Householder transformation I'm trying to make this Householder method run as quickly as possible. It runs fine now, but I'm still a bit new to java, is there anything you guys see to make this run faster?

public static ArrayList&lt;double[][]&gt; householder(double[][] A) {
    int m = A.length; // rows
    int n = A[0].length; // cols
    byte sign; 
    double sum, A0,tmp;
    ArrayList&lt;double[][]&gt; QR = new ArrayList&lt;&gt;();
    double[] v = new double[m];
    double[][] Q = identity(m);
    for(int k = 0; k &lt; n; k++) {
        sum = 0;
        sign = 1;
        A0 = A[k][k];
        if(A0 &lt; 0){
            sign = -1;
        }
        for(int i = k; i &lt; m; i++) {
            sum += A[i][k]*A[i][k];
        }
        tmp = sign*Math.sqrt(sum);
        v[k] = tmp + A0;          
        tmp = Math.sqrt(2*(sum + A0*tmp));
        v[k] /= tmp;
        for(int i = k+1; i &lt; m; i++) {
            v[i] = A[i][k]/tmp;
        }                        
        for(int j = 0; j &lt; n; j++) {
            sum = 0;
            for(int i = k; i &lt; m; i++) {
                sum += v[i]*A[i][j];
            }
            for(int i = k; i &lt; m; i++) {
                A[i][j] -= 2*v[i]*sum;
            }
        }            
        for(int j = 0; j &lt; m; j++) {
            sum = 0;
            for(int i = k; i &lt; m; i++) {
                sum += v[i]*Q[i][j];
            }
            for(int i = k; i &lt; m; i++) {
                Q[i][j] -= 2*v[i]*sum;
            }
        }                
    }
    QR.add(transpose(Q));
    QR.add(A);
    return QR;
}


I'm mostly concerned with the portion of the code that multiplies the vectors to get the Q and A matrices. 

Here are some ideas I had but wasn't sure would help much.


if there's a better way to return the values than arraylist, like if double[][][] would be faster. I have had issues with recasting double[][][] arrays to double[][] however.
changing all the ints to shorts, i was concerned with casting
I only need sign to be a bit 
should I write sum = 0.0?
should I initialize the for loop int i's and j's outside of the loop to save memory?
byte sign; 
double sum, A0,tmp;



Keep variables to the most inner scope where they can be defined.


sign = 1;
A0 = A[k][k];
if(A0 &lt; 0){
    sign = -1;
}



sign can be assigned to a ternary expression

sign = A0 &lt; 0 ? -1 : 1;



tmp = sign*Math.sqrt(sum);
v[k] = tmp + A0;          
tmp = Math.sqrt(2*(sum + A0*tmp));
v[k] /= tmp;



EDIT Oops screwed up, the maths on v[k] evaluation... too many times...

EDIT 2 changing tmp to have the right value for the inner loop.

I would reduce the calculation to this:

    double sqrSum = sign*Math.sqrt(sum);
    double tmp = Math.sqrt(2*(sum + A0*sqrSum));        
    v[k] = (sqrSum  + A0) / tmp;


This is trying to store the value in v[k] just once, instead of accessing it twice.


  if there's a better way to return the values than arraylist, like if
  double[][][] would be faster. I have had issues with recasting
  double[][][] arrays to double[][] however.


Well in the end you know that you are returning an Array with two entries, the first being transpose(Q), and the second being A. This is pretty simple to achieve both on the implementation and the caller as long the method is properly documented.

