Creating Data Structure for L/U

1. Creating Data Structure for L/U Lecture #10 EEE 574 Dr. Dan Tylavsky

2. Triangular factorization of a sparse matrix: • Assume: Pivoting is not necessary. • Fact. code can be broken into two segments: Symbolic & Numeric • Symbolic • Ordering: Reorder rows to minimize fill.(We’ll talk about this later.) • Create data structure for L, (D), U factors. • Numeric • Perform numerical factorization storing result in data structure created in symbolic segment.

3. Symbolic Segment (Specific Example) • Assumptions: • Matrix is incidence symmetric • Matrix is stored in RR(C)U • No Pivoting is needed. • Matrix is stored in an unordered form • Desired Data Structure: • L stored by columns (ordered) • U stored by rows (ordered) • Diagonal values stored in compact form (ordered). • ( LC\O, D\O, UR\O)

6. 1 4 3 7 5 6 2 Ex: Assuming no pivoting, construct arrays of pointers and indices for the L and U factors of the A matrix (stored in RR(C)\U) shown below. Assume the L/U factors will be stored in an LC\U-UR\U-D\O storage scheme.

7. Step 0: Zero ‘Link’,MinNod & ‘Switch’. i=1 Step 1: Copy from Orig Matrix (use ‘Switch’), update ‘MinNod’, ‘ERPU’. Step 2: Use ‘Link’ to Adjust for fill. Update ‘MinNod’, ‘ERPU’. Step 3: Add row 1 to Link node (MinNod). Reset ‘MinNod’. Step 1: i=1+1,Copy from Orig Matrix (use Switch), update MinNod, ERPU. Step 2: Use ‘Link’ to Adjust for fill. Update ‘MinNod’, ‘ERPU’. Step 3: Add row 2 to Link node (MinNod). Reset ‘MinNod’. Step 1: i=2+1,Copy from Orig Matrix (use Switch), update MinNod, ERPU. Step 2: Use ‘Link’ to Adjust for fill. Update ‘MinNod’, ‘ERPU’. Step 3: Add row 3 to Link node (MinNod). Reset ‘MinNod’. Step 1: i=3+1,Copy from Orig Matrix (use Switch), update MinNod, ERPU. Step 2: Use ‘Link’ to Adjust for fill. Update ‘MinNod’, ‘ERPU’. Step 3: Add row 4 to Link node (MinNod). Reset ‘MinNod’. Step 1: i=4+1,Copy from Orig Matrix (use Switch), update MinNod, ERPU. Step 2: Use ‘Link’ to Adjust for fill. Update ‘MinNod’, ‘ERPU’. Step 3: Add row 5 to Link node (MinNod). Reset ‘MinNod’. Step 1: i=5+1,Copy from Orig Matrix (use Switch), update MinNod, ERPU. Step 2: Use ‘Link’ to Adjust for fill. Update ‘MinNod’, ‘ERPU’. Step 3: Add row 6 to Link node (MinNod). Reset ‘MinNod’. Step 4: Since i=NoBus-1=6, terminate.

8. Team Problem: Verify that the data structure derived for the matrix shown below is correct.

9. Note that the L/U data structure is unordered. • In our LU factorization algorithm, it will be convenient to order this matrix. • we’ll use a by-row-to-by-column conversion routine to order the matrix. • The routine converts from storage by rows (cols.) to storage by cols. (rows.) • We’ll need to use the routine twice.

11. Step 0: Initialize pointer to beginnings of each column. (Accumulate number of elements in each column. Perform running sum on accumulator array starting with 1 and right shift results.) Step 2&3&4: For each row, enter row index in appropriate pos. Update NxColPt. Step 5: Subtract 1 from NxColPt to get ‘ECPU’.

12. Team Problem: Verify that the data structure derived in the previous example is correct. Then reconvert matrix to get U(L) stored by row (cols.)

13. The End