texhtml">''T''''h''''e''''n''</span> <span class="texhtml">''w''''e''</span> <span class="texhtml">''h''''a''''v''''e'',</span>
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      <math>d^{(1)} = -H_1 g^{(0)} = - \begin{bmatrix}
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  \frac{26}{25} & -\frac{7}{25} \\
 +
  -\frac{7}{25} & \frac{23}{50}
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\end{bmatrix} \begin{bmatrix}
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  -\frac{1}{2}  \\
 +
  1
 +
\end{bmatrix} = \begin{bmatrix}
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  \frac{4}{5}  \\
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  -\frac{3}{5}
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\end{bmatrix}</math>
 +
 
 +
      <math>\alpha_1 = argminf(x^{(1)} + \alpha d^{(1)}) = - \frac{g^{(1)^T}d^{(1)}}{d^{(1)^T}Qd^{(1)}} = - \frac{\begin{bmatrix}
 +
  -2 & 1
 +
\end{bmatrix}\begin{bmatrix}
 +
  \frac{4}{5}  \\
 +
  -\frac{3}{5}
 +
\end{bmatrix}}{\begin{bmatrix}
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  \frac{4}{5} & -\frac{3}{5}\end{bmatrix}\begin{bmatrix}
 +
  1 & 1 \\
 +
  1 & 2
 +
\end{bmatrix}\begin{bmatrix}
 +
  \frac{4}{5}  \\
 +
  -\frac{3}{5}
 +
\end{bmatrix}} = \frac{5}{2}</math>
 +
 
 +
      <math>x^{(2)} = x^{(1)} + \alpha_1 d^{(1)} = \begin{bmatrix}
 +
  1  \\
 +
  \frac{1}{2}
 +
\end{bmatrix} + \frac{5}{2}\begin{bmatrix}
 +
  \frac{4}{5}  \\
 +
  -\frac{3}{5}
 +
\end{bmatrix} = \begin{bmatrix}
 +
  3  \\
 +
  -1
 +
\end{bmatrix} </math>
 +
 
 +
      <math>\Delta x^{(1)} = x^{(2)} - x^{(1)} = \begin{bmatrix}
 +
  2  \\
 +
  -\frac{3}{2}
 +
\end{bmatrix}</math>
 +
      <math>g^{(2)} = \begin{bmatrix}
 +
  1 & 1 \\
 +
  1 & 2
 +
\end{bmatrix} x^{(0)} - \begin{bmatrix}
 +
  2  \\
 +
  1
 +
\end{bmatrix} = \begin{bmatrix}
 +
  0  \\
 +
  0
 +
\end{bmatrix}</math>
 +
 
 +
      <span class="texhtml">''N''''o''''t''''e''</span> <span class="texhtml">''t''''h''''a''''t''</span> <span class="texhtml">''w''''e''</span> <span class="texhtml">''h''''a''''v''''e''</span> <math>d^{(0)^T}Qd^{(0)} = 0;</math>
 +
      <span class="texhtml">''t''''h''''a''''t''</span> <span class="texhtml">''i''''s'',</span> <math>d^{(0)} = \begin{bmatrix}
 +
  2  \\
 +
  1
 +
\end{bmatrix}</math> <span class="texhtml">''a''''n''''d''</span> <math>d^{(1)}  = \begin{bmatrix}
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  \frac{4}{5}  \\
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  -\frac{3}{5}
 +
\end{bmatrix}</math> <span class="texhtml">''a''''r''''e''</span> <span class="texhtml">''Q'' − ''c''''o''''n''''j''''u''''g''''a''''t''''e''</span> <span class="texhtml">''d''''i''''r''''e''''c''''t''''i''''o''''n''''s''.</span>
 +
 
 +
<br>
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Revision as of 19:01, 25 January 2013

QE2012_AC-3_ECE580-2

2. (20pts) Employ the DFP method to construct a set of Q-conjugate directions using the function 

      $ f = \frac{1}{2}x^TQx - x^Tb+c  $
       $   =\frac{1}{2}x^T  \begin{bmatrix}   1 & 1 \\   1 & 2  \end{bmatrix}x-x^T\begin{bmatrix}   2  \\   1  \end{bmatrix} + 3. $

Where x(0) is arbitrary.


