/**
* 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).
*/
function 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>";
}
/**
* det
* Calculate determinant
* @return float Determinant
*/
function det()
{
$L = new LUDecomposition($this);
return $L->det();
}
function fitcircle($X, $Y)
{
$points = sizeof($X);
if ($points < 3) {
return array("cx" => 0, "cy" => 0, "r" => 0, "err" => 100);
}
// Mean of points
$meanX = 0;
$meanY = 0;
for ($i = 0; $i < sizeof($X); $i++) {
$meanX += $X[$i];
$meanY += $Y[$i];
}
$meanX /= $points;
$meanY /= $points;
// Auxiliary means
$mXx = 0;
$mXy = 0;
$mYy = 0;
$mRx = 0;
$mRy = 0;
for ($i = 0; $i < sizeof($X); $i++) {
$x = $X[$i];
$y = $Y[$i];
$r = $x * $x + $y * $y;
$mXx += $x * ($x - $meanX);
$mXy += $x * ($y - $meanY);
$mYy += $y * ($y - $meanY);
$mRx += $r * ($x - $meanX);
$mRy += $r * ($y - $meanY);
}
$mXx /= $points;
$mXy /= $points;
$mYy /= $points;
$mRx /= $points;
$mRy /= $points;
// Assemble matrix
$A[0][0] = $mXx;
$A[0][1] = 2.0 * $mXy;
$A[1][0] = 0.0;
$A[1][1] = $mYy;
$matrixA = new Matrix($A);
// A=A+transpose(A)
$matrixA = $matrixA->plus($matrixA->transpose());
// Assemble B vector
$b[0][0] = $mRx;
$b[1][0] = $mRy;
$matrixB = new Matrix($b);
// Calculate center Solve A*x=b for x,then center=x
$LU = new LUDecomposition($matrixA);
if ($LU->isNonsingular()) {
$C = $matrixA->solve($matrixB);
$cx = $C->get(0, 0);
$cy = $C->get(1, 0);
// Calculate radius
$meanR2 = 0;
$maxR2 = 0;
$minR2 = 10000;
for ($i = 0; $i < sizeof($X); $i++) {
$x = $X[$i];
$y = $Y[$i];
$r2 = ($x - $cx) * ($x - $cx) + ($y - $cy) * ($y - $cy);
if ($r2 > $maxR2) {
$maxR2 = $r2;
}
if ($r2 < $minR2) {
$minR2 = $r2;
}
$meanR2 += $r2;
}
$meanR2 /= $points;
$radius = sqrt($meanR2);
// Calculate error
$err = sqrt($maxR2) / sqrt($minR2);
// Return
return array("cx" => $cx, "cy" => $cy, "r" => $radius, "err" => $err);
} else {
return array("cx" => 0, "cy" => 0, "r" => 0, "err" => 100);
}
}
