Matrices over Finite Fields


Detailed Description

In the MeatAxe, a matrix over a finite field is represented by a Matrix_t structure. Matrices can be created in many ways, for example

The application is responsible for releasing matrices which are no longer needed. Matrices can consume large amounts of memory, so it always a good idea to delete a matrix as early as possible. There is only one possibility of deleting a matrix: calling MatFree().

Echelon form and pivot tables

A matrix A with entries (aij) is said to be in echelon form if the following conditions are satisfied:

If a matrix is in echelon form, the column indexes of its pivot elements are called the pivot columns of the matrix. The list of all pivot columns is called the pivot table of the matrix.

The Matrix_t structure, which represents a matrix, has a PivotTable field which is used to store the pivot table. When different from 0, PivotTable is a pointer to an array of integers containing first the pivot columns and then the non-pivot columns. This means, the size of the array is always Noc: the first Nor elements contain the pivot columns and the remaining Noc-Nor elements contain the non-pivot columns in arbitrary order. Note that for a matrix in echelon form Nor is always less or equal to Noc.

Data Structures

class  Matrix_t
 A matrix over a finite field. More...

Functions

Matrix_tMatAdd (Matrix_t *dest, const Matrix_t *src)
 Sum of two matrices.
int MatClean (Matrix_t *mat, const Matrix_t *sub)
 Clean a matrix.
int MatCompare (const Matrix_t *a, const Matrix_t *b)
 Compare two matrices If the matrices are equal, the return value is 0.
int MatCopyRegion (Matrix_t *dest, int destrow, int destcol, const Matrix_t *src, int row1, int col1, int nrows, int ncols)
 Copy a rectangular region of a matrix This function copies a rectangular region of src tp dest.
int MatIsValid (const Matrix_t *mat)
 Check if the matrix is valid.
Matrix_tMatAlloc (int field, int nor, int noc)
 Create a new matrix.
PTR MatGetPtr (const Matrix_t *mat, int row)
 Pointer to a row of a matrix.
void Mat_DeletePivotTable (Matrix_t *mat)
 Delete the pivot table of a matrix.
int MatFree (Matrix_t *mat)
 Delete a matrix.
Matrix_tMatCut (const Matrix_t *src, int row1, int col1, int nrows, int ncols)
 Cut a rectangle out of a matrix.
Matrix_tMatCutRows (const Matrix_t *src, int row1, int nrows)
 Copy a range of rows of a matrix.
Matrix_tMatDup (const Matrix_t *src)
 Duplicate a matrix This function creates a copy of an existing matrix.
int MatEchelonize (Matrix_t *mat)
 Reduce to echelon form This function performs a Gaussian elimination on the matrix |mat|.
long MatNullity (const Matrix_t *mat)
 Nullity of a matrix.
long MatNullity__ (Matrix_t *mat)
 Nullity of a matrix.
Matrix_tMatId (int fl, int nor)
 Identity matrix This function creates an identity matrix with nor nows over GF(fl).
Matrix_tMatInsert_ (Matrix_t *mat, const Poly_t *pol)
 Insert a matrix into a polynomial Given a square matrix A and a polynomial p over the same field, this functions calculates p(A).
Matrix_tMatInsert (const Matrix_t *mat, const Poly_t *pol)
 Insert a matrix into a polynomial Given a square matrix A and a polynomial p over the same field, this functions calculates p(A).
Matrix_tMatInverse (const Matrix_t *mat)
 Inverse of a matrix This function calculates the inverse of a matrix.
Matrix_tMatMul (Matrix_t *dest, const Matrix_t *src)
 Multiply matrices This function multiplies dest from the right by src.
Matrix_tMatNullSpace_ (Matrix_t *mat, int flags)
 Null-space of a matrix This function calculates the null-space of a matrix.
Matrix_tMatNullSpace (const Matrix_t *mat)
 Null-space of a matrix This function calculates the null-space of a matrix.
Matrix_tMatNullSpace__ (Matrix_t *mat)
 Null-space of a matrix This function calculates the null-space of a matrix and deletes the original matrix.
int MatOrder (const Matrix_t *mat)
 Order of a matrix.


Function Documentation

void Mat_DeletePivotTable ( Matrix_t mat  ) 

Delete the pivot table of a matrix.

This function deletes the pivot table associated with a matrix. It is used internally, applications should never call this function directly.

Parameters:
mat Pointer to the matrix.

Matrix_t* MatAdd ( Matrix_t dest,
const Matrix_t src 
)

Sum of two matrices.

This function adds src to dest, overwriteing the previos value in dest. The matrices must be over the same field and have the same dimensions.

Returns:
dest on success, 0 on error.

Matrix_t* MatAlloc ( int  field,
int  nor,
int  noc 
)

Create a new matrix.

This function creates a new matrix with given dimensions over a given field.

