/usr/share/singular/LIB/matrix.lib is in singular-data 4.0.3+ds-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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version="version matrix.lib 4.0.0.0 Jun_2013 "; // $Id: 92849631fbe0c7b8e6b2b1315b241bbd69578007 $
category="Linear Algebra";
info="
LIBRARY: matrix.lib Elementary Matrix Operations
PROCEDURES:
compress(A); matrix, zero columns from A deleted
concat(A1,A2,..); matrix, concatenation of matrices A1,A2,...
diag(p,n); matrix, nxn diagonal matrix with entries poly p
dsum(A1,A2,..); matrix, direct sum of matrices A1,A2,...
flatten(A); ideal, generated by entries of matrix A
genericmat(n,m[,id]); generic nxm matrix [entries from id]
is_complex(c); 1 if list c is a complex, 0 if not
outer(A,B); matrix, outer product of matrices A and B
power(A,n); matrix/intmat, n-th power of matrix/intmat A
skewmat(n[,id]); generic skew-symmetric nxn matrix [entries from id]
submat(A,r,c); submatrix of A with rows/cols specified by intvec r/c
symmat(n[,id]); generic symmetric nxn matrix [entries from id]
tensor(A,B); matrix, tensor product of matrices A nd B
unitmat(n); unit square matrix of size n
gauss_col(A); transform a matrix into col-reduced Gauss normal form
gauss_row(A); transform a matrix into row-reduced Gauss normal form
addcol(A,c1,p,c2); add p*(c1-th col) to c2-th column of matrix A, p poly
addrow(A,r1,p,r2); add p*(r1-th row) to r2-th row of matrix A, p poly
multcol(A,c,p); multiply c-th column of A with poly p
multrow(A,r,p); multiply r-th row of A with poly p
permcol(A,i,j); permute i-th and j-th columns
permrow(A,i,j); permute i-th and j-th rows
rowred(A[,any]); reduction of matrix A with elementary row-operations
colred(A[,any]); reduction of matrix A with elementary col-operations
linear_relations(E); find linear relations between homogeneous vectors
rm_unitrow(A); remove unit rows and associated columns of A
rm_unitcol(A); remove unit columns and associated rows of A
headStand(A); A[n-i+1,m-j+1]:=A[i,j]
symmetricBasis(n,k[,s]); basis of k-th symmetric power of n-dim v.space
exteriorBasis(n,k[,s]); basis of k-th exterior power of n-dim v.space
symmetricPower(A,k); k-th symmetric power of a module/matrix A
exteriorPower(A,k); k-th exterior power of a module/matrix A
(parameters in square brackets [] are optional)
";
LIB "inout.lib";
LIB "ring.lib";
LIB "random.lib";
LIB "general.lib"; // for sort
LIB "nctools.lib"; // for superCommutative
///////////////////////////////////////////////////////////////////////////////
proc compress (def A)
"USAGE: compress(A); A matrix/ideal/module/intmat/intvec
RETURN: same type, zero columns/generators from A deleted
(if A=intvec, zero elements are deleted)
EXAMPLE: example compress; shows an example
"
{
if( typeof(A)=="matrix" ) { return(matrix(simplify(A,2))); }
if( typeof(A)=="intmat" or typeof(A)=="intvec" )
{
ring r=0,x,lp;
if( typeof(A)=="intvec" ) { intmat C=transpose(A); kill A; intmat A=C; }
module m = matrix(A);
if ( size(m) == 0)
{ intmat B; }
else
{ intmat B[nrows(A)][size(m)]; }
int i,j;
for( i=1; i<=ncols(A); i++ )
{
if( m[i]!=[0] )
{
j++;
B[1..nrows(A),j]=A[1..nrows(A),i];
}
}
if( defined(C) ) { return(intvec(B)); }
return(B);
}
return(simplify(A,2));
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z),ds;
matrix A[3][4]=1,0,3,0,x,0,z,0,x2,0,z2,0;
print(A);
print(compress(A));
module m=module(A); show(m);
show(compress(m));
intmat B[3][4]=1,0,3,0,4,0,5,0,6,0,7,0;
compress(B);
intvec C=0,0,1,2,0,3;
compress(C);
}
///////////////////////////////////////////////////////////////////////////////
proc concat (list #)
"USAGE: concat(A1,A2,..); A1,A2,... matrices
RETURN: matrix, concatenation of A1,A2,.... Number of rows of result matrix
is max(nrows(A1),nrows(A2),...)
EXAMPLE: example concat; shows an example
"
{
int i;
for( i=size(#);i>0; i-- ) { #[i]=module(#[i]); }
module B=#[1..size(#)];
return(matrix(B));
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z),ds;
matrix A[3][3]=1,2,3,x,y,z,x2,y2,z2;
matrix B[2][2]=1,0,2,0; matrix C[1][4]=4,5,x,y;
print(A);
print(B);
print(C);
print(concat(A,B,C));
}
///////////////////////////////////////////////////////////////////////////////
proc diag (list #)
"USAGE: diag(p,n); p poly, n integer
diag(A); A matrix
RETURN: diag(p,n): diagonal matrix, p times unit matrix of size n.
