Lower triangular matrix-vector product

问题

For a programming exercise, I was given the lower triangular elements of a symmetric 3x3 matrix saved as an array

|1 * *|

|2 4 *| => [1,2,3,4,5,6]

|3 5 6|

I need to make the product C(i)=C(i)+M(i,j)V(j) where M is the symmetric matrix and V is a vector.

V =>[A,B,C]
C(1)=1*A + 2*B + 3*C
C(2)=2*A + 4*B + 5*C
C(3)=3*A + 5*B + 6*C

I am trying to make an efficient algorithm that can perform this product

I can easily generate all the product I need for C(3) However, I have a problem when I try to generate the values C(1), C(2) and I don't know how to get around this without using extra memory.

This is what I have done

k=6
n=3

  DO 1 j = n,1,-1
    l= k
    DO 2 i = n,j + 1,-1
       C(i) = C(i) + V(j)*M(l)
       l = l - 1           
2   enddo             
    C(j) = V(j)*M(k-n+j)
    k = k - (n-j+1)
1 enddo

The problem I have is that I can no generate and add the 2*B for C(1) and the 5*C for C(2). The goal of the exercise is to use as few steps and as little array space as possible.

Any suggestions?

回答1:

There are multiple issues with your code:

In the outer loop, you assign C(n) (probably the diagonal entries), so the computation of inner loop is not used at all
You are looping over the lower left triangle from back to front, if you would reverse that, the indexing inside the vectorized matrix would be much simpler
The calculation of the position inside the matrix (k and l) is wrong
You do not calculate the products of the mirrored elements

Here is my solution that honors above points:

  ! Loop over all elements in the lower left triangle
  k = 0
  do j=1,n
    ! Increment the position inside the unrolled matrix
    k = k+1
    ! diagonal entries, i = j
    c(j) = c(j) + v(j)*M(k)

    ! off-diagonal entries
    do i=j+1,n
      ! Increment the position inside the unrolled matrix
      k = k+1
      ! Original entry
      c(i) = c(i) + v(j)*M(k)
      ! Mirrored one
      c(j) = c(j) + v(i)*M(k)
    enddo !i
  enddo !j

来源：https://stackoverflow.com/questions/45093764/lower-triangular-matrix-vector-product

标签

fortran

matrix-multiplication