# ApCoCoA-1:Slinalg.SGEF

## Slinalg.SGEF

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

### Syntax

```Slinalg.SSEF(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 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.
```

This function allows you to compute a (reduced) row echelon form of M over a finite field. If you want to use the first version without the parameter P, the components of the input matrix M must be of type ZMOD and your current working ring must be the same ring over which M has been defined. The second version of this function lets you compute a (reduced) row echelon form of M mod P and the components of M must be of type INT.

• @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.

• @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]]

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

```