# ApCoCoA-1:Slinalg.SGEF

## 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:

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

(2) Declare some additional light columns to be heavy, chossing the heaviest ones.

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

(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 above four steps we apply usuall 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 sturctured Gaussian Elimination you want to use.

If CSteps is set to "GE" Then this function is the same as slinalg.SEF(NRow, NCol, SMat).

If CSteps is set to "GE_v2" Then this function is the same as slinalg.SEF_v2(NRow, NCol, SMat).

If CSteps is set to "SGE0" Then it performs the follwing:

loop

```  Step 2
Step 4
```

End

and at the end it perfoms usuall Gaussian Elimination.

If CSteps is set to "SGE1" Then it performs the follwing:

Step 1

loop

```  Step 2
Step 4
```

End

and at the end it performs usuall Gaussian Elimination.

If CSteps is set to "SGE2" Then it performs the follwing:

Step 1

loop

```  Step 2
Step 4
```

End

Step 1

Step 3

and at the end it perfoms usuall Gaussian Elimination.

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

#### Example

```Use ZZ/(2)[x];
NRow:=10;
NCol:=13;
M := [[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]];

Slinalg.SEF(NRow, NCol, M);
[[1,2,6,7],
[2,3],
[3,4,6,7,9,10],
[4,5,7],
[5,6,8],
[6,8,9,10,12],
[8,9,11,13],
[10,11],
[11,13]]

-------------------------------

```