def ALGO4(As,Bs,Cs,Ds,do_test):
#-------------------STEP 1------------------------------
if not is_row_proper(As):
Us,Ast=row_proper(As)
else:
Us,Ast=eye(As.rows),As
Bst=Us*Bs
Bst=expand(Bst)
r=Ast.cols
#-------------------STEP 2------------------------------
K=simplify(Ast.inv()*Bst) #very important
Ys=zeros(K.shape)
for i,j in product(range(K.rows),range(K.cols)):
Ys[i,j],q=div(numer(K[i,j]),denom(K[i,j]))
B_hat=Bst-Ast*Ys
#-------------------END STEP 2------------------------------
#-------------------STEP 3------------------------------
Psi=diag(*[[s**( mc.row_degrees(Ast,s)[j] -i -1) for i in range( mc.row_degrees(Ast,s)[j])] for j in range(r)]).T
S=diag(*[s**(rho) for rho in mc.row_degrees(Ast,s)])
Ahr=mc.highest_row_degree_matrix(Ast,s)
Help=Ast-S*Ahr
SOL={}
numvar=Psi.rows*Psi.cols
alr=symbols('a0:%d'%numvar)
Alr=Matrix(Psi.cols,Psi.rows,alr)
RHS=Psi*Alr
for i,j in product(range(Help.rows),range(Help.cols)): #diagonal explain later
SOL.update(solve_undetermined_coeffs(Eq(Help[i,j],RHS[i,j]),alr,s))
Alr=Alr.subs(SOL) #substitute(SOL)
Aoc=Matrix(BlockDiagMatrix(*[Matrix(rho, rho, lambda i,j: KroneckerDelta(i+1,j))for rho in mc.row_degrees(Ast,s)]))
Boc=eye(sum(mc.row_degrees(Ast,s)))
Coc=Matrix(BlockDiagMatrix(*[SparseMatrix(1,rho,{(x,0):1 if x==0 else 0 for x in range(rho)}) for rho in mc.row_degrees(Ast,s)]))
A0=Aoc-Alr*Ahr.inv()*Matrix(Coc)
C0=Ahr.inv()*Coc
SOL={}
numvar=Psi.cols*Bst.cols
b0=symbols('b0:%d'%numvar)
B0=Matrix(Psi.cols,Bst.cols,b0)
RHS=Psi*B0
for i,j in product(range(B_hat.rows),range(B_hat.cols)): #diagonal explain later
SOL.update(solve_undetermined_coeffs(Eq(B_hat[i,j],RHS[i,j]),b0,s))
B0=B0.subs(SOL) #substitute(SOL)
LHS_matrix=simplify(Cs*C0) #left hand side of the equation (1)
sI_A=s*eye(A0.cols)- A0
max_degree=mc.find_degree(LHS_matrix,s) #get the degree of the matrix at the LHS
#which is also the maximum degree for the coefficients of Λ(s)
#---------------------------Creating Matrices Λ(s) and C -------------------------------------
Lamda=[]
numvar=((max_degree))*A0.cols
a=symbols('a0:%d'%numvar)
for i in range(A0.cols): # paratirisi den douleuei to prin giat;i otra oxi diagonios
p=sum(a[n +i*(max_degree)]*s**n for n in range(max_degree)) # we want variables one degree lower because we are multiplying by first order monomials
Lamda.append(p)
Lamda=Matrix(Cs.rows,A0.cols,Lamda) #convert the list to Matrix
c=symbols('c0:%d'%(Lamda.rows*Lamda.cols))
C=Matrix(Lamda.rows,Lamda.cols,c)
#-----------------------------------------
RHS_matrix=Lamda*sI_A +C #right hand side of the equation (1)
'''
-----------Converting equation (1) to a system of linear -----------
-----------equations, comparing the coefficients of the -----------
-----------polynomials in both sides of the equation (1) -----------
'''
EQ=[Eq(LHS_matrix[i,j],expand(RHS_matrix[i,j])) for i,j in product(range(LHS_matrix.rows),range(LHS_matrix.cols)) ]
def ALGO21(As,Bs,Cs,Ds,do_test):
#-------------------STEP 1------------------------------
r=As.rows
p=Cs.rows
m=Ds.cols
Ts=BlockMatrix([[As,Bs,zeros(As.rows,p)],
[-Cs,Ds,eye(p)],
[zeros(m,Cs.cols),-eye(m),zeros(m,p)]]).as_mutable()
rt=r+p+m
T_over_I=BlockMatrix([[Ts],[eye(rt)]]).as_mutable()
H=reduction.column_proper(T_over_I)
Qs_over_Rs=BlockMatrix([[H[1][0:rt,0:rt]],[H[0]]]).as_mutable()
#-------------------END STEP 1------------------------------
#-------------------STEP 2------------------------------
qi=mc.column_degrees(Qs_over_Rs,s)
Ss=BlockDiagMatrix(*[Matrix(q+1,1,[s**(q-k) for k in range(q+1)]) for q in qi]).as_mutable() ## POS??POSS?
#find Qc
numvar=rt*Ss.rows
a=symbols('a0:%d'%numvar)
Qc=Matrix(rt,Ss.rows,a)
RHS_matrix=(Qc*Ss).as_mutable()
Qs=Qs_over_Rs[0:rt,0:rt]
SOL=dict()
for i,j in product(range(Qs.rows),range(Qs.cols)):
SOL.update(solve_undetermined_coeffs(Eq(Qs[i,j],RHS_matrix[i,j]),a,s))
Qc=Qc.subs(SOL)
#find Rc
numvar=rt*Ss.rows
Rc=Matrix(rt,Ss.rows,a)
RHS_matrix=(Rc*Ss).as_mutable()
Rs=H[0]
SOL=dict()
for i,j in product(range(Rs.rows),range(Rs.cols)):
SOL.update(solve_undetermined_coeffs(Eq(Rs[i,j],RHS_matrix[i,j]),a,s))
Rc=Rc.subs(SOL)
#-------------------END STEP 2------------------------------
#-------------------STEP 3------------------------------
def E0block(i,j):
if i==j and i!=0:
return 1
else:
return 0
E0=BlockDiagMatrix(*[Matrix(q+1,q+1, lambda i,j: E0block(i,j)) for q in qi]).as_mutable()
A0=BlockDiagMatrix(*[Matrix(q+1,q+1, lambda i,j: KroneckerDelta(i-1,j)) for q in qi]).as_mutable()
B0=BlockDiagMatrix(*[Matrix(q+1,1, lambda i,j: KroneckerDelta(i,j)) for q in qi]).as_mutable()
n=sum(qi)+rt
C0=eye(n)
V=BlockMatrix([[zeros(p,r),zeros(p,m),eye(p)]]).as_mutable()
U=BlockMatrix([[zeros(r,m)],[zeros(p,m)],[eye(m)]]).as_mutable()
E=SparseMatrix(E0)
A=SparseMatrix(A0-B0*Qc)
B=SparseMatrix(B0*U)
C=SparseMatrix(V*Rc)
#E=E0.as_mutable()
#pprint(B0*Qc)
#A=(simplify(A0-B0*Qc)).as_mutable()
#B=(B0*U).as_mutable()
#C=(V*Rc).as_mutable()
#-------------------END STEP 3------------------------------
if do_test==True:
test_result=test(B0,Ts,E,A,B,C,U,V,Rc,Rs,Ss,p,m,rt)
else:
test_result= 'not done'
D=zeros(p,m)
return E,A,B,C,D,test_result