1. Triangular Linear Equations Lecture #5 EEE 574 Dr. Dan Tylavsky

2. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky 4 Sparse Matrix Equations –Solving Sparse Matrix Equations is one goal of this course. –Let’s look a solving a special case:Lx=b L is dense and lower triangular. (Forward Substitution) –(L is stored by rows, RR(C)U/O.)

3. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky –Term under the summation sign is a dot product. –Conceptually construct. –Using dot product algorithm. Remember: no symbolic step necessary since b/x is dense. –Approach works if data is stored by rows.

4. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky IR=IR+1 Construct almost dot product of row IR with b IR. Replace b IR IR-1 with the result. IR=N? End No IR=0 N Y

5. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky 4 Individual Problem: Solve the following for x.

6. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky –We often have L (in Lx=b), stored as CR(C)O/U (x,b dense, stored in ordered compact form.) Column 1Column 2Column k Column n

7. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky 4 Individual Problem: Solve the following for x assuming L is store by columns.

8. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky –Let’s look at solving a special case:Ux=b U is dense and upper triangular. (Backward Substitution) –(U is stored by rows, RR(C)U/O.)

9. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky –Conceptually construct. –Approach works if data is stored by rows. –Term under the summation sign is a dot product.

10. Triangular Linear Equations © Copyright 1999 Daniel Tylavsky 4 Individual Problem: Solve the following for x assuming U is store by rows.

11. The End