Solution: 

      $ f = \frac{1}{2}x^TQx - x^Tb+c  $
     U's'e i'n'i't'i'a'l  p'o'i'n't x(0) = [0,0]Ta'n'd H0 = I2
     I'n t'h'i's c'a's'e,
     $ g^{(k)} = \begin{bmatrix}   1 & 1 \\   1 & 2  \end{bmatrix} x^{(k)} - \begin{bmatrix}   2  \\   1  \end{bmatrix} $

      H'e'n'c'e, 
     $ g^{(0)} = \begin{bmatrix}   -2  \\   -1  \end{bmatrix}, $  $ d^{(0)} = -H_0g^{(0)} =- \begin{bmatrix}   1 & 0 \\   0 & 1  \end{bmatrix}\begin{bmatrix}   -2  \\   -1  \end{bmatrix} = \begin{bmatrix}   2  \\   1  \end{bmatrix} $



      B'e'c'a'u's'e f i's a q'u'a'd'r'a't'i'c f'u'n'c't'i'o'n,

      $ \alpha_0 = argminf(x^{(0)} + \alpha d^{(0)}) = - \frac{g^{(0)^T}d^{(0)}}{d^{(0)^T}Qd^{(0)}} = - \frac{\begin{bmatrix}   -2 & -1   \end{bmatrix}\begin{bmatrix}   2  \\   1  \end{bmatrix}}{\begin{bmatrix}   2 & 1\end{bmatrix}\begin{bmatrix}   1 & 1 \\   1 & 2  \end{bmatrix}\begin{bmatrix}   2  \\   1  \end{bmatrix}} = \frac{1}{2} $

      $ x^{(1)} = x^{(0)} + \alpha d^{(0)} = \frac{1}{2} \begin{bmatrix}   2  \\   1  \end{bmatrix} = \begin{bmatrix}   1  \\   \frac{1}{2}  \end{bmatrix} $

      $ \Delta x^{(0)} = x^{(1)}- x^{(0)} = \begin{bmatrix}   1  \\   \frac{1}{2}  \end{bmatrix} $
     $ g^{(1)} =\begin{bmatrix}   1 & 1 \\   1 & 2  \end{bmatrix} x^{(1)} - \begin{bmatrix}   2  \\   1  \end{bmatrix}= \begin{bmatrix}   -\frac{1}{2}  \\   1  \end{bmatrix} $

      $ \Delta g^{(0)} = g^{(1)} - g^{(0)} = \begin{bmatrix}   -\frac{3}{2}  \\   2  \end{bmatrix}  $