@* diag(A) : n*m x n*m diagonal matrix with entries all the entries of
the nxm matrix A, taken from the 1st row, 2nd row etc of A
EXAMPLE: example diag; shows an example
"
{
if( size(#)==2 ) { return(matrix(#[1]*freemodule(#[2]))); }
if( size(#)==1 )
{
int i; ideal id=#[1];
int n=ncols(id); matrix A[n][n];
for( i=1; i<=n; i++ ) { A[i,i]=id[i]; }
}
return(A);
}
example
{ "EXAMPLE:"; echo = 2;
ring r = 0,(x,y,z),ds;
print(diag(xy,4));
matrix A[3][2] = 1,2,3,4,5,6;
print(A);
print(diag(A));
}
///////////////////////////////////////////////////////////////////////////////
proc dsum (list #)
"USAGE: dsum(A1,A2,..); A1,A2,... matrices
RETURN: matrix, direct sum of A1,A2,...
EXAMPLE: example dsum; shows an example
"
{
int i,N,a;
list L;
for( i=1; i<=size(#); i++ ) { N=N+nrows(#[i]); }
for( i=1; i<=size(#); i++ )
{
matrix B[N][ncols(#[i])];
B[a+1..a+nrows(#[i]),1..ncols(#[i])]=#[i];
a=a+nrows(#[i]);
L[i]=B; kill B;
}
return(concat(L));
}
example
{ "EXAMPLE:"; echo = 2;
ring r = 0,(x,y,z),ds;
matrix A[3][3] = 1,2,3,4,5,6,7,8,9;
matrix B[2][2] = 1,x,y,z;
print(A);
print(B);
print(dsum(A,B));
}
///////////////////////////////////////////////////////////////////////////////
proc flatten (def A)
"USAGE: flatten(A); A matrix
RETURN: ideal, generated by all entries from A
EXAMPLE: example flatten; shows an example
"
{
return(ideal(A));
}
example
{ "EXAMPLE:"; echo = 2;
ring r = 0,(x,y,z),ds;
matrix A[2][3] = 1,2,x,y,z,7;
print(A);
flatten(A);
}
///////////////////////////////////////////////////////////////////////////////
proc genericmat (int n,int m,list #)
"USAGE: genericmat(n,m[,id]); n,m=integers, id=ideal
RETURN: nxm matrix, with entries from id.
NOTE: if id has less than nxm elements, the matrix is filled with 0's,
(default: id=maxideal(1)).
genericmat(n,m); creates the generic nxm matrix
EXAMPLE: example genericmat; shows an example
"
{
if( size(#)==0 ) { ideal id=maxideal(1); }
if( size(#)==1 ) { ideal id=#[1]; }
if( size(#)>=2 ) { "// give 3 arguments, 3-rd argument must be an ideal"; }
matrix B[n][m]=id;
return(B);
}
example
{ "EXAMPLE:"; echo = 2;
ring R = 0,x(1..16),lp;
print(genericmat(3,3)); // the generic 3x3 matrix
ring R1 = 0,(a,b,c,d),dp;
matrix A = genericmat(3,4,maxideal(1)^3);
print(A);
int n,m = 3,2;
ideal i = ideal(randommat(1,n*m,maxideal(1),9));
print(genericmat(n,m,i)); // matrix of generic linear forms
}
///////////////////////////////////////////////////////////////////////////////
proc is_complex (list c)
"USAGE: is_complex(c); c = list of size-compatible modules or matrices
RETURN: 1 if c[i]*c[i+1]=0 for all i, 0 if not, hence checking whether the
list of matrices forms a complex.
NOTE: Ideals are treated internally as 1-line matrices.
If printlevel > 0, the position where c is not a complex is shown.
EXAMPLE: example is_complex; shows an example
"
{
int i;
module @test;
for( i=1; i<=size(c)-1; i++ )
{
c[i]=matrix(c[i]); c[i+1]=matrix(c[i+1]);
@test=c[i]*c[i+1];
if (size(@test)!=0)
{
dbprint(printlevel-voice+2,"// not a complex at position " +string(i));
return(0);
}
}
return(1);
}
example
{ "EXAMPLE:"; echo = 2;
ring r = 32003,(x,y,z),ds;
ideal i = x4+y5+z6,xyz,yx2+xz2+zy7;
list L = nres(i,0);
is_complex(L);
L[4] = matrix(i);
is_complex(L);
}
///////////////////////////////////////////////////////////////////////////////
proc outer (matrix A, matrix B)
"USAGE: outer(A,B); A,B matrices
RETURN: matrix, outer (tensor) product of A and B
EXAMPLE: example outer; shows an example
"
{
int i,j; list L;
int triv = nrows(B)*ncols(B);
if( triv==1 )
{
return(B[1,1]*A);
}
else
{
int N = nrows(A)*nrows(B);
matrix C[N][ncols(B)];
for( i=ncols(A);i>0; i-- )
{
for( j=1; j<=nrows(A); j++ )
{
C[(j-1)*nrows(B)+1..j*nrows(B),1..ncols(B)]=A[j,i]*B;
}
L[i]=C;
}
return(concat(L));
}
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),ds;
matrix A[3][3]=1,2,3,4,5,6,7,8,9;
matrix B[2][2]=x,y,0,z;
print(A);
print(B);
print(outer(A,B));
}
///////////////////////////////////////////////////////////////////////////////
proc power (def A, int n)
"USAGE: power(A,n); A a square-matrix of type intmat or matrix, n=integer>=0
RETURN: intmat resp. matrix, the n-th power of A
NOTE: for A=intmat and big n the result may be wrong because of int overflow
EXAMPLE: example power; shows an example
"
{
//---------------------------- type checking ----------------------------------
if( typeof(A)!="matrix" and typeof(A)!="intmat" )
{
ERROR("no matrix or intmat!");
}
if( ncols(A) != nrows(A) )
{
ERROR("not a square matrix!");
}
//---------------------------- trivial cases ----------------------------------
int ii;
if( n <= 0 )
{
if( typeof(A)=="matrix" )
{
return (unitmat(nrows(A)));
}
if( typeof(A)=="intmat" )
{
