## smrdivide

Safely produce c = a / b for matrices a and b of appropriate dimensions. This is fit for code generation.

c = smrdivide(a, b);

We can do this by taking a singular value decomposition of b, inverting its singular values, transposing it, and multiplying it on the left by a.

[u, s, v] = svd(b);
inv(b) == v * diag(diag(s).^-1) * u.'
a * inv(b) == a * v * diag(diag(s).^-1) * u.'

Of course, to be safe, we don't want to invert the singular values when they're zero, so we check for this.

Two optional tolerances can be used to determine if the singular values are big enough for inversion: an absolute tolerance and a relative tolerance. For an absolute tolerance of, e.g., 1e-15, no singular value less than 1e-15 will be inverted. For a relative tolerance of 1e-10, no singular value less than the largest singular value times this relative tolerance will be inverted. If both are provided, the maximum tolerance of the two will be used.

Clearly, this matrix doesn't really produce a / b -- that's undefined where b has null space. However, the intended use of this function is for safety in critical applications, where b should be full rank but for whatever reason isn't. It keeps the program from crashing with an error about division by 0.

## Inputs

a Any matrix The matrix by which to right-divide a Absolute tolerance; singular values smaller than this number won't be inverted. Can be left empty. Relative tolerance; singular values smaller than this number times the largest singular value won't be inverted. By default, 1e-15.

## Outputs

c The result of a / b, ignoring nullspace

## Example

Make one normal matrix, a, and a matrix with null space, b.

a = randn(3, 5);
a(:, 2) = 0;
b = randn(5);
b(:, 2) = 0;
b(2, :) = 0;

Show that the rank of b is 4, not 5 we'd need for full inversion.

rank(b)
ans =
4

Try a normal divide -- doesn't work.

c1 = a / b
[> In ml2jade (line 254)
In deploy_web (line 62)]
[Warning: Matrix is singular to working precision.]
c1 =
NaN   NaN   NaN   NaN   NaN
NaN   NaN   NaN   NaN   NaN
NaN   NaN   NaN   NaN   NaN

Divide manually, ignoring null space.

c2(:, [1 3 4 5]) = a(:, [1 3 4 5]) / b([1 3 4 5], [1 3 4 5])
c2 =
0.7546         0   -0.9916   -1.1112   -0.3590
-0.8776         0    0.7010    1.1744    0.4129
0.7195         0   -0.3537   -0.7209   -0.2368

Show that the safe divide gives the same answer as ignoring null space.

c3 = smrdivide(a, b)
c3 =
0.7546   -0.0000   -0.9916   -1.1112   -0.3590
-0.8776    0.0000    0.7010    1.1744    0.4129
0.7195   -0.0000   -0.3537   -0.7209   -0.2368