17.5 Banded Matrices (.NET, C#, CSharp, VB, Visual Basic, F#)
A banded matrix is a matrix
that has all its non-zero entries near the diagonal. Entries farther
above the diagonal than the upper bandwidth, or farther
below the diagonal than the lower bandwidth, are
defined to be zero. That is, if 
is the upper bandwidth, and 
is the lower bandwidth,
then 
whenever 
or 
.
For example, this is a 7 x 7 banded matrix with upper bandwidth 1 and lower bandwidth 3:
NMath provides banded matrix classes for four datatypes: single- and double-precision floating point numbers, and single- and double-precision complex numbers. The classnames are FloatBandMatrix, DoubleBandMatrix, FloatComplexBandMatrix, and DoubleComplexBandMatrix.
For efficiency, zero elements outside the bandwidth
are not stored. Instead, matrix values are stored in a vector column
by column. Blank entries are inserted in the data vector so that the
each column takes up the same number of elements, 
, in the vector. For example,
the following 8 x 8 matrix with an upper bandwidth of 2 and a lower bandwidth
of 1:
is stored in a data vector as:
v = [x x a00 a10
x a01 a11 a21
a02 a12 a22 a32
a13 a23 a33 a43
a24 a34 a44 a54
a35 a45 a55 a65
a46 a56 a66 a76
a57 a67 a77 x ]
where x denotes an unused location.