intmat B[nrows(A)][nrows(A)];
for( ii=1; ii<=nrows(A); ii++ )
{
B[ii,ii] = 1;
}
return (B);
}
}
if( n == 1 ) { return (A); }
//---------------------------- sub procedure ----------------------------------
proc matpow (def A, int n)
{
def B = A*A;
int ii= 2;
int jj= 4;
while( jj <= n )
{
B=B*B;
ii=jj;
jj=2*jj;
}
return(B,n-ii);
}
//----------------------------- main program ----------------------------------
list L = matpow(A,n);
def B = L[1];
ii = L[2];
while( ii>=2 )
{
L = matpow(A,ii);
B = B*L[1];
ii= L[2];
}
if( ii == 0) { return(B); }
if( ii == 1) { return(A*B); }
}
example
{ "EXAMPLE:"; echo = 2;
intmat A[3][3]=1,2,3,4,5,6,7,8,9;
print(power(A,3));"";
ring r=0,(x,y,z),dp;
matrix B[3][3]=0,x,y,z,0,0,y,z,0;
print(power(B,3));"";
}
///////////////////////////////////////////////////////////////////////////////
proc skewmat (int n, list #)
"USAGE: skewmat(n[,id]); n integer, id ideal
RETURN: skew-symmetric nxn matrix, with entries from id
(default: id=maxideal(1))
skewmat(n); creates the generic skew-symmetric matrix
NOTE: if id has less than n*(n-1)/2 elements, the matrix is
filled with 0's,
EXAMPLE: example skewmat; shows an example
"
{
matrix B[n][n];
if( size(#)==0 ) { ideal id=maxideal(1); }
else { ideal id=#[1]; }
id = id,B[1..n,1..n];
int i,j;
for( i=0; i<=n-2; i++ )
{
B[i+1,i+2..n]=id[j+1..j+n-i-1];
j=j+n-i-1;
}
matrix A=transpose(B);
B=B-A;
return(B);
}
example
{ "EXAMPLE:"; echo = 2;
ring R=0,x(1..5),lp;
print(skewmat(4)); // the generic skew-symmetric matrix
ring R1 = 0,(a,b,c),dp;
matrix A=skewmat(4,maxideal(1)^2);
print(A);
int n=3;
ideal i = ideal(randommat(1,n*(n-1) div 2,maxideal(1),9));
print(skewmat(n,i)); // skew matrix of generic linear forms
kill R1;
}
///////////////////////////////////////////////////////////////////////////////
proc submat (matrix A, intvec r, intvec c)
"USAGE: submat(A,r,c); A=matrix, r,c=intvec
RETURN: matrix, submatrix of A with rows specified by intvec r
and columns specified by intvec c.
EXAMPLE: example submat; shows an example
"
{
matrix B[size(r)][size(c)]=A[r,c];
return(B);
}
example
{ "EXAMPLE:"; echo = 2;
ring R=32003,(x,y,z),lp;
matrix A[4][4]=x,y,z,0,1,2,3,4,5,6,7,8,9,x2,y2,z2;
print(A);
intvec v=1,3,4;
matrix B=submat(A,v,1..3);
print(B);
}
///////////////////////////////////////////////////////////////////////////////
proc symmat (int n, list #)
"USAGE: symmat(n[,id]); n integer, id ideal
RETURN: symmetric nxn matrix, with entries from id (default: id=maxideal(1))
NOTE: if id has less than n*(n+1)/2 elements, the matrix is filled with 0's,
symmat(n); creates the generic symmetric matrix
EXAMPLE: example symmat; shows an example
"
{
matrix B[n][n];
if( size(#)==0 ) { ideal id=maxideal(1); }
else { ideal id=#[1]; }
id = id,B[1..n,1..n];
int i,j;
for( i=0; i<=n-1; i++ )
{
B[i+1,i+1..n]=id[j+1..j+n-i];
j=j+n-i;
}
matrix A=transpose(B);
for( i=1; i<=n; i++ ) { A[i,i]=0; }
B=A+B;
return(B);
}
example
{ "EXAMPLE:"; echo = 2;
ring R=0,x(1..10),lp;
print(symmat(4)); // the generic symmetric matrix
ring R1 = 0,(a,b,c),dp;
matrix A=symmat(4,maxideal(1)^3);
print(A);
int n=3;
ideal i = ideal(randommat(1,n*(n+1) div 2,maxideal(1),9));
print(symmat(n,i)); // symmetric matrix of generic linear forms
kill R1;
}
///////////////////////////////////////////////////////////////////////////////
proc tensor (matrix A, matrix B)
"USAGE: tensor(A,B); A,B matrices
RETURN: matrix, tensor product of A and B
EXAMPLE: example tensor; shows an example
"
{
if (ncols(A)==0)
{
int q=nrows(A)*nrows(B);
matrix D[q][0];
return(D);
}
int i,j;
matrix C,D;
for( i=1; i<=nrows(A); i++ )
{
C = A[i,1]*B;
for( j=2; j<=ncols(A); j++ )
{
C = concat(C,A[i,j]*B);
}
D = concat(D,transpose(C));
}
D = transpose(D);
return(submat(D,2..nrows(D),1..ncols(D)));
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),(c,ds);
matrix A[3][3]=1,2,3,4,5,6,7,8,9;
matrix B[2][2]=x,y,0,z;
print(A);
print(B);
print(tensor(A,B));
}
///////////////////////////////////////////////////////////////////////////////
proc unitmat (int n)
"USAGE: unitmat(n); n integer >= 0
RETURN: nxn unit matrix
NOTE: needs a basering, diagonal entries are numbers (=1) in the basering
EXAMPLE: example unitmat; shows an example
"
{
return(matrix(freemodule(n)));
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),lp;
print(xyz*unitmat(4));
print(unitmat(5));
}
///////////////////////////////////////////////////////////////////////////////
proc gauss_col (matrix A, list #)
"USAGE: gauss_col(A[,e]); A a matrix, e any type
RETURN: - a matrix B, if called with one argument; B is the complete column-
reduced upper-triangular normal form of A if A is constant,
(resp. as far as this is possible if A is a polynomial matrix;
no division by polynomials).
