3. TRIANGULAR
FACTORIZATION
By Syed Zulqadar Hassan
CIIT Abbottabad Campus

4. TRIANGULAR FACTORIZATION
• It involves three steps:
• Step 1 Triangular Factorization
• Step 2 Forward Substitution
• Step 3 Back Substitution

7. TRIANGULAR FACTORIZATION Count…
• So we can write
• We Know that

8. • Finally we get
TRIANGULAR FACTORIZATION Count…

9. ADVANTAGES
• Solution of a linear system by triangular factorization and subsequent forward
and back substitution is very popular because of the many advantages of the
method:
• Efficiency
• Ability to preserve sparsity of the matrix

10. Sparsity
• The fraction of zero elements (non-zero elements) in a matrix is called
the sparsity (density).

11. Sparsity in Power System
• Let us analyze the requirements for a 1000 node/2000 branch circuit.
• For this network, the admittance matrix Y will have approximately 5000
nonzero elements. The table of factors for this matrix will have 5000Rs
nonzero elements.
• If Rs = 2.5, then 12,500 nonzero elements need to be stored.

12. Sparsity in Power System Count…
• The sparsity preservation index also impacts the efficiency of the method.
• This becomes obvious by considering the fact that the forward and back
substitutions require as many multiply-adds as the number of non-zeros
entries in the table of factors.
• If Rs = 2.5, then only 12,500 multiply-adds are required in the forward and
back substitution, a small number compared with the required multiply-
adds for the operation inverse of Y.
• The inverse of Y have 10,000,000 Multiply-adds while Factorization have
900,000 Multiply-adds.

13. References
• Power System Modeling, Analysis and Control By A. P. Sakis Meliopoulos
(Page 18)
• https://en.wikipedia.org/wiki/Sparse_matrix
• “Triangular Factorization Method for Power Flow Analysis” by Y.Okamoto
Published in journal “Electrical Engineering in Japan” Vol 96, No 1, January
1976, pp 31-35