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Contents

  1. LU Factorization
    1. Definition
    2. Theorems, Lemmas, and Corollaries
    3. How is it used?
  2. Algorithms
    1. Partitioned Matrix Expression
    2. Loop-invariants
    3. Algorithmic variants

LU Factorization

Definition

Definition: Given matrix

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/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

, with

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

, its LU factorization is given by

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

where

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

is unit lower trapezoidal and

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/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

is upper triangular.


Theorems, Lemmas, and Corollaries

  •  

Theorem: Given a matrix

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

,

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

, its LU factorization exists if every leading principle submatrix of

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

is nonsingular.


How is it used?

The LU factorization is commonly used when solving a square system of linear equations

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

where

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

and

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

are given and

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

is to be computed. If

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

, then

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

means

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

or

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

where

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

so that

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

can be computed by first solving the lower triangular system of equations

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

(often referred to as forward substitution) and then solving the upper triangular system

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

(often referred to as backward substitution).

Algorithms

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/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

denotes the operation that overwrites

         the LowerTrapezoidal part of

         latex error! exitcode was 127 (signal 0), transscript follows:
          
         /lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

with the UnitLowerTrapezoidal matrix

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

, and

         the UpperTriangular part of

         latex error! exitcode was 127 (signal 0), transscript follows:
          
         /lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

with the UpperTriangular matrix

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

.

The diagonal of

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

, which consists of unit elements, is not stored.

Partitioned Matrix Expression

The PME for this operation is given by

         latex error! exitcode was 127 (signal 0), transscript follows:
          
         /lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory
  •  

Here

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

.

Details of how to derive the PME for this operation.

Loop-invariants

The dependencies in the above PME allow for five different loop-invariant to be identified:

  •  

Invariant 1

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

Invariant 2

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

Invariant 3

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

Invariant 4

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

Invariant 5

latex error! exitcode was 127 (signal 0), transscript follows:
 
/lusr/share/software/tex/2005/bin/i686-pc-linux-gnu/latex: error while loading shared libraries: libstdc++.so.5: cannot open shared object file: No such file or directory

Algorithmic variants

 

 

linear maths