@* - a list L of two matrices, if called with two arguments;
L satisfies L[1] = A * L[2] with L[1] the column-reduced form of A
and L[2] the transformation matrix.
NOTE: * The procedure just applies interred to A with ordering (C,dp).
The transformation matrix is obtained by applying 'lift'.
This should be faster than the procedure colred.
@* * It should only be used with exact coefficient field (there is no
pivoting and rounding error treatment).
@* * Parameters are allowed. Hence, if the entries of A are parameters,
B is the column-reduced form of A over the rational function field.
SEE ALSO: colred
EXAMPLE: example gauss_col; shows an example
"
{
def R=basering; int u;
string mp = string(minpoly);
int n = nrows(A);
int m = ncols(A);
module M = A;
intvec v = option(get);
//------------------------ change ordering if necessary ----------------------
if( ordstr(R) != ("C,dp("+string(nvars(R))+")") )
{
def @R=changeord(list(list("C",0:1),list("dp",1:nvars(R))),R);
setring @R; u=1;
if (mp!="0") { execute("minpoly="+mp+";");}
matrix A = imap(R,A);
module M = A;
}
//------------------------------ start computation ---------------------------
option(redSB);
M = simplify(interred(M),1);
if(size(#) != 0)
{
module N = lift(A,M);
}
//--------------- reset ring and options and return --------------------------
if ( u==1 )
{
setring R;
M=imap(@R,M);
if (size(#) != 0)
{
module N = imap(@R,N);
}
kill @R;
}
option(set,v);
// M = sort(M,size(M)..1)[1];
A = matrix(M,n,m);
if (size(#) != 0)
{
list L= A,matrix(N,m,m);
return(L);
}
return(matrix(M,n,m));
}
example
{ "EXAMPLE:"; echo = 2;
ring r=(0,a,b),(A,B,C),dp;
matrix m[8][6]=
0, 2*C, 0, 0, 0, 0,
0, -4*C,a*A, 0, 0, 0,
b*B, -A, 0, 0, 0, 0,
-A, B, 0, 0, 0, 0,
-4*C, 0, B, 2, 0, 0,
2*A, B, 0, 0, 0, 0,
0, 3*B, 0, 0, 2b, 0,
0, AB, 0, 2*A,A, 2a;"";
list L=gauss_col(m,1);
print(L[1]);
print(L[2]);
ring S=0,x,(c,dp);
matrix A[5][4] =
3, 1, 1, 1,
13, 8, 6,-7,
14,10, 6,-7,
7, 4, 3,-3,
2, 1, 0, 3;
print(gauss_col(A));
}
///////////////////////////////////////////////////////////////////////////////
proc gauss_row (matrix A, list #)
"USAGE: gauss_row(A [,e]); A matrix, e any type
RETURN: - a matrix B, if called with one argument; B is the complete row-
reduced lower-triangular normal form of A if A is constant,
(resp. as far as this is possible if A is a polynomial matrix;
no division by polynomials).
@* - a list L of two matrices, if called with two arguments;
L satisfies transpose(L[2])*A=transpose(L[1])
with L[1] the row-reduced form of A
and L[2] the transformation matrix.
NOTE: * This procedure just applies gauss_col to the transposed matrix.
The transformation matrix is obtained by applying lift.
This should be faster than the procedure rowred.
@* * It should only be used with exact coefficient field (there is no
pivoting and rounding error treatment).
@* * Parameters are allowed. Hence, if the entries of A are parameters,
B is the row-reduced form of A over the rational function field.
