restart;
Rational Number Reconstruction:Cf := proc (x,tol,it)
#=======================================
# x : a floating point
# tol : tolerance
# it : number of the loop passes in the main proc
# (the tolearance is decresed as the number of iterations increases)
# Output : rational approximation of x using continued fractions
#=======================================
local a, p, q, n, k, B, tol2,max, C, err;
max:=100; #max size of the quotient sequence
a:=numtheory[cfrac](x,max,'quotients');
n:=nops(a);
p[1] := a[1]; if nops(a)=1 then return a[1] end if;
p[2] := a[1]*a[2]+1;
q[1] := 1;
q[2] := a[2];
C[1] := p[1]/q[1];
err[1] := abs(x-C[1]);
#Setting bounds on error and denominator growth
tol2:=tol;
B := ceil(evalf(1/sqrt(2*tol)));
if err[1] <= tol2 and q[1] <= B then
return C[1];
end if;
C[2] := p[2]/q[2];
err[2] := abs(x-C[2]);
if err[2] <= tol2 and q[2] <= B then
return C[2];
end if;
for k from 3 to n do
p[k] := a[k]*p[k-1]+p[k-2];
q[k] := a[k]*q[k-1]+q[k-2];
C[k] := p[k]/q[k];
err[k] := abs(x-C[k]);
if err[k] = 0 then
return C[k];
elif err[k] <= tol2 and q[k] <= B then
return C[k];
elif err[k] = err[k-1] and err[k-1]=err[k-2] then
return C[k];
end if;
end do;
return C[n];
end proc:
#debug(Cf);Complex Rational Number Reconstruction:ComplexCf := proc (x, tol,it)
#=======================================
# x : a complex floating point
# tol : tolerance
# it : number of the loop passes in the main proc
# (the tolearance is decresed as the number of iterations increases)
# Output : rational approximation of x using continued fractions
#=======================================
local y;
if type(x, complex) then
if Im(x) < tol then
y := Cf(Re(x),tol,it);
else
y := Cf(Re(x),tol,it)+I*Cf(Im(x),tol,it) ;
end if;
else
y := Cf(x,tol,it);
end if;
end proc:
#debug(ComplexCf);Vectorel Substitution (an auxiliary proc):subsv := proc (f, x, r)
#===================================
# f : list of polynomials in x
# x : list of variables
# r : list of polynomials in T
# Output : list of f in terms of v
#===================================
local i, j, up_f;
up_f := f;
for i to nops(x) do
up_f := simplify(subs(x[i] = r[i], up_f));
end do;
end proc:
#debug(subsv);Polynomial Rationalization (an auxiliary proc):PolRat := proc (q,tol,it)
#===================================
# q : a polynomial in T
# tol : tolerance
# it : number of the loop passes in the main proc
# (the tolearance is decresed as the number of iterations increases)
# Output : polynomial q with rationalized coefficients
#===================================
local j, cv, RatV;
cv := PolynomialTools[CoefficientVector](q, T);
for j to degree(q, T)+1 do
cv[j] := ComplexCf(cv[j],tol,it);
end do;
RatV := PolynomialTools[FromCoefficientVector](cv, T);
end proc:
#debug(PolRat):Homotopy Solution Matrix (an optional proc):SolMat := proc (f,x)
#===================================
# f : a list of polynomials (square system)
# x : a list of variables
# Output : Homotopy solutions of a square random subsystem
#===================================
local n, i, j, k, Sol, S;
n := nops(x);
Sol := RootFinding[Homotopy](f); #Numerical Solutions
#if Sol=[] then Sol:=fsolve(f,x); end if;
#print(Sol);
k := nops(Sol); # Number of solutions
S:=Matrix(k,n);
for i to k do
for j to n do
S[i, j] := rhs(Sol[i][j]);
end do
end do;
return(S);
#Sol := convert(Sol, Matrix); #Matrix of solutions -only numbers
end proc:
#debug(SolMat);Approximation Solution Matrix -Root Selection\134Elimination (an auxiliary proc):ApproxSolMat := proc (f, x, Sol, tol)
#===================================
# f : list of polynomials (overdetermined system)
# x : list of variables
# Sol : solution matrix
# tol : tolerance
# Output : a matