Hey so I'm making a factoring program and I'm wondering if anyone can give me any ideas on an efficient way to find what two numbers multiple to a specified number, and also add to a specified number.
for example I may have
(a)(b) = 6
a + b = 5
So essentially i just need a way to find the a and b values. In this case they would be 2 and 3.
Can anyone give me any ideas on where to start? Negative numbers must also be considered for use.
There is no need to loop, just use simple math to solve this equation system:
a*b = i;
a+b = j;
a = j/b;
a = i-b;
j/b = i-b; so:
b + j/b + i = 0
b^2 + i*b + j = 0
From here, its a quadratic equation, and it's trivial to find b (just implement the quadratic equation formula) and from there get the value for a.
There you go:
function finder($add,$product)
{
$inside_root = $add*$add - 4*$product;
if($inside_root >=0)
{
$b = ($add + sqrt($inside_root))/2;
$a = $add - $b;
echo "$a+$b = $add and $a*$b=$product\n";
}else
{
echo "No real solution\n";
}
}
Real live action:
http://codepad.org/JBxMgHBd
Here is how I would do that:
$sum = 5;
$product = 6;
$found = FALSE;
for ($a = 1; $a < $sum; $a++) {
$b = $sum - $a;
if ($a * $b == $product) {
$found = TRUE;
break;
}
}
if ($found) {
echo "The answer is a = $a, b = $b.";
} else {
echo "There is no answer where a and b are both integers.";
}
Basically, start at $a = 1 and $b = $sum - $a, step through it one at a time since we know then that $a + $b == $sum is always true, and multiply $a and $b to see if they equal $product. If they do, that's the answer.
See it working
Whether that is the most efficient method is very much debatable.
With the multiplication, I recommend using the modulo operator (%) to determine which numbers divide evenly into the target number like:
$factors = array();
for($i = 0; $i < $target; $i++){
if($target % $i == 0){
$temp = array()
$a = $i;
$b = $target / $i;
$temp["a"] = $a;
$temp["b"] = $b;
$temp["index"] = $i;
array_push($factors, $temp);
}
}
This would leave you with an array of factors of the target number.
That's basically a set of 2 simultaneous equations:
x*y = a
X+y = b
(using the mathematical convention of x and y for the variables to solve and a and b for arbitrary constants).
But the solution involves a quadratic equation (because of the x*y), so depending on the actual values of a and b, there may not be a solution, or there may be multiple solutions.
Related
I've been practicing a lot of algorithms recently for an interview. I was wondering if there was another way to solve this problem. I wrote it in a way where I only increment it positively, because I know from basic math that two negatives multiplied by each other would result to a positive number, so I would just have to make the integer that would satisfy the condition to negative.
Is there a way to write this elegantly where you didn't have the knowledge of multiplying two negative numbers result to a positive?
<?php
# Z = {integers}
# B = {x:x, x is an element of Z, x^2 + 1 = 10}
$numNotFound = true;
$x = 0;
$b = [];
while ($numNotFound) {
if ($x*$x + 1 == 10) {
array_push($b, $x, $x*-1);
$numNotFound = false;
}
$x++;
}
echo json_encode($b); #[3, -3]
Updated
This solution does not use the fact that -1 * -1 = 1. It will output the first number found as the first element in the array. If x=-3 then [-3,3] or if x=3 [3,-3].
$numNotFound = TRUE;
$x = 0;
$b = [];
Do{
if ((pow($x, 2) + 1) === 10) {
array_push($b, $x, 0 - $x);
$numNotFound = FALSE;
}
$x++;
}while($numNotFound);
echo json_encode($b); //[3, -3]
I need to find the value of x where the variance of two results (which take x into account) is the closest to 0. The problem is, the only way to do this is to cycle through all possible values of x. The equation uses currency, so I have to check in increments of 1 cent.
This might make it easier:
$previous_var = null;
$high_amount = 50;
for ($i = 0.01; $i <= $high_amount; $i += 0.01) {
$val1 = find_out_1($i);
$val2 = find_out_2();
$var = variance($val1, $val2);
if ($previous_var == null) {
$previous_var = $var;
}
// If this variance is larger, it means the previous one was the closest to
// 0 as the variance has now started increasing
if ($var > $previous_var) {
$l_s -= 0.01;
break;
}
}
$optimal_monetary_value = $i;
I feel like there is a mathematical formula that would make the "cycling through every cent" more optimal? It works fine for small values, but if you start using 1000's as the $high_amount it takes quite a few seconds to calculate.
Based on the comment in your code, it sounds like you want something similar to bisection search, but a little bit different:
function calculate_variance($i) {
$val1 = find_out_1($i);
$val2 = find_out_2();
return variance($val1, $val2);
}
function search($lo, $loVar, $hi, $hiVar) {
// find the midpoint between the hi and lo values
$mid = round($lo + ($hi - $lo) / 2, 2);
if ($mid == $hi || $mid == $lo) {
// we have converged, so pick the better value and be done
return ($hiVar > $loVar) ? $lo : $hi;
}
$midVar = calculate_variance($mid);
if ($midVar >= $loVar) {
// the optimal point must be in the lower interval
return search($lo, $loVar, $mid, $midVar);
} elseif ($midVar >= $hiVar) {
// the optimal point must be in the higher interval
return search($mid, $midVar, $hi, $hiVar);
} else {
// we don't know where the optimal point is for sure, so check
// the lower interval first
$loBest = search($lo, $loVar, $mid, $midVar);
if ($loBest == $mid) {
// we can't be sure this is the best answer, so check the hi
// interval to be sure
return search($mid, $midVar, $hi, $hiVar);
} else {
// we know this is the best answer
return $loBest;
}
}
}
$optimal_monetary_value = search(0.01, calculate_variance(0.01), 50.0, calculate_variance(50.0));
This assumes that the variance is monotonically increasing when moving away from the optimal point. In other words, if the optimal value is O, then for all X < Y < O, calculate_variance(X) >= calculate_variance(Y) >= calculate_variance(O) (and the same with all > and < flipped). The comment in your code and the way have you have it written make it seem like this is true. If this isn't true, then you can't really do much better than what you have.
Be aware that this is not as good as bisection search. There are some pathological inputs that will make it take linear time instead of logarithmic time (e.g., if the variance is the same for all values). If you can improve the requirement that calculate_variance(X) >= calculate_variance(Y) >= calculate_variance(O) to be calculate_variance(X) > calculate_variance(Y) > calculate_variance(O), you can improve this to be logarithmic in all cases by checking to see how the variance for $mid compares the the variance for $mid + 0.01 and using that to decide which interval to check.
Also, you may want to be careful about doing math with currency. You probably either want to use integers (i.e., do all math in cents instead of dollars) or use exact precision numbers.
If you known nothing at all about the behavior of the objective function, there is no other way than trying all possible values.
On the opposite if you have a guarantee that the minimum is unique, the Golden section method will converge very quickly. This is a variant of the Fibonacci search, which is known to be optimal (require the minimum number of function evaluations).
Your function may have different properties which call for other algorithms.
Why not implementing binary search ?
<?php
$high_amount = 50;
// computed val2 is placed outside the loop
// no need te recalculate it each time
$val2 = find_out_2();
$previous_var = variance(find_out_1(0.01), $val2);
$start = 0;
$end = $high_amount * 100;
$closest_variance = NULL;
while ($start <= $end) {
$section = intval(($start + $end)/2);
$cursor = $section / 100;
$val1 = find_out_1($cursor);
$variance = variance($val1, $val2);
if ($variance <= $previous_var) {
$start = $section;
}
else {
$closest_variance = $cursor;
$end = $section;
}
}
if (!is_null($closest_variance)) {
$closest_variance -= 0.01;
}
I have the following PHP code in which I want to compare two decimal numbers. I have read in the PHP documentation that floating point numbers have limited precision.
$a = 0.0;
for ($i = 0; $i < 10; $i++) {
$a += 0.1;
}
var_dump($a);
echo gettype($a);
if ($a === 1.0) {
echo "IF";
} else {
echo "ELSE";
}
When I compare variable $a with 1.0, it always returns false, and the result will be 'ELSE'. My question is how I can get the code above working properly.
you can just do it this way:
//just a check if it is float, than round it to 1 decimal number and compare
if(is_float($a)){
echo 'not a float';
$a = round($a,1);
}
and output will be 'IF'
This question has been asked before, see Compare floats in PHP.
Basically you need to calculate the difference and see if it is small enough to be acceptable as "equal".
Try something like this:
$a = 0.0;
for ($i = 0; $i < 10; $i++) {
$a += 0.1;
$a=number_format($a,1);
//echo gettype($a);
//echo $a.'<br>';
if (floatval($a) === 1.0)
echo "IF";
else
echo "ELSE";
}
In order to make the comparison work I would truncate $a to one decimal place and format $a to a string with the required precision and compare it to "1.0".
To truncate $a I suggest reading this answer .
Or you can use $a as an integer. Them instead of incrementing by 0.1 use 1. And use 10 in the final comparison.
I've searched through a number of similar questions, but unfortunately I haven't been able to find an answer to this problem. I hope someone can point me in the right direction.
I need to come up with a PHP function which will produce a random number within a set range and mean. The range, in my case, will always be 1 to 100. The mean could be anything within the range.
For example...
r = f(x)
where...
r = the resulting random number
x = the mean
...running this function in a loop should produce random values where the average of the resulting values should be very close to x. (The more times we loop the closer we get to x)
Running the function in a loop, assuming x = 10, should produce a curve similar to this:
+
+ +
+ +
+ +
+ +
Where the curve starts at 1, peeks at 10, and ends at 100.
Unfortunately, I'm not well versed in statistics. Perhaps someone can help me word this problem correctly to find a solution?
interesting question. I'll sum it up:
We need a funcion f(x)
f returns an integer
if we run f a million times the average of the integer is x(or very close at least)
I am sure there are several approaches, but this uses the binomial distribution: http://en.wikipedia.org/wiki/Binomial_distribution
Here is the code:
function f($x){
$min = 0;
$max = 100;
$curve = 1.1;
$mean = $x;
$precision = 5; //higher is more precise but slower
$dist = array();
$lastval = $precision;
$belowsize = $mean-$min;
$abovesize = $max-$mean;
$belowfactor = pow(pow($curve,50),1/$belowsize);
$left = 0;
for($i = $min; $i< $mean; $i++){
$dist[$i] = round($lastval*$belowfactor);
$lastval = $lastval*$belowfactor;
$left += $dist[$i];
}
$dist[$mean] = round($lastval*$belowfactor);
$abovefactor = pow($left,1/$abovesize);
for($i = $mean+1; $i <= $max; $i++){
$dist[$i] = round($left-$left/$abovefactor);
$left = $left/$abovefactor;
}
$map = array();
foreach ($dist as $int => $quantity) {
for ($x = 0; $x < $quantity; $x++) {
$map[] = $int;
}
}
shuffle($map);
return current($map);
}
You can test it out like this(worked for me):
$results = array();
for($i = 0;$i<100;$i++){
$results[] = f(20);
}
$average = array_sum($results) / count($results);
echo $average;
It gives a distribution curve that looks like this:
I'm not sure if I got what you mean, even if I didn't this is still a pretty neat snippet:
<?php
function array_avg($array) { // Returns the average (mean) of the numbers in an array
return array_sum($array)/count($array);
}
function randomFromMean($x, $min = 1, $max = 100, $leniency = 3) {
/*
$x The number that you want to get close to
$min The minimum number in the range
$max Self-explanatory
$leniency How far off of $x can the result be
*/
$res = [mt_rand($min,$max)];
while (true) {
$res_avg = array_avg($res);
if ($res_avg >= ($x - $leniency) && $res_avg <= ($x + $leniency)) {
return $res;
break;
}
else if ($res_avg > $x && $res_avg < $max) {
array_push($res,mt_rand($min, $x));
}
else if ($res_avg > $min && $res_avg < $x) {
array_push($res, mt_rand($x,$max));
}
}
}
$res = randomFromMean(22); // This function returns an array of random numbers that have a mean close to the first param.