SEE ALSO: rowred
EXAMPLE: example gauss_row; shows an example
"
{
if(size(#) > 0)
{
list L = gauss_col(transpose(A),1);
return(L);
}
A = gauss_col(transpose(A));
return(transpose(A));
}
example
{ "EXAMPLE:"; echo = 2;
ring r=(0,a,b),(A,B,C),dp;
matrix m[6][8]=
0, 0, b*B, -A,-4C,2A,0, 0,
2C,-4C,-A,B, 0, B, 3B,AB,
0,a*A, 0, 0, B, 0, 0, 0,
0, 0, 0, 0, 2, 0, 0, 2A,
0, 0, 0, 0, 0, 0, 2b, A,
0, 0, 0, 0, 0, 0, 0, 2a;"";
print(gauss_row(m));"";
ring S=0,x,dp;
matrix A[4][5] = 3, 1,1,-1,2,
13, 8,6,-7,1,
14,10,6,-7,1,
7, 4,3,-3,3;
list L=gauss_row(A,1);
print(L[1]);
print(L[2]);
}
///////////////////////////////////////////////////////////////////////////////
proc addcol (matrix A, int c1, poly p, int c2)
"USAGE: addcol(A,c1,p,c2); A matrix, p poly, c1, c2 positive integers
RETURN: matrix, A being modified by adding p times column c1 to column c2
EXAMPLE: example addcol; shows an example
"
{
int k=nrows(A);
A[1..k,c2]=A[1..k,c2]+p*A[1..k,c1];
return(A);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),lp;
matrix A[3][3]=1,2,3,4,5,6,7,8,9;
print(A);
print(addcol(A,1,xy,2));
}
///////////////////////////////////////////////////////////////////////////////
proc addrow (matrix A, int r1, poly p, int r2)
"USAGE: addrow(A,r1,p,r2); A matrix, p poly, r1, r2 positive integers
RETURN: matrix, A being modified by adding p times row r1 to row r2
EXAMPLE: example addrow; shows an example
"
{
int k=ncols(A);
A[r2,1..k]=A[r2,1..k]+p*A[r1,1..k];
return(A);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),lp;
matrix A[3][3]=1,2,3,4,5,6,7,8,9;
print(A);
print(addrow(A,1,xy,3));
}
///////////////////////////////////////////////////////////////////////////////
proc multcol (matrix A, int c, poly p)
"USAGE: multcol(A,c,p); A matrix, p poly, c positive integer
RETURN: matrix, A being modified by multiplying column c by p
EXAMPLE: example multcol; shows an example
"
{
int k=nrows(A);
A[1..k,c]=p*A[1..k,c];
return(A);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),lp;
matrix A[3][3]=1,2,3,4,5,6,7,8,9;
print(A);
print(multcol(A,2,xy));
}
///////////////////////////////////////////////////////////////////////////////
proc multrow (matrix A, int r, poly p)
"USAGE: multrow(A,r,p); A matrix, p poly, r positive integer
RETURN: matrix, A being modified by multiplying row r by p
EXAMPLE: example multrow; shows an example
"
{
int k=ncols(A);
A[r,1..k]=p*A[r,1..k];
return(A);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),lp;
matrix A[3][3]=1,2,3,4,5,6,7,8,9;
print(A);
print(multrow(A,2,xy));
}
///////////////////////////////////////////////////////////////////////////////
proc permcol (matrix A, int c1, int c2)
"USAGE: permcol(A,c1,c2); A matrix, c1,c2 positive integers
RETURN: matrix, A being modified by permuting columns c1 and c2
EXAMPLE: example permcol; shows an example
"
{
matrix B=A;
int k=nrows(B);
B[1..k,c1]=A[1..k,c2];
B[1..k,c2]=A[1..k,c1];
return(B);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),lp;
matrix A[3][3]=1,x,3,4,y,6,7,z,9;
print(A);
print(permcol(A,2,3));
}
///////////////////////////////////////////////////////////////////////////////
proc permrow (matrix A, int r1, int r2)
"USAGE: permrow(A,r1,r2); A matrix, r1,r2 positive integers
RETURN: matrix, A being modified by permuting rows r1 and r2
EXAMPLE: example permrow; shows an example
"
{
matrix B=A;
int k=ncols(B);
B[r1,1..k]=A[r2,1..k];
B[r2,1..k]=A[r1,1..k];
return(B);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=32003,(x,y,z),lp;
matrix A[3][3]=1,2,3,x,y,z,7,8,9;
print(A);
print(permrow(A,2,1));
}
///////////////////////////////////////////////////////////////////////////////
proc rowred (matrix A,list #)
"USAGE: rowred(A[,e]); A matrix, e any type
RETURN: - a matrix B, being the row reduced form of A, if rowred is called
with one argument.
(as far as this is possible over the polynomial ring; no division
by polynomials)
@* - a list L of two matrices, such that L[1] = L[2] * A with L[1]
the row-reduced form of A and L[2] the transformation matrix
(if rowred is called with two arguments).
ASSUME: The entries of A are in the base field. It is not verified whether
this assumption holds.
NOTE: * This procedure is designed for teaching purposes mainly.
@* * The straight forward Gaussian algorithm is implemented in the
library (no standard basis computation).
The transformation matrix is obtained by concatenating a unit
matrix to A. proc gauss_row should be faster.
@* * It should only be used with exact coefficient field (there is no
pivoting) over the polynomial ring (ordering lp or dp).
@* * Parameters are allowed. Hence, if the entries of A are parameters
the computation takes place over the field of rational functions.
SEE ALSO: gauss_row
EXAMPLE: example rowred; shows an example
"
{
int m,n=nrows(A),ncols(A);
int i,j,k,l,rk;
poly p;
matrix d[m][n];
for (i=1;i<=m;i++)
{ for (j=1;j<=n;j++)
{ p = A[i,j];
if (ord(p)==0)
{ if (deg(p)==0) { d[i,j]=p; }
}
}
}
matrix b = A;
if (size(#) != 0) { b = concat(b,unitmat(m)); }
for (l=1;l<=n;l=l+1)
{
k = findfirst(ideal(d[l]),rk+1);
if (k)
{ rk = rk+1;
b = permrow(b,rk,k);
p = b[rk,l]; p = 1/p;
b = multrow(b,rk,p);
for (i=1;i<=m;i++)
{
if (rk-i) { b = addrow(b,rk,-b[i,l],i);}
}
d = 0;
for (i=rk+1;i<=m;i++)
{ for (j=l+1;j<=n;j++)
{ p = b[i,j];
if (ord(p)==0)
{ if (deg(p)==0) { d[i,j]=p; }
}
}
}
}
}
d = submat(b,1..m,1..n);
if (size(#))
{
list L=d,submat(b,1..m,n+1..n+m);
return(L);
}
return(d);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=(0,a,b),(A,B,C),dp;
matrix m[6][8]=
0, 0, b*B, -A,-4C,2A,0, 0,
2C,-4C,-A,B, 0, B, 3B,AB,
0,a*A, 0, 0, B, 0, 0, 0,
0, 0, 0, 0, 2, 0, 0, 2A,
0, 0, 0, 0, 0, 0, 2b, A,
0, 0, 0, 0, 0, 0, 0, 2a;"";
print(rowred(m));"";
list L=rowred(m,1);
print(L[1]);
print(L[2]);
}
///////////////////////////////////////////////////////////////////////////////
proc colred (matrix A,list #)
"USAGE: colred(A[,e]); A matrix, e any type
RETURN: - a matrix B, being the column reduced form of A, if colred is
called with one argument.
