17.2 Upper Triangular Matrices (.NET, C#, CSharp, VB, Visual Basic, F#)
An upper triangular matrix
is a square matrix with all elements below the main diagonal equal to
zero. That is, 
for
. For example, this
is a 4 x 4 upper triangular matrix:
Like lower triangular matrices, upper triangular matrices often arise at an intermediate stage in solving systems of equations and inverting matrices.
NMath provides upper triangular matrix classes for four datatypes: single- and double-precision floating point numbers, and single- and double-precision complex numbers. The classnames are FloatUpperTriMatrix, DoubleUpperTriMatrix, FloatComplexUpperTriMatrix, and DoubleComplexUpperTriMatrix.
For efficiency, zero elements below the main diagonal are not stored. Instead, matrix values are stored in a vector column by column. For example, the following 5 x 5 upper triangular matrix:
is stored in a data vector as:
v = [ a00 a01 a11 a02 a12 a22 a03 a13 a23 a33 a04 a14 a24 a34 a44 ]
In general, the relationship between matrix and vector indices is:
A[i,j] = v[i + j(j+1)/2]