Difference between revisions of "ApCoCoA-1:Slinalg.SGEF"

From ApCoCoAWiki
Line 10: Line 10:
 
<em>Structured Gaussian Elimination:</em> Structured Gaussian Elimination has the following four steps:
 
<em>Structured Gaussian Elimination:</em> Structured Gaussian Elimination has the following four steps:
 
<itemize>
 
<itemize>
<item>Delete all columns that have a single non-zero coefficient and the rows in which those columns have non-zero coefficients.</item>
+
<item><em>Step 1:</em> Delete all columns that have a single non-zero coefficient and the rows in which those columns have non-zero coefficients.</item>
<item>Declare some additional light columns to be heavy, chossing the heaviest ones.</item>
+
<item><em>Step 2:</em> Declare some additional light columns to be heavy, chossing the heaviest ones.</item>
<item>Delete some of the rows, selecting those which have the largest number of non-zero elements in the light columns.</item>
+
<item><em>Step 3:</em> Delete some of the rows, selecting those which have the largest number of non-zero elements in the light columns.</item>
<item>For any row which has only a single non-zero coefficient equal to 1 in the light column, subtract appropriate multiples of that row from all other rows that have non-zero coefficients on that column so as to make those coefficients 0.</item>
+
<item><em>Step 4:</em> For any row which has only a single non-zero coefficient equal to 1 in the light column, subtract appropriate multiples of that row from all other rows that have non-zero coefficients on that column so as to make those coefficients 0.</item>
 
</itemize>
 
</itemize>
 
After performing the four steps above we apply usual Gaussian Elimination, specially on heavy part of the matrix.
 
After performing the four steps above we apply usual Gaussian Elimination, specially on heavy part of the matrix.

Revision as of 15:51, 14 October 2009

Slinalg.SGEF

Computes the row echelon form of a sparse matrix over F2 using Structured Gaussian Elimination.

Syntax

Slinalg.SGEF(NRow:INT ,NCol:INT, Mat:LIST, CSteps:STRING):LIST of LIST

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Structured Gaussian Elimination: Structured Gaussian Elimination has the following four steps:

  • Step 1: Delete all columns that have a single non-zero coefficient and the rows in which those columns have non-zero coefficients.

  • Step 2: Declare some additional light columns to be heavy, chossing the heaviest ones.

  • Step 3: Delete some of the rows, selecting those which have the largest number of non-zero elements in the light columns.

  • Step 4: For any row which has only a single non-zero coefficient equal to 1 in the light column, subtract appropriate multiples of that row from all other rows that have non-zero coefficients on that column so as to make those coefficients 0.

After performing the four steps above we apply usual Gaussian Elimination, specially on heavy part of the matrix.

  • @param NRow: Number of rows of the matrix.

  • @param NCol: Number of Columns of the matrix.

  • @param Mat: List of lists containing positions of non zero elements.

  • @param CSteps: The parameter CSetps lets you specify which steps of the structured Gaussian Elimination you want to use.

  • @return A list of lists containing the row echelon form of the matrix.

We have to distinguish the following cases:

  • CSteps is set to "GE": Then this function is the same as Slinalg.SEF.

  • CSteps is set to "GE_v2": Then this function is the same as Slinalg.SEF.

  • CSteps is set to "SGE0": Then it performs the following: {loop Step 2, Step 4 End} and at the end it performs usual Gaussian Elimination.

  • CSteps is set to "SGE1": Then it performs the following: {Step 1, {loop Step 2, Step 4 End}} and at the end it performs usual Gaussian Elimination.

  • CSteps is set to "SGE2": Then it performs the following: {Step 1, {loop Step 2, Step 4 End}, Step 1, Step 3} and at the end it performs usual Gaussian Elimination.

Example

Use ZZ/(2)[x];
NRow := 10;
NCol := 13;
CSteps:=<quotes>GE_v2</quotes>;
Mat := [[1, 2, 6, 7],
      [1, 2, 4, 5, 6], 
      [2, 3], 
      [2, 3, 10, 11], 
      [2, 4, 6, 7, 9, 10], 
      [2, 10, 11, 13], 
      [5, 6, 8],
      [ 6, 8, 9,10,12],
      [6, 10, 12], 
      [10, 13]];
  
  $apcocoa/slinalg.SGEF(NRow, NCol, Mat, CSteps);
  [[1, 2, 6, 7],
   [2, 3], 
   [3, 10, 11, 13],
   [4, 5, 7],
   [5, 6, 8],
   [6, 10, 12],
   [8, 9],
   [10, 11],
   [11, 13]]	
-------------------------------

Example

NRow := 10;
NCol := 13;
CSteps:=<quotes>SGE1</quotes>;
Mat := [[1, 2, 6, 7],
      [ 2, 4, 5, 6], 
      [2, 3], 
      [2, 3, 10, 11], 
      [2, 4, 6, 7, 9, 10], 
      [2, 10, 11, 13], 
      [5, 6, 8],
      [ 6, 8, 9,10,12],
      [6, 10, 12], 
      [10, 13]];
  
  $apcocoa/slinalg.SGEF(NRow, NCol, Mat, CSteps);
  [[2, 3],
   [3, 13],
   [10, 11],
   [11, 13]]


Example

NRow := 10;
NCol := 13;
CSteps:=<quotes>SGE2</quotes>;
Mat := [[1, 2, 6, 7],
      [ 2, 4, 5, 6], 
      [2, 3], 
      [2, 3, 10, 11], 
      [2, 4, 6, 7, 9, 10], 
      [2, 10, 11, 13], 
      [5, 6, 8],
      [ 6, 8, 9,10,12],
      [6, 10, 12], 
      [10, 13]];
  
  $apcocoa/slinalg.SGEF(NRow, NCol, Mat, CSteps);
   [ ]




See also

Introduction to CoCoAServer

IML.REF

LinAlg.REF