(as far as this is possible over the polynomial ring;
no division by polynomials)
@* - a list L of two matrices, such that L[1] = A * L[2] with L[1]
the column-reduced form of A and L[2] the transformation matrix
(if colred is called with two arguments).
ASSUME: The entries of A are in the base field. It is not verified whether
this assumption holds.
NOTE: * This procedure is designed for teaching purposes mainly.
@* * It applies rowred to the transposed matrix.
proc gauss_col should be faster.
@* * It should only be used with exact coefficient field (there is no
pivoting) over the polynomial ring (ordering lp or dp).
@* * Parameters are allowed. Hence, if the entries of A are parameters
the computation takes place over the field of rational functions.
SEE ALSO: gauss_col
EXAMPLE: example colred; shows an example
"
{
A = transpose(A);
if (size(#))
{ list L = rowred(A,1); return(transpose(L[1]),transpose(L[2]));}
else
{ return(transpose(rowred(A)));}
}
example
{ "EXAMPLE:"; echo = 2;
ring r=(0,a,b),(A,B,C),dp;
matrix m[8][6]=
0, 2*C, 0, 0, 0, 0,
0, -4*C,a*A, 0, 0, 0,
b*B, -A, 0, 0, 0, 0,
-A, B, 0, 0, 0, 0,
-4*C, 0, B, 2, 0, 0,
2*A, B, 0, 0, 0, 0,
0, 3*B, 0, 0, 2b, 0,
0, AB, 0, 2*A,A, 2a;"";
print(colred(m));"";
list L=colred(m,1);
print(L[1]);
print(L[2]);
}
//////////////////////////////////////////////////////////////////////////////
proc linear_relations(module M)
"USAGE: linear_relations(M);
M: a module
ASSUME: All non-zero entries of M are homogeneous polynomials of the same
positive degree. The base field must be an exact field (not real
or complex).
It is not checked whether these assumptions hold.
RETURN: a maximal module R such that M*R is formed by zero vectors.
EXAMPLE: example linear_relations; shows an example.
"
{ int n = ncols(M);
def BaseR = basering;
def br = changeord(list(list("dp",1:nvars(basering))));
setring br;
module M = imap(BaseR,M);
ideal vars = maxideal(1);
ring tmpR = 0, ('y(1..n)), dp;
def newR = br + tmpR;
setring newR;
module M = imap(br,M);
ideal vars = imap(br,vars);
attrib(vars,"isSB",1);
for (int i = 1; i<=n; i++) {
M[i] = M[i] + 'y(i)*gen(1);
}
M = interred(M);
module redM = NF(M,vars);
module REL;
int sizeREL;
int j;
for (i=1; i<=n; i++) {
if (M[i][1]==redM[i][1]) { //-- relation found!
sizeREL++;
REL[sizeREL]=0;
for (j=1; j<=n; j++) {
REL[sizeREL] = REL[sizeREL] + (M[i][1]/'y(j))*gen(j);
}
}
}
setring BaseR;
return(minbase(imap(newR,REL)));
}
example
{ "EXAMPLE:"; echo = 2;
ring r = (3,w), (a,b,c,d),dp;
minpoly = w2-w-1;
module M = [a2,b2],[wab,w2c2+2b2],[(w-2)*a2+wab,wb2+w2c2];
module REL = linear_relations(M);
pmat(REL);
pmat(M*REL);
}
//////////////////////////////////////////////////////////////////////////////
static proc findfirst (ideal i,int t)
{
int n,k;
for (n=t;n<=ncols(i);n=n+1)
{
if (i[n]!=0) { k=n;break;}
}
return(k);
}
//////////////////////////////////////////////////////////////////////////////
proc rm_unitcol(matrix A)
"USAGE: rm_unitcol(A); A matrix (being row-reduced)
RETURN: matrix, obtained from A by deleting unit columns (having just one 1
and else 0 as entries) and associated rows
EXAMPLE: example rm_unitcol; shows an example
"
{
int l,j;
intvec v;
for (j=1;j<=ncols(A);j++)
{
if (gen(l+1)==module(A)[j]) {l=l+1;}
else { v=v,j;}
}
if (size(v)>1)
{ v = v[2..size(v)];
return(submat(A,l+1..nrows(A),v));
}
else
{ return(0);}
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(A,B,C),dp;
matrix m[6][8]=
0, 0, A, 0, 1,0, 0,0,
0, 0, -C2, 0, 0,0, 1,0,
0, 0, 0,1/2B, 0,0, 0,1,
0, 0, B, -A, 0,2A, 0,0,
2C,-4C, -A, B, 0,B, 0,0,
0, A, 0, 0, 0,0, 0,0;
print(rm_unitcol(m));
}
//////////////////////////////////////////////////////////////////////////////
proc rm_unitrow (matrix A)
"USAGE: rm_unitrow(A); A matrix (being col-reduced)
RETURN: matrix, obtained from A by deleting unit rows (having just one 1
and else 0 as entries) and associated columns
EXAMPLE: example rm_unitrow; shows an example
"
{
int l,j;
intvec v;
module M = transpose(A);
for (j=1; j <= nrows(A); j++)
{
if (gen(l+1) == M[j]) { l=l+1; }
else { v=v,j; }
}
if (size(v) > 1)
{ v = v[2..size(v)];
return(submat(A,v,l+1..ncols(A)));
}
else
{ return(0);}
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(A,B,C),dp;
matrix m[8][6]=
0,0, 0, 0, 2C, 0,
0,0, 0, 0, -4C,A,
A,-C2,0, B, -A, 0,
0,0, 1/2B,-A,B, 0,
1,0, 0, 0, 0, 0,
0,0, 0, 2A,B, 0,
0,1, 0, 0, 0, 0,
0,0, 1, 0, 0, 0;
print(rm_unitrow(m));
}
