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Numerical Triangular Factorization

Numerical Triangular Factorization. Lecture #11 EEE 574 Dr. Dan Tylavsky. Triangular factorization of a sparse matrix: Created ordered data structure L/U. Numerical Program Segment: Variable Assignment: A - Matrix to be factored stored in RR(C)U. CIndxA - Column indices of A.

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Numerical Triangular Factorization

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  1. Numerical Triangular Factorization Lecture #11 EEE 574 Dr. Dan Tylavsky

  2. Triangular factorization of a sparse matrix: • Created ordered data structure L/U. • Numerical Program Segment: • Variable Assignment: • A - Matrix to be factored stored in RR(C)U. • CIndxA - Column indices of A. • URO - Upper triangular factor matrix stored by rows (ordered). • LCO - Lower triangular factor matrix stored by cols. (ordered). • CIndxUO - Column Indices or URO (Row Indices of LCO). • Diag - Diagonal elements (of D matrix of LDU factors) stored in compact form.

  3. Variable Assignment (cont’d): • Rindex - Row Index of current row being processed. • ExAcum - Expanded Accumulator used in update process. • ICPL - Initial column pointers to L matrix. (Updated as L matrix is created.) • Link - Self-referential linked list used to associate the disjoint sets of rows needed in the reduction of each row.

  4. Let’s look at this algorithm in more detail.

  5. Let’s factorize the following matrix.

  6. Row 1 Step 0: Initialize RIndx, ICPL, Link. Step 1.1a,b: Zero ExAcum using Link,CIndxUO. Step 1.1c: Write row Rindx values into ExAcum. Step 2.1a: Initialize RX=0. Step 2.1b,c: Search Link for next greater value. If no greater value go to step 3. Step 3.1: Zero all links in Link associated with RIndx. Step 4.1: Invert ExAcum(RIndx), store in Diag(Rindx). Step 5.1: Multiply active ExAcumby Diag(1) & store in URO. Step 6.1: Add Rindx=1 to Link list associated with CIndxUO(ICPL(Rindx)).

  7. Row 2 Step 1.2a,b: Zero ExAcum using Link, CIndxUO. Step 1.2c: Write row Rindx values into ExAcum. Step 2.2a: Initialize RX=0. Step 2.2b,c: Search Link for next greater value. If no greater value go to step 3. Step 3.2: Zero all links in Link associated with RIndx. Step 4.2: Invert ExAcum(2), store in Diag(2). Step 5.2: Multiply active ExAcumentries by Diag(2) & store in URO. Step 6.2: Add Rindx=2 to Link list associated with CIndxUO(ICPL(Rindx)).

  8. Row 3 Step 2.6: Add Rindx=2 to Link list associated with CIndxUO(ICPL(Rindx)).m Step 3.1a,b: Zero ExAcum using Link, CIndxUO. Step 3.1c: Write row 3 values into ExAcum. Step 3.2a: Initialize RX=0. Step 3.2b,c: Search Link for next greater value. If no greater value go to step 3. Step 3.3: Zero all links in Link associated with RIndx. Step 3.4: Invert ExAcum(3), store in Diag(3). Step 3.5: Multiply active ExAcumentries by Diag(3) & store in URO. Step 3.6: Add Rindx to Link list associated with CIndxUO(ICPL(Rindx)).

  9. Row 4 Step 4.2ef1: Mult. ExAcum(RX) by Diag(RX). Store result in ICPL(RX). Incr. ICPL(RX). Step 3.6: Add Rindx to Link list associated with CIndxUO(ICPL(Rindx)). Step 4.1a,b: Incr Rindx. Zero ExAcum using Link, CIndxUO. Step 4.1c: Write row 4 values into ExAcum. Step 4.2a: Initialize RX=0. Step 4.2b,c1: Search Link for next greater value. If no greater value go to step 3. Step 4.2d1: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 4.2g1: Add row RX to link list associated with CincdxUO(ICPL(RX)).

  10. Row 4(b) Step 4.2g1: Add row RX to link list associated with CincdxUO(ICPL(RX)). Step 4.2b,c2: Search Link for next greater value. If no greater value go to step 3. Step 4.2d2: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 4.2ef2: Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)).Incr. ICPL(RX). Step 4.2g2: Add row RX to link list associated with CincdxUO(ICPL(RX)). Step 4.2b,c3: Search Link for next greater value. If no greater value go to step 3. Step 4.3: Zero all links in Link associated with RIndx. Step 4.4: Invert ExAcum(4), store in Diag(4). Step 4.5: Multiply active ExAcumentries by Diag(4) & store in URO. Step 4.6: Add Rindx=4 to Link list associated with CIndxUO(ICPL(Rindx)).

  11. Row 5 Step 5.2ef1:Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)). Incr. ICPL(RX). Step 4.6: Add Rindx=4 to Link list associated with CIndxUO(ICPL(Rindx)). Step 5.1a,b: Incr Rindx. Zero ExAcum using Link, CIndxUO. Step 5.1c: Write row 5 values into ExAcum. Step 5.2a: Initialize RX=0. Step 5.2b,c1: Search Link for next greater value. If no greater value go to step 3. Step 5.2d1: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 5.2g1: Add row RX to link list associated with CincdxUO(ICPL(RX)).

  12. Row 5(b) Step 5.2ef2: Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)).Incr. ICPL(RX). Step 5.2g1: Add row RX to link list associated with CincdxUO(ICPL(RX)). Step 5.2b,c2: Search Link for next greater value. If no greater value go to step 3. Step 5.2d2: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 5.2g2: Add row RX to link list associated with CincdxUO(ICPL(RX)). Step 5.2b,c3: Search Link for next greater value. If no greater value go to step 3. Step 5.3: Zero all links in Link associated with RIndx. Step 5.4: Invert ExAcum(5), store in Diag(5). Step 5.5: Multiply active ExAcumentries by Diag(5) & store in URO. Step 5.6: Add RIndx=5 to Link list associated with CIndxUO(ICPL(Rindx)).