Attention:
To destroy a matrix, use MatFree(), not SysFree().
Parameters:
field Field order.
nor Number of rows.
noc Number of columns.
Returns:
Pointer to the new matrix or 0 on error.

int MatClean ( Matrix_t mat,
const Matrix_t sub 
)

Clean a matrix.

This function "cleans" a matrix with a space, i.e., it adds suitable linear combinations of the rows in sub to the rows of mat such that all pivot columns in mat are zero. Both matrices must be over the same field and have the same number of colums. The second matrix, sub, must be in echelon form. The cleaned matrix is reduced to echelon form.

Returns:
Rank of the cleaned matrix, or -1 on error.

int MatCompare ( const Matrix_t a,
const Matrix_t b 
)

Compare two matrices If the matrices are equal, the return value is 0.

Otherwise the return value is positive, if a is "greater" than b and negative, if a is "less" than b. The ordering matrices is defined as follows:

  • If the matrices are over different fields, the matrix over the smaller field is smaller.
  • Otherwise, if the matrices have different number of columns, the matrix with the smaller number of columns is smaller.
  • Otherwise, if the matrices have different number of rows, the matrix with the smaller number of rows is smaller.
  • Otherwise, the relation is determined by the return value of FfCmpRow() on the first row that is not equal in both matrices.

In case an error occurs, the return value is -1. But note that a return value of -1 does not necessarily mean that an error has occured.

Parameters:
a First matrix.
b Second matrix.
Returns:
0 if the matrices are equal, nonzero otherwise (see description).

int MatCopyRegion ( Matrix_t dest,
int  destrow,
int  destcol,
const Matrix_t src,
int  row1,
int  col1,
int  nrows,
int  ncols 
)

Copy a rectangular region of a matrix This function copies a rectangular region of src tp dest.

The source region is defined by its upper left corner and dimensions, the destination region is specified by its upper left corner and has the same dimensions. Both nrows and ncols can be given as -1. In this case the region extends up to the last row or last column, respectively. The two matrices must be over the same field. Both source and destination region must not exceed the matrices' dimensions. In particular, it is not possible to extend the destination matrix by using MatCopyRegion().

Parameters:
dest Pointer to the destination matrix.
destrow Destination row.
destcol Destination column.
src Pointer to the source matrix.
row1 First row in region.
col1 First column in region.
nrows Number of rows to copy. -1 means as many rows as possible.
ncols Number of columns to copy. -1 means as many columns as possible.
Returns:
0 on success, -1 on error.

Matrix_t* MatCut ( const Matrix_t src,
int  row1,
int  col1,
int  nrows,
int  ncols 
)

Cut a rectangle out of a matrix.

This function creates a new matrix containing a copy of a rectangular region of the source matrix. The region, defined by row1, col1, nrows and ncols, must not exceed the matrix. However, both nrows and ncols may be -1. In this case the region extends up to the last row or last column, respectivly. For example, to extract the first 10 rows from a matrix independently of the number of columns, you could say

 MatCut(mat,0,0,10,-1)
See also:
MatCopyRegion MatCutRows
Parameters:
src Pointer to the matrix.
row1 First row in region.
col1 First column in region.
nrows Number of rows to cut. -1 means as many rows as possible.
ncols Number of columns to cut. -1 means as many columns as possible.
Returns:
Pointer to a new matrix containing the specified region, or 0 on error.

Matrix_t* MatCutRows ( const Matrix_t src,
int  row1,
int  nrows 
)

Copy a range of rows of a matrix.

This function creates a new matrix containing a range of consecutive rows of the source matrix. The range must now exceed the matrix's dimensions. However, nrows may be given as -1, meaning "up to the last row".

See also:
MatCopyRegion MatCutRows
Parameters:
src Pointer to the matrix.
row1 First row in region.
nrows Number of rows to cut. -1 means as many rows as possible.
Returns:
A new matrix containing the specified rows of src, or 0 on error.

Matrix_t* MatDup ( const Matrix_t src  ) 

Duplicate a matrix This function creates a copy of an existing matrix.

The caller is responsible for destroying the copy with MatFree() when it is no longer needed.

Returns:
A copy of the source Matrix, or 0 on error.

int MatEchelonize ( Matrix_t mat  ) 

Reduce to echelon form This function performs a Gaussian elimination on the matrix |mat|.

On return, |mat| is in semi echelon form and a pivot table has been attatched to the matrix. If the rank of |mat| was smaller than the number of rows, some rows are removed during the process. This function can also be used to rebuild the pivot table after the matrix has been modified.

Parameters:
mat Pointer to the matrix.
Returns:
Rank of mat, or -1 on error.

int MatFree ( Matrix_t mat  ) 

Delete a matrix.

This function frees a matrix which has beed created by MatAlloc(). Freeing includes the internal data buffers as well as the Matrix_t structure itself.

Parameters:
mat Pointer to the matrix.
Returns:
0 on success, -1 on error.

