Need help with update

This program is to be used at the end of each month to begin
the month with fresh values. When executed there are no
errors, the message “Records have been updated” is displayed,
but no records are changed?

$stat = mysql_connect("localhost","root","");
$stat = mysql_select_db("prerentdb");
$query = "SELECT name FROM payments Where apt='$apt'";
$stat = @mysql_fetch_assoc(mysql_query($query));
echo $stat["name"];
$result= mysql_query("select * from payments");
// if no payment or partial payment, add $10 to latechg field
// and amount not paid to prevbal field
if ($paidsum < $rentdue)
$latechg = $latechg + 10;
$owe = $rentdue - $paidsum;
$prevbal = $prevbal + $owe;
// if over-payment and late field not "L", subtract over-payment
// from prevbal field
if ($paidsum > $rentdue && $late |= 'L')
$sub = $paidsum - $rentdue;
$prevbal = $prevbal - $sub;
// refresh every record - give every record the below values
$amtpaid = 0;
$hudpay = 0;
$tentpay = 0;
$datepaid = ' ';
$late = ' ';
$paidsum = 0;
$comments = ' ';

$sql = "UPDATE payments SET
$amtpaid, $prevbal, $hudpay, $tentpay, $datepaid, $late, $comments, $paidsum";

mysql_query($sql) ;
echo "Records have been updated";

As far as I know, this is not the syntax of an Update statement.
It should be something like:

UPDATE payment
WHERE ....

Check the returned value from mysql_query before echoing that records have been updated.
If the returned value is FALSE, type mysql_error().

UPDATE payments SET $amtpaid, $prevbal, $hudpay, $tentpay, $datepaid, $late, $comments, $paidsum

Isn’t a valid query.