?>
If you then var_dump($res), You get something like this:
array (size=4)
0 => int 18
1 => int 54
2 => int 22
3 => int 4
EDIT: Using a low value for $leniency (like 1 or 2) will result in huge arrays, since testing, I recommend a leniency of around 3.
So I've read the two related questions for calculating a trend line for a graph, but I'm still lost.
I have an array of xy coordinates, and I want to come up with another array of xy coordinates (can be fewer coordinates) that represent a logarithmic trend line using PHP.
I'm passing these arrays to javascript to plot graphs on the client side.
Logarithmic Least Squares
Since we can convert a logarithmic function into a line by taking the log of the x values, we can perform a linear least squares curve fitting. In fact, the work has been done for us and a solution is presented at Math World.
In brief, we're given $X and $Y values that are from a distribution like y = a + b * log(x). The least squares method will give some values aFit and bFit that minimize the distance from the parametric curve to the data points given.
Here is an example implementation in PHP:
First I'll generate some random data with known underlying distribution given by $a and $b
// True parameter valaues
$a = 10;
$b = 5;
// Range of x values to generate
$x_min = 1;
$x_max = 10;
$nPoints = 50;
// Generate some random points on y = a * log(x) + b
$X = array();
$Y = array();
for($p = 0; $p < $nPoints; $p++){
$x = $p / $nPoints * ($x_max - $x_min) + $x_min;
$y = $a + $b * log($x);
$X[] = $x + rand(0, 200) / ($nPoints * $x_max);
$Y[] = $y + rand(0, 200) / ($nPoints * $x_max);
}
Now, here's how to use the equations given to estimate $a and $b.
// Now convert to log-scale for X
$logX = array_map('log', $X);
// Now estimate $a and $b using equations from Math World
$n = count($X);
$square = create_function('$x', 'return pow($x,2);');
$x_squared = array_sum(array_map($square, $logX));
$xy = array_sum(array_map(create_function('$x,$y', 'return $x*$y;'), $logX, $Y));
$bFit = ($n * $xy - array_sum($Y) * array_sum($logX)) /
($n * $x_squared - pow(array_sum($logX), 2));
$aFit = (array_sum($Y) - $bFit * array_sum($logX)) / $n;
You may then generate points for your Javascript as densely as you like:
$Yfit = array();
foreach($X as $x) {
$Yfit[] = $aFit + $bFit * log($x);
}
In this case, the code estimates bFit = 5.17 and aFit = 9.7, which is quite close for only 50 data points.
For the example data given in the comment below, a logarithmic function does not fit well.
The least squares solution is y = -514.734835478 + 2180.51562281 * log(x) which is essentially a line in this domain.
I would recommend using library: http://www.drque.net/Projects/PolynomialRegression/
Available by Composer: https://packagist.org/packages/dr-que/polynomial-regression.
In case anyone is having problems with the create_function, here is how I edited it. (Though I wasn't using logs, so I did take those out.)
I also reduced the number of calculations and added an R2. It seems to work so far.
function lsq(){
$X = array(1,2,3,4,5);
$Y = array(.3,.2,.7,.9,.8);
// Now estimate $a and $b using equations from Math World
$n = count($X);
$mult_elem = function($x,$y){ //anon function mult array elements
$output=$x*$y; //will be called on each element
return $output;
};
$sumX2 = array_sum(array_map($mult_elem, $X, $X));
$sumXY = array_sum(array_map($mult_elem, $X, $Y));
$sumY = array_sum($Y);
$sumX = array_sum($X);
$bFit = ($n * $sumXY - $sumY * $sumX) /
($n * $sumX2 - pow($sumX, 2));
$aFit = ($sumY - $bFit * $sumX) / $n;
echo ' intercept ',$aFit,' ';
echo ' slope ',$bFit,' ' ;
//r2
$sumY2 = array_sum(array_map($mult_elem, $Y, $Y));
$top=($n*$sumXY-$sumY*$sumX);
$bottom=($n*$sumX2-$sumX*$sumX)*($n*$sumY2-$sumY*$sumY);
$r2=pow($top/sqrt($bottom),2);
echo ' r2 ',$r2;
}