20.5 The Differences in Treatment of Zero Elements

Making diagonal and permutation matrices special matrix objects in their own right and the consequent usage of smarter algorithms for certain operations implies, as a side effect, small differences in treating zeros. The contents of this section applies also to sparse matrices, discussed in the following chapter.

The IEEE standard defines the result of the expressions 0*Inf and 0*NaN as NaN, as it has been generally agreed that this is the best compromise. Numerical software dealing with structured and sparse matrices (including Octave) however, almost always makes a distinction between a "numerical zero" and an "assumed zero". A "numerical zero" is a zero value occurring in a place where any floating-point value could occur. It is normally stored somewhere in memory as an explicit value. An "assumed zero", on the contrary, is a zero matrix element implied by the matrix structure (diagonal, triangular) or a sparsity pattern; its value is usually not stored explicitly anywhere, but is implied by the underlying data structure.

The primary distinction is that an assumed zero, when multiplied by any number, or divided by any nonzero number, yields *always* a zero, even when, e.g., multiplied by Inf or divided by NaN. The reason for this behavior is that the numerical multiplication is not actually performed anywhere by the underlying algorithm; the result is just assumed to be zero. Equivalently, one can say that the part of the computation involving assumed zeros is performed symbolically, not numerically.

This behavior not only facilitates the most straightforward and efficient implementation of algorithms, but also preserves certain useful invariants, like:

• scalar * diagonal matrix is a diagonal matrix
• sparse matrix / scalar preserves the sparsity pattern
• permutation matrix * matrix is equivalent to permuting rows

all of these natural mathematical truths would be invalidated by treating assumed zeros as numerical ones.

Note that certain competing software does not strictly follow this principle and converts assumed zeros to numerical zeros in certain cases, while not doing so in other cases. As of today, there are no intentions to mimic such behavior in Octave.

Examples of effects of assumed zeros vs. numerical zeros:

Inf * eye (3)
⇒
   Inf     0     0
     0   Inf     0
     0     0   Inf

Inf * speye (3)
⇒
Compressed Column Sparse (rows = 3, cols = 3, nnz = 3 [33%])

  (1, 1) -> Inf
  (2, 2) -> Inf
  (3, 3) -> Inf

Inf * full (eye (3))
⇒
   Inf   NaN   NaN
   NaN   Inf   NaN
   NaN   NaN   Inf

diag(1:3) * [NaN; 1; 1]
⇒
   NaN
     2
     3

sparse(1:3,1:3,1:3) * [NaN; 1; 1]
⇒
   NaN
     2
     3
[1,0,0;0,2,0;0,0,3] * [NaN; 1; 1]
⇒
   NaN
   NaN
   NaN