//////////////////////////////////////////////////////////////////////////////
proc headStand(matrix M)
"USAGE: headStand(M); M matrix
RETURN: matrix B such that B[i][j]=M[n-i+1,m-j+1], n=nrows(M), m=ncols(M)
EXAMPLE: example headStand; shows an example
"
{
int i,j;
int n=nrows(M);
int m=ncols(M);
matrix B[n][m];
for(i=1;i<=n;i++)
{
for(j=1;j<=m;j++)
{
B[n-i+1,m-j+1]=M[i,j];
}
}
return(B);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(A,B,C),dp;
matrix M[2][3]=
0,A, B,
A2, B2, C;
print(M);
print(headStand(M));
}
//////////////////////////////////////////////////////////////////////////////
// Symmetric/Exterior powers thanks to Oleksandr Iena for his persistence ;-)
proc symmetricBasis(int n, int k, list #)
"USAGE: symmetricBasis(n, k[,s]); n int, k int, s string
RETURN: ring, poynomial ring containing the ideal \"symBasis\",
being a basis of the k-th symmetric power of an n-dim vector space.
NOTE: The output polynomial ring has characteristics 0 and n variables
named \"S(i)\", where the base variable name S is either given by the
optional string argument(which must not contain brackets) or equal to
"e" by default.
SEE ALSO: exteriorBasis
KEYWORDS: symmetric basis
EXAMPLE: example symmetricBasis; shows an example"
{
//------------------------ handle optional base variable name---------------
string S = "e";
if( size(#) > 0 )
{
if( typeof(#[1]) != "string" )
{
ERROR("Wrong optional argument: must be a string");
}
S = #[1];
if( (find(S, "(") + find(S, ")")) > 0 )
{
ERROR("Wrong optional argument: must be a string without brackets");
}
}
//------------------------- create ring container for symmetric power basis-
execute("ring @@@SYM_POWER_RING_NAME=(0),("+S+"(1.."+string(n)+")),dp;");
//------------------------- choose symmetric basis -------------------------
ideal symBasis = maxideal(k);
//------------------------- export and return -------------------------
export symBasis;
return(basering);
}
example
{ "EXAMPLE:"; echo = 2;
// basis of the 3-rd symmetricPower of a 4-dim vector space:
def R = symmetricBasis(4, 3, "@e"); setring R;
R; // container ring:
symBasis; // symmetric basis:
}
//////////////////////////////////////////////////////////////////////////////
proc exteriorBasis(int n, int k, list #)
"USAGE: exteriorBasis(n, k[,s]); n int, k int, s string
RETURN: qring, an exterior algebra containing the ideal \"extBasis\",
being a basis of the k-th exterior power of an n-dim vector space.
NOTE: The output polynomial ring has characteristics 0 and n variables
named \"S(i)\", where the base variable name S is either given by the
optional string argument(which must not contain brackets) or equal to
"e" by default.
SEE ALSO: symmetricBasis
KEYWORDS: exterior basis
EXAMPLE: example exteriorBasis; shows an example"
{
//------------------------ handle optional base variable name---------------
string S = "e";
if( size(#) > 0 )
{
if( typeof(#[1]) != "string" )
{
ERROR("Wrong optional argument: must be a string");
}
S = #[1];
if( (find(S, "(") + find(S, ")")) > 0 )
{
ERROR("Wrong optional argument: must be a string without brackets");
}
}
//------------------------- create ring container for symmetric power basis-
execute("ring @@@EXT_POWER_RING_NAME=(0),("+S+"(1.."+string(n)+")),dp;");
//------------------------- choose exterior basis -------------------------
def T = superCommutative(); setring T;
ideal extBasis = simplify( NF(maxideal(k), std(0)), 1 + 2 + 8 );
//------------------------- export and return -------------------------
export extBasis;
return(basering);
}
example
{ "EXAMPLE:"; echo = 2;
// basis of the 3-rd symmetricPower of a 4-dim vector space:
def r = exteriorBasis(4, 3, "@e"); setring r;
r; // container ring:
extBasis; // exterior basis:
}
//////////////////////////////////////////////////////////////////////////////
static proc chooseSafeVarName(string prefix, string suffix)
"USAGE: give appropreate prefix for variable names
RETURN: safe variable name (repeated prefix + suffix)
"
{
string V = varstr(basering);
string S = suffix;
while( find(V, S) > 0 )
{
S = prefix + S;
}
return(S);
}
//////////////////////////////////////////////////////////////////////////////
static proc mapPower(int p, module A, int k, def Tn, def Tm)
"USAGE: by both symmetric- and exterior-Power"
NOTE: everything over the basering!
module A (matrix of the map), int k (power)
rings Tn is source- and Tm is image-ring with bases
resp. Ink and Imk.
