Numerical triangular factorization
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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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Numerical triangular factorization

Numerical Triangular Factorization

Lecture #11

EEE 574

Dr. Dan Tylavsky


Numerical triangular factorization

  • 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.


Numerical triangular factorization

  • 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.


Numerical triangular factorization

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


Numerical triangular factorization

  • Let’s factorize the following matrix.


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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.


Numerical triangular factorization

Row 7

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


The end

The End


Numerical triangular factorization

  • Subsequent Slides are to be ignored.


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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)).


Numerical triangular factorization

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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