      O'b's'e'r'v'e t'h'a't:
     $ \Delta x^{(0)} \Delta x^{(0)^T} = \begin{bmatrix}   1  \\   \frac{1}{2}   \end{bmatrix} \begin{bmatrix}   1  & \frac{1}{2}   \end{bmatrix} = \begin{bmatrix}   1 & \frac{1}{2}  \\   \frac{1}{2}  & \frac{1}{4}   \end{bmatrix}  $
     $  \Delta x^{(0)^T} \Delta g^{(0)} = \begin{bmatrix}   1  & \frac{1}{2}   \end{bmatrix}\begin{bmatrix}   \frac{3}{2}   \\   2   \end{bmatrix}  = \frac{5}{2} $
     $ H_0 \Delta g^{(0)} = \begin{bmatrix}   1 & 0 \\   0 & 1  \end{bmatrix} \begin{bmatrix}   \frac{3}{2}   \\   2   \end{bmatrix} = \begin{bmatrix}   \frac{3}{2}   \\   2   \end{bmatrix}, $ $ (H_0 \Delta g^{(0)})(H_0 \Delta g^{(0)})^T = \begin{bmatrix}   \frac{9}{4}  & 3 \\   3 & 4  \end{bmatrix} $
     $ \Delta g^{(0)^T}H_0 \Delta g^{(0)} = \begin{bmatrix}   \frac{3}{2}  & 2   \end{bmatrix} \begin{bmatrix}   1 & 0 \\   0 & 1  \end{bmatrix} \begin{bmatrix}   \frac{3}{2}  \\ 2   \end{bmatrix} = \frac{25}{4} $
     U's'i'n'g t'h'e a'b'o'v'e, n'o'w w'e c'o'm'p'u't'e
     $ H_1 = H_0 + \frac{\Delta x^{(0)} \Delta x^{(0)^T}}{\Delta x^{(0)^T} \Delta g^{(0)}} - \frac{(H_0 \Delta g^{(0)})(H_0 \Delta g^{(0)})^T}{\Delta g^{(0)^T}H_0 \Delta g^{(0)} }  = \begin{bmatrix}   1 & 0 \\   0 & 1  \end{bmatrix} + \begin{bmatrix}   \frac{2}{5} & \frac{1}{5} \\   \frac{1}{5} & \frac{1}{10}  \end{bmatrix} - \frac{25}{4}\begin{bmatrix}   \frac{9}{4} & 3 \\   3 & 4  \end{bmatrix} = \begin{bmatrix}   \frac{26}{25} & -\frac{7}{25} \\   -\frac{7}{25} & \frac{23}{50}  \end{bmatrix} $

      T'h'e'n w'e h'a'v'e,
     $ d^{(1)} = -H_1 g^{(0)} = - \begin{bmatrix}   \frac{26}{25} & -\frac{7}{25} \\   -\frac{7}{25} & \frac{23}{50}  \end{bmatrix} \begin{bmatrix}   -\frac{1}{2}  \\   1  \end{bmatrix} = \begin{bmatrix}   \frac{4}{5}  \\   -\frac{3}{5}  \end{bmatrix} $

      $ \alpha_1 = argminf(x^{(1)} + \alpha d^{(1)}) = - \frac{g^{(1)^T}d^{(1)}}{d^{(1)^T}Qd^{(1)}} = - \frac{\begin{bmatrix}   -2 & 1   \end{bmatrix}\begin{bmatrix}   \frac{4}{5}  \\   -\frac{3}{5}  \end{bmatrix}}{\begin{bmatrix}   \frac{4}{5} & -\frac{3}{5}\end{bmatrix}\begin{bmatrix}   1 & 1 \\   1 & 2  \end{bmatrix}\begin{bmatrix}   \frac{4}{5}  \\   -\frac{3}{5}  \end{bmatrix}} = \frac{5}{2} $

      $ x^{(2)} = x^{(1)} + \alpha_1 d^{(1)} = \begin{bmatrix}   1  \\   \frac{1}{2}  \end{bmatrix} + \frac{5}{2}\begin{bmatrix}   \frac{4}{5}  \\   -\frac{3}{5}  \end{bmatrix} = \begin{bmatrix}   3  \\   -1  \end{bmatrix}  $

      $ \Delta x^{(1)} = x^{(2)} - x^{(1)} = \begin{bmatrix}   2  \\   -\frac{3}{2}  \end{bmatrix} $
     $ g^{(2)} = \begin{bmatrix}   1 & 1 \\   1 & 2  \end{bmatrix} x^{(0)} - \begin{bmatrix}   2  \\   1  \end{bmatrix} = \begin{bmatrix}   0  \\   0  \end{bmatrix} $

      N'o't'e t'h'a't w'e h'a'v'e $ d^{(0)^T}Qd^{(0)} = 0; $
     t'h'a't i's, $ d^{(0)} = \begin{bmatrix}   2  \\   1  \end{bmatrix} $ a'n'd $ d^{(1)}  = \begin{bmatrix}   \frac{4}{5}  \\   -\frac{3}{5}  \end{bmatrix} $ a'r'e Q − c'o'n'j'u'g'a't'e d'i'r'e'c't'i'o'n's.




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