NMath User's Guide

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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]