PTR MatGetPtr ( const Matrix_t mat,
int  row 
)

Pointer to a row of a matrix.

This function returns a pointer to the specified row of a matrix. Row numbers start from 0. The current row size is not changed.

Parameters:
mat Pointer to the matrix.
row Row index.
Returns:
Pointer to the selected row or 0 on error.

Matrix_t* MatId ( int  fl,
int  nor 
)

Identity matrix This function creates an identity matrix with nor nows over GF(fl).

Parameters:
fl Field order.
nor Number of rows.
Returns:
Pointer to the matrix, or 0 on error.
See also:
MatAlloc

Matrix_t* MatInsert ( const Matrix_t mat,
const Poly_t pol 
)

Insert a matrix into a polynomial Given a square matrix A and a polynomial p over the same field, this functions calculates p(A).

Unlike MatInsert_() this function returns a new matrix and does not modify the original matrix.

Parameters:
mat Pointer to the matrix.
pol Pointer to the polynomial.
Returns:
pol(mat), or 0 on error.

Matrix_t* MatInsert_ ( Matrix_t mat,
const Poly_t pol 
)

Insert a matrix into a polynomial Given a square matrix A and a polynomial p over the same field, this functions calculates p(A).

Unlike MatInsert() this function is destructive. The result is stored in the original matrix and the old value is lost.

Parameters:
mat Pointer to the matrix.
pol Pointer to the polynomial.
Returns:
The function returns mat, or 0 on error.

Matrix_t* MatInverse ( const Matrix_t mat  ) 

Inverse of a matrix This function calculates the inverse of a matrix.

mat must be a non-singular square matrix. The inverse matrix is returned in a newly allocated Matrix_t structure, and the original matrix remains unchanged.

Parameters:
mat Pointer to the matrix.
Returns:
Inverse matrix or 0 on error.

int MatIsValid ( const Matrix_t mat  ) 

Check if the matrix is valid.

This function checks if the argument is a pointer to a valid matrix. If the matrix is o.k., the function returns 1. Otherwise, an error is signalled and, if the error handler does not terminate the program, the function returns 0.

Parameters:
mat Matrix to check.
Returns:
1 if the matrix is valid, 0 otherwise.

Matrix_t* MatMul ( Matrix_t dest,
const Matrix_t src 
)

Multiply matrices This function multiplies dest from the right by src.

The matrices must be compatible for multiplication, i.e. they must be over the same field, and the number of columns of dest must be equal to the number of rows of src. The result of the multiplication is stored in dest, overwriting the original contents.

See also:
MatPower()
Parameters:
dest Left factor and result.
src Right factor.
Returns:
The function returns dest, or 0 on error.

long MatNullity ( const Matrix_t mat  ) 

Nullity of a matrix.

This function calculates the dimension of the null-space of a matrix. Unlike MatNullity__() this function does not modify the matrix.

Parameters:
mat Pointer to the matrix.
Returns:
Nullity of the matrix, or -1 on error.

long MatNullity__ ( Matrix_t mat  ) 

Nullity of a matrix.

This function calculates the dimension of the null-space of a matrix and deletes the matrix.

Parameters:
mat Pointer to the matrix.
Returns:
Nullity of mat, or -$ on error.

Matrix_t* MatNullSpace ( const Matrix_t mat  ) 

Null-space of a matrix This function calculates the null-space of a matrix.

Unlike MatNullSpace_() and MatNullSpace__(), this function does not change the original matrix, but it allocates a temporary copy of the matrix and thus needs more memory.

Parameters:
mat Pointer to the matrix.
Returns:
Pointer to the null-space, or 0 on error.

Matrix_t* MatNullSpace_ ( Matrix_t mat,
int  flags 
)

Null-space of a matrix This function calculates the null-space of a matrix.

Unlike MatNullSpace(), this function modifies the orginal matrix, but uses less memory since no temporary workspace is allocated. The result is in echelon form.

Parameters:
mat Pointer to the matrix.
flags If nonzero, the null-space is not reduced to echelon form.
Returns:
Pointer to the null-space of |mat|, or |NULL| on error.

Matrix_t* MatNullSpace__ ( Matrix_t mat  ) 

Null-space of a matrix This function calculates the null-space of a matrix and deletes the original matrix.

See also:
MatNullSpace_(), MatNullSpace()
Parameters:
mat Pointer to the matrix.
Returns:
Pointer to the null-space, or 0 on error.

int MatOrder ( const Matrix_t mat  ) 

Order of a matrix.

This function calculates the order of a matrix. mat must be a non-singular, square matrix. Even if mat is non-singular, the function may fail. This happens if the order is greater than 1000000, or if the order on any cyclic subspace is greater than 1000.

Parameters:
mat Pointer to the matrix.
Returns:
The order of mat, or 1 on error.


MeatAxe 2.4 documentation, generated on Tue May 12 01:29:55 2009