  13. Row 6 Step 6.2ef1:Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)). Incr. ICPL(RX). Step 5.6: Add RIndx=5 to Link list associated with CIndxUO(ICPL(Rindx)). Step 6.1a,b: Incr RIndx. Zero ExAcum using Link, CIndxUO. Step 6.1c: Write row 6 values into ExAcum. Step 6.2a: Initialize RX=0. Step 6.2b,c1: Search Link for next greater value. If no greater value go to step 3. Step 6.2d1: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 6.2g1: Add row RX to link list associated with CIncdxUO(ICPL(RX)).

  14. Row 6(b) Step 6.2ef2:Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)). Incr. ICPL(RX). Step 6.2g1: Add row RX to link list associated with CIncdxUO(ICPL(RX)). Step 6.2b,c2: Search Link for next greater value. If no greater value go to step 3. Step 6.2d2: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 6.2g2: Add row RX to link list associated with CincdxUO(ICPL(RX)). (ERR!) Step 6.2b,c3: Search Link for next greater value. If no greater value go to step 3. Step 6.3: Zero all links in Link associated with RIndx. Step 6.4: Invert ExAcum(6), store in Diag(6). Step 6.5: Multiply active ExAcumentries by Diag(6) & store in URO. Step 6.6: Add Rindx=6 to Link list associated with CIndxUO(ICPL(Rindx)). One job you have in your midterm is to correct this non-disjoint linked list problem.

  15. Row 7 Team Problem: Complete the factorization using the data structure below. (Finish Row 7).

  16. The End

  17. Subsequent Slides are to be ignored.

  18. Step 5.2g1: Add row RX to link list associated with CincdxUO(ICPL(RX)). Step 5.2b,c2: Search Link for next greater value. If no greater value go to step 3. Step 5.2d2: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 5.2ef2: Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)).Incr. ICPL(RX). Step 5.2g2: Add row RX to link list associated with CincdxUO(ICPL(RX)). Step 6.2b,c3: Search Link for next greater value. If no greater value go to step 3. Step 6.3: Zero all links in Link associated with RIndx. Step 6.4: Invert ExAcum(4), store in Diag(4). Step 6.5: Multiply active ExAcumentries by Diag(4) & store in URO. Step 6.6: Add Rindx=4 to Link list associated with CIndxUO(ICPL(Rindx)).

  19. Step 6.2b,c2: Search Link for next greater value. If no greater value go to step 3. Step 6.2d2: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 6.2ef2:Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)). Incr. ICPL(RX). Step 6.2g2: Add row RX to link list associated with CincdxUO(ICPL(RX)).

  20. Step 6.1a,b: Incr Rindx. Zero ExAcum using Link, CIndxUO. Step 6.1c: Write row 5 values into ExAcum. Step 6.2a: Initialize RX=0. Step 6.2b,c1: Search Link for next greater value. If no greater value go to step 3. Step 6.2d1: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 6.2ef1:Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)). Incr. ICPL(RX). Step 6.2g1: Add row RX to link list associated with CincdxUO(ICPL(RX)).

  21. Step 5.2g1: Add row RX to link list associated with CincdxUO(ICPL(RX)). Step 5.2b,c2: Search Link for next greater value. If no greater value go to step 3. Step 5.2d2: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 5.2ef2: Mult. ExAcum(RX) by Diag(RX).Store result in LCO(ICPL(RX)).Incr. ICPL(RX). Step 5.2g2: Add row RX to link list associated with CincdxUO(ICPL(RX)). Step 5.2b,c3: Search Link for next greater value. If no greater value go to step 3. Step 5.3: Zero all links in Link associated with RIndx. Step 5.4: Invert ExAcum(4), store in Diag(4). Step 5.5: Multiply active ExAcumentries by Diag(4) & store in URO. Step 5.6: Add Rindx=4 to Link list associated with CIndxUO(ICPL(Rindx)).

  22. Row 5 Step 6: Add Rindx to Link list associated with CIndxUO(ICPL(Rindx)). Step 1a,b: Incr Rindx. Zero ExAcum using Cindx, Link, CIndxUO. Step 1c: Write row 5 values into ExAcum. Step 2a: Initialize RX=0. Step 2b,c: Search Link for next greater value. If no greater value go to step 3. Step 2d: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 2ef: Mult. ExAcum(RX) by Diag(RX). Store result in ICPL(RX). Incr. ICPL(RX). Step 2g: Add row RX to link list associated with Mult. CincdxUO(ICPL(RX)). Step 2b,c: Search Link for next greater value. If no greater value go to step 3. Step 2d: Mult. Row RX of URO by ExAcum(RX) and subtract from ExAcum. Step 2ef: Mult. ExAcum(RX) by Diag(RX). Store result in ICPL(RX). Incr. ICPL(RX). Step 2g: Add row RX to link list associated with Mult. CincdxUO(ICPL(RX)). Step 2b,c: Search Link for next greater value. If no greater value go to step 3. Step 3: Zero all links in Link associated with RIndx. Step 4: Invert ExAcum(4), store in Diag(4). Step 4: Invert ExAcum(2), store in Diag(2). Step 5: Multiply active ExAcumentries by Diag(4) & store in URO. Step 6: Add Rindx to Link list associated with CIndxUO(ICPL(Rindx)).

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