Algorithm as Equation

Question

I have developed an algorithm that counts the number of times a particular block (within a 2D Matrix) crosses zero. Here's an example:

Matrix = {
            1, 2, -1, -0.1,
            4, 3, -6, -12,
            12, 2, -5, 19,
            8, -1, 12, -9,
          }

Then the matrix is split into sub-matrices (or blocks)

B1 = {
        1, 2,
        4, 3
     }

B2 = {
        -1, -0.1,
        -6, -12,
     }

...,

..., 

...

An example summing up each of the blocks is:

Ex=∑n|x[n]|2

Find the signum value of each element within the block (will return "1", "-1", "0") respectively.

If the signum value returns -1 then count increments by 1.

This will repeat until the there is no blocks, however, will only produce 1 value per block.

I am looking for a way to put all this process into an equation so I can demonstrate this rather than having to explain the processes in written text everytime. Is this possible?

Answer 1

Suppose your data is $x_1,...,x_n$, then let $C(x) = | \{i | i=0,...,n-1,\ \mathbb{sgn}\, x_i \neq \mathbb{sgn}\, x_{i+1} \} |$.

Unfortunately the above is wrong when $x$ contains zeros. The fix is cumbersome: $$ C(x) = | \{i | i < n,\, \exists j >i,\, j\leq n,\, |\mathbb{sgn}\, x_i -\mathbb{sgn}\, x_j|=2, \, x_{i+1} = \cdots = x_{j-1} = 0 \} |$$ This presumes that $\mathbb{sgn}\, 0 = 0$.

Answer 2

If data is $B=(b_1, b_2, \ldots, b_n)$ and all numbers are nonzero then the number of zero cross-overs will be

$N(B)=\sum_{i=1}^{n-1} \frac{1}{2}|\mathrm{sgn} b_i - \mathrm{sgn} b_{i+1} |$

If you have zeros in the data, you should remove them before applying this, otherwise a closed-form equation would be too cumbersome.