MagicSquareExample Class Reference

Example of use of Matrix Class, featuring magic squares. More...

Collaboration diagram for MagicSquareExample:

Public Member Functions
	magic ($n)
	Generate magic square test matrix.
	microtime_float ()
	Simple function to replicate PHP 5 behaviour.
	main ()
	Tests LU, QR, SVD and symmetric Eig decompositions.

Detailed Description

Example of use of Matrix Class, featuring magic squares.

Definition at line 11 of file MagicSquareExample.php.

Member Function Documentation

MagicSquareExample::magic

(

$n

)

Generate magic square test matrix.

Parameters

int	n dimension of matrix

Definition at line 17 of file MagicSquareExample.php.

References $n, and $t.

Referenced by main().

                     {

    // Odd order

    if (($n % 2) == 1) {
      $a = ($n+1)/2;
      $b = ($n+1);
      for ($j = 0; $j < $n; ++$j)
        for ($i = 0; $i < $n; ++$i)
          $M[$i][$j] = $n*(($i+$j+$a) % $n) + (($i+2*$j+$b) % $n) + 1;

    // Doubly Even Order

    } else if (($n % 4) == 0) {
      for ($j = 0; $j < $n; ++$j) {
        for ($i = 0; $i < $n; ++$i) {
          if ((($i+1)/2)%2 == (($j+1)/2)%2)
            $M[$i][$j] = $n*$n-$n*$i-$j;
          else
            $M[$i][$j] = $n*$i+$j+1;
        }
      }

    // Singly Even Order

    } else {

      $p = $n/2;
      $k = ($n-2)/4;
      $A = $this->magic($p);
      $M = array();
      for ($j = 0; $j < $p; ++$j) {
        for ($i = 0; $i < $p; ++$i) {
          $aij = $A->get($i,$j);
          $M[$i][$j]       = $aij;
          $M[$i][$j+$p]    = $aij + 2*$p*$p;
          $M[$i+$p][$j]    = $aij + 3*$p*$p;
          $M[$i+$p][$j+$p] = $aij + $p*$p;
        }
      }

      for ($i = 0; $i < $p; ++$i) {
        for ($j = 0; $j < $k; ++$j) {
          $t = $M[$i][$j];
          $M[$i][$j] = $M[$i+$p][$j];
          $M[$i+$p][$j] = $t;
        }
        for ($j = $n-$k+1; $j < $n; ++$j) {
          $t = $M[$i][$j];
          $M[$i][$j] = $M[$i+$p][$j];
          $M[$i+$p][$j] = $t;
        }
      }

      $t = $M[$k][0];  $M[$k][0]  = $M[$k+$p][0];  $M[$k+$p][0]  = $t;
      $t = $M[$k][$k]; $M[$k][$k] = $M[$k+$p][$k]; $M[$k+$p][$k] = $t;

    }

    return new Matrix($M);

  }

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MagicSquareExample::main

(

)

Tests LU, QR, SVD and symmetric Eig decompositions.

n = order of magic square. trace = diagonal sum, should be the magic sum, (n^3 + n)/2. max_eig = maximum eigenvalue of (A + A')/2, should equal trace. rank = linear algebraic rank, should equal n if n is odd, be less than n if n is even. cond = L_2 condition number, ratio of singular values. lu_res = test of LU factorization, norm1(L*U-A(p,:))/(n*eps). qr_res = test of QR factorization, norm1(Q*R-A)/(n*eps).

Definition at line 100 of file MagicSquareExample.php.

References $d, $n, $res, $t, magic(), main(), microtime_float(), and n.

Referenced by main().

                  {
    ?>
    <p>Test of Matrix Class, using magic squares.</p>
    <p>See MagicSquareExample.main() for an explanation.</p>
    <table border='1' cellspacing='0' cellpadding='4'>
      <tr>
        <th>n</th>
        <th>trace</th>
        <th>max_eig</th>
        <th>rank</th>
        <th>cond</th>
        <th>lu_res</th>
        <th>qr_res</th>
      </tr>
      <?php
      $start_time = $this->microtime_float();
      $eps = pow(2.0,-52.0);
      for ($n = 3; $n <= 6; ++$n) {
        echo "<tr>";

        echo "<td align='right'>$n</td>";

        $M = $this->magic($n);
        $t = (int) $M->trace();

        echo "<td align='right'>$t</td>";

        $O = $M->plus($M->transpose());
        $E = new EigenvalueDecomposition($O->times(0.5));
        $d = $E->getRealEigenvalues();

        echo "<td align='right'>".$d[$n-1]."</td>";

        $r = $M->rank();

        echo "<td align='right'>".$r."</td>";

        $c = $M->cond();

        if ($c < 1/$eps)
          echo "<td align='right'>".sprintf("%.3f",$c)."</td>";
        else
          echo "<td align='right'>Inf</td>";

        $LU = new LUDecomposition($M);
        $L = $LU->getL();
        $U = $LU->getU();
        $p = $LU->getPivot();
        // Java version: R = L.times(U).minus(M.getMatrix(p,0,n-1));
        $S = $L->times($U);
        $R = $S->minus($M->getMatrix($p,0,$n-1));
        $res = $R->norm1()/($n*$eps);

        echo "<td align='right'>".sprintf("%.3f",$res)."</td>";

        $QR = new QRDecomposition($M);
        $Q = $QR->getQ();
        $R = $QR->getR();
        $S = $Q->times($R);
        $R = $S->minus($M);
        $res = $R->norm1()/($n*$eps);

        echo "<td align='right'>".sprintf("%.3f",$res)."</td>";

        echo "</tr>";

     }
     echo "<table>";
     echo "<br />";

     $stop_time = $this->microtime_float();
     $etime = $stop_time - $start_time;

     echo "<p>Elapsed time is ". sprintf("%.4f",$etime) ." seconds.</p>";

  }

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MagicSquareExample::microtime_float

(

)

Simple function to replicate PHP 5 behaviour.

Definition at line 83 of file MagicSquareExample.php.

Referenced by main().

                             {
    list($usec, $sec) = explode(" ", microtime());
    return ((float)$usec + (float)$sec);
  }

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The documentation for this class was generated from the following file:

• Services/Excel/classes/PHPExcel/Shared/JAMA/examples/MagicSquareExample.php