Vectorize function to avoid loop

Question

I'm trying to speed up my code because it's running very long. I already found out where the problem lies. Consider the following example:

x<-c((2+2i),(3+1i),(4+1i),(5+3i),(6+2i),(7+2i))
P<-matrix(c(2,0,0,3),nrow=2)
out<-sum(c(0.5,0.5)%*%mtx.exp(P%*%(matrix(c(x,0,0,x),nrow=2)),5))

I have a vector x with complex values, the vector has 12^11 entries and then I want to calculate the sum in the third row. (I need the function mtx.exp because it's a complex matrix power (the function is in the package Biodem). I found out that the %^% function does not support complex arguments.)

So my problem is that if I try

sum(c(0.5,0.5)%*%mtx.exp(P%*%(matrix(c(x,0,0,x),nrow=2)),5))

I get an error: "Error in pot %*% pot : non-conformable arguments." So my solution was to use a loop:

tmp<-NULL
for (i in 1:length(x)){
  tmp[length(tmp)+1]<-sum(c(0.5,0.5)%*%mtx.exp(P%*%matrix(c(x[i],0,0,x[i]),nrow=2),5))
}

But as said, this takes very long. Do you have any ideas how to speed up the code? I also tried sapply but that takes just as long as the loop.

I hope you can help me, because i have to run this function approximatly 500 times and this took in first try more than 3 hours. Which is not very satisfying..

Thank u very much

Answer 1

The code can be sped up by pre-allocating your vector,

tmp <- rep(NA,length(x))

but I do not really understand what you are trying to compute: in the first example,
you are trying to take the power of a non-square matrix, in the second, you are taking the power of a diagonal matrix (which can be done with ^).

The following seems to be equivalent to your computations:

sum(P^5/2) * x^5

EDIT

If P is not diagonal and C not scalar, I do not see any easy simplification of mtx.exp( P %*% C, 5 ).

You could try something like

y <- sapply(x, function(u) 
  sum( 
    c(0.5,0.5) 
    %*% 
    mtx.exp( P %*% matrix(c(u,0,0,u),nrow=2), 5 )
  )
)

but if your vector really has 12^11 entries, that will take an insanely long time.

Alternatively, since you have a very large number of very small (2*2) matrices, you can explicitely compute the product P %*% C and its 5th power (using some computer algebra system: Maxima, Sage, Yacas, Maple, etc.) and use the resulting formulas: these are just (50 lines of) straightforward operations on vectors.

/* Maxima code */ 
p: matrix([p11,p12], [p21,p22]);
c: matrix([c1,0],[0,c2]);
display2d: false;
factor(p.c . p.c . p.c . p.c . p.c);

I then copy and paste the result in R:

c1 <- dnorm(abs(x),0,1); # C is still a diagonal matrix
c2 <- dnorm(abs(x),1,3);
p11 <- P[1,1]
p12 <- P[1,2]
p21 <- P[2,1]
p22 <- P[2,2]
# Result of the Maxima computations: 
# I just add all the elements of the resulting 2*2 matrix,
# but you may want to do something slightly different with them.

          c1*(c2^4*p12*p21*p22^3+2*c1*c2^3*p11*p12*p21*p22^2
                                +2*c1*c2^3*p12^2*p21^2*p22
                                +3*c1^2*c2^2*p11^2*p12*p21*p22
                                +3*c1^2*c2^2*p11*p12^2*p21^2
                                +4*c1^3*c2*p11^3*p12*p21+c1^4*p11^5)
          +
          c2*p12
            *(c2^4*p22^4+c1*c2^3*p11*p22^3+3*c1*c2^3*p12*p21*p22^2
                        +c1^2*c2^2*p11^2*p22^2+4*c1^2*c2^2*p11*p12*p21*p22
                        +c1^3*c2*p11^3*p22+c1^2*c2^2*p12^2*p21^2
                        +3*c1^3*c2*p11^2*p12*p21+c1^4*p11^4)
         +
         c1*p21
            *(c2^4*p22^4+c1*c2^3*p11*p22^3+3*c1*c2^3*p12*p21*p22^2
                        +c1^2*c2^2*p11^2*p22^2+4*c1^2*c2^2*p11*p12*p21*p22
                        +c1^3*c2*p11^3*p22+c1^2*c2^2*p12^2*p21^2
                        +3*c1^3*c2*p11^2*p12*p21+c1^4*p11^4)
         +
         c2*(c2^4*p22^5+4*c1*c2^3*p12*p21*p22^3
                        +3*c1^2*c2^2*p11*p12*p21*p22^2
                        +3*c1^2*c2^2*p12^2*p21^2*p22
                        +2*c1^3*c2*p11^2*p12*p21*p22
                        +2*c1^3*c2*p11*p12^2*p21^2+c1^4*p11^3*p12*p21)