M = max dim of Image, N - dim. of source
SEE ALSO: symmetricPower, exteriorPower"
{
def save = basering;
int n = nvars(save);
int M = nrows(A);
int N = ncols(A);
int i, j;
//------------------------- compute matrix of single images ------------------
def Rm = save + Tm; setring Rm;
dbprint(p-2, "Temporary Working Ring", Rm);
module A = imap(save, A);
ideal B; poly t;
for( i = N; i > 0; i-- )
{
t = 0;
for( j = M; j > 0; j-- )
{
t = t + A[i][j] * var(n + j);
}
B[i] = t;
}
dbprint(p-1, "Matrix of single images", B);
//------------------------- compute image ---------------------
// apply S^k(A): Tn -> Rm to Source basis vectors Ink:
map TMap = Tn, B;
ideal C = NF(TMap(Ink), std(0));
dbprint(p-1, "Image Matrix: ", C);
//------------------------- write it in Image basis ---------------------
ideal Imk = imap(Tm, Imk);
module D; poly lm; vector tt;
for( i = ncols(C); i > 0; i-- )
{
t = C[i];
tt = 0;
while( t != 0 )
{
lm = leadmonom(t);
// lm;
for( j = ncols(Imk); j > 0; j-- )
{
if( lm / Imk[j] != 0 )
{
tt = tt + (lead(t) / Imk[j]) * gen(j);
break;
}
}
t = t - lead(t);
}
D[i] = tt;
}
//------------------------- map it back and return ---------------------
setring save;
return( imap(Rm, D) );
}
//////////////////////////////////////////////////////////////////////////////
proc symmetricPower(module A, int k)
"USAGE: symmetricPower(A, k); A module, k int
RETURN: module: the k-th symmetric power of A
NOTE: the chosen bases and most of intermediate data will be shown if
printlevel is big enough
SEE ALSO: exteriorPower
KEYWORDS: symmetric power
EXAMPLE: example symmetricPower; shows an example"
{
int p = printlevel - voice + 2;
def save = basering;
int M = nrows(A);
int N = ncols(A);
string S = chooseSafeVarName("@", "@_e");
//------------------------- choose source basis -------------------------
def Tn = symmetricBasis(N, k, S); setring Tn;
ideal Ink = symBasis;
export Ink;
dbprint(p-3, "Temporary Source Ring", basering);
dbprint(p, "S^k(Source Basis)", Ink);
//------------------------- choose image basis -------------------------
def Tm = symmetricBasis(M, k, S); setring Tm;
ideal Imk = symBasis;
export Imk;
dbprint(p-3, "Temporary Image Ring", basering);
dbprint(p, "S^k(Image Basis)", Imk);
//------------------------- compute and return S^k(A) in chosen bases --
setring save;
return(mapPower(p, A, k, Tn, Tm));
}
example
{ "EXAMPLE:"; echo = 2;
ring r = (0),(a, b, c, d), dp; r;
module B = a*gen(1) + c* gen(2), b * gen(1) + d * gen(2); print(B);
// symmetric power over a commutative K-algebra:
print(symmetricPower(B, 2));
print(symmetricPower(B, 3));
// symmetric power over an exterior algebra:
def g = superCommutative(); setring g; g;
module B = a*gen(1) + c* gen(2), b * gen(1) + d * gen(2); print(B);
print(symmetricPower(B, 2)); // much smaller!
print(symmetricPower(B, 3)); // zero! (over an exterior algebra!)
}
//////////////////////////////////////////////////////////////////////////////
proc exteriorPower(module A, int k)
"USAGE: exteriorPower(A, k); A module, k int
RETURN: module: the k-th exterior power of A
NOTE: the chosen bases and most of intermediate data will be shown if
printlevel is big enough. Last rows will be invisible if zero.
SEE ALSO: symmetricPower
KEYWORDS: exterior power
EXAMPLE: example exteriorPower; shows an example"
{
int p = printlevel - voice + 2;
def save = basering;
int M = nrows(A);
int N = ncols(A);
string S = chooseSafeVarName("@", "@_e");
//------------------------- choose source basis -------------------------
def Tn = exteriorBasis(N, k, S); setring Tn;
ideal Ink = extBasis;
export Ink;
dbprint(p-3, "Temporary Source Ring", basering);
dbprint(p, "E^k(Source Basis)", Ink);
//------------------------- choose image basis -------------------------
def Tm = exteriorBasis(M, k, S); setring Tm;
ideal Imk = extBasis;
export Imk;
dbprint(p-3, "Temporary Image Ring", basering);
dbprint(p, "E^k(Image Basis)", Imk);
//------------------------- compute and return E^k(A) in chosen bases --
setring save;
return(mapPower(p, A, k, Tn, Tm));
}
example
{ "EXAMPLE:"; echo = 2;
ring r = (0),(a, b, c, d, e, f), dp;
r; "base ring:";
module B = a*gen(1) + c*gen(2) + e*gen(3),
b*gen(1) + d*gen(2) + f*gen(3),
e*gen(1) + f*gen(3);
print(B);
print(exteriorPower(B, 2));
print(exteriorPower(B, 3));
def g = superCommutative(); setring g; g;
module A = a*gen(1), b * gen(1), c*gen(2), d * gen(2);
print(A);
print(exteriorPower(A, 2));
module B = a*gen(1) + c*gen(2) + e*gen(3),
b*gen(1) + d*gen(2) + f*gen(3),
e*gen(1) + f*gen(3);
print(B);
print(exteriorPower(B, 2));
print(exteriorPower(B, 3));
}
//////////////////////////////////////////////////////////////////////////////
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