(octave.info)Zeros Treatment

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21.5 Differences in Treatment of Zero Elements
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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 apply also to sparse matrices,
discussed in the following chapter.  (Note: Sparse Matrices)

   The IEEE floating point standard defines the result of the
expressions ‘0*Inf’ and ‘0*NaN’ as ‘NaN’.  This is widely agreed to be a
good 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 MATLAB 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