Return on implementation

return new double[][][]{
    transpose(Q),
    A
}


Use on caller

var result = householder(...);
var transpose = result[0]; //documentation says entry 0 has the transpose
var a = result[1]; //documnentation says entry 1 has A


Resulting code

public static double[][] householder(double[][] A) {
    int m = A.length; // rows
    int n = A[0].length; // cols
    double[] v = new double[m];
    double[][] Q = identity(m);
    for(int k = 0; k &lt; n; k++) {
        double sum = 0;
        byte sign = A0 &lt; 0 ? -1 : 1;
        double A0 = A[k][k];
        for(int i = k; i &lt; m; i++) {
            sum += A[i][k]*A[i][k];
        }
        double sqrSum = sign*Math.sqrt(sum);
        double tmp = Math.sqrt(2*(sum + A0*sqrSum));        
        v[k] = (sqrSum  + A0) / tmp;
        for(int i = k+1; i &lt; m; i++) {
            v[i] = A[i][k]/tmp;
        }                        
        for(int j = 0; j &lt; n; j++) {
            sum = 0;
            for(int i = k; i &lt; m; i++) {
                sum += v[i]*A[i][j];
            }
            for(int i = k; i &lt; m; i++) {
                A[i][j] -= 2*v[i]*sum;
            }
        }            
        for(int j = 0; j &lt; m; j++) {
            sum = 0;
            for(int i = k; i &lt; m; i++) {
                sum += v[i]*Q[i][j];
            }
            for(int i = k; i &lt; m; i++) {
                Q[i][j] -= 2*v[i]*sum;
            }
        }                
    }
    return new double[][][]{
        transpose(Q),
        A
    }
}



  Changing all the ints to shorts, i was concerned with casting


Don't bother with it. In the end all of it will be running mostly on cpu registers which can hold 32 bits, incrementing 16 or 32 bits costs mostly the same and you shouldn't be bothered by that.


  I only need sign to be a bit


This is almost effectively the same as evaluating sign to be +1 or -1 once. keeping this on a variable makes you code cleaner. Pre-optimization is the rule of all evil, first you need to make sure your code is maintainable, understandable, follows design principles, and optimally that you chose the best data structures and kept the algorithm running time to a minimum, although that also falls in the optimization hole is not as aggressive as going into the last details like you referring to.


  should I write sum = 0.0?


Meh floating point is not my area of expertise but what I think is even if there is a difference on the representation between 0 and 0.0 which there isn't, you end up with slightly errors of accuracy.


  should I initialize the for loop int i's and j's outside of the loop
  to save memory?


No. Compilers do a great job optimizing trivial things for you, such as this and also the ones mentioned before. In the end the cpu will need to increment a variable, hopefully that variable will be on the cpu register or in cache, if not will be on the stack (ram). The variations between this can be different depending on how you write your code but it is the sort of things that you if you are truly worried about then you go into onto other languages like C and ultimately assembly.



On variable declaration and assignment. 

This seems to be a topic a little bit out of your knowledge, let's get some of those doubts sorted.

Historically, in languages like C, variable names were declared at the beginning of a method, or scope. The reason being that if you didn't do that way the compiler would give you a warning. I suspect that this warning could be due to possible inefficiency on the usage of the stack, but I'm not totally positive for this (at least this seem to be a plausible reason for those warnings).

Thus, like I told you, things on the internals can change a little bit depending how/where you declare you variables. Compilers can use registers differently, stack can be managed differently, cache can be used differently and so on. However in Java this is thing that is mostly out of your control, unless you go to take a peek to the bytecode and how that code would translate into assembly you really can't do much about it. That's why you move onto C or assembly if you are concerned about those things. 

Some algorithm pseudo-codes also follow the same strategy.

This is primarily why you will often see declaration of variables at the beginning of a method or scope. 

However things work differently on Java and some other languages (I would say most of them but I think that would push the boundaries). In Java the compiler does not give you a warning for where you are declaring a variable. However for readability variables tend to be declared at the beginning the scope, notice that I said scope and not method this time. Variables can also be declared in the first moment when you have everything needed to assign them.

Like I told you tmp in this algorithm can be defined in the first for body, however it could also be defined on the first inner for body. There are two reasons however to why you should declare on the outer for. 


tmp is needed to assign a value to v[k].
The evaluation of Math.sqrt(2*(sum + A0*sign*Math.sqrt(sum))) is expensive and only needs to be evaluated once. To make sure that this is only evaluated once for each k, this variable needs to be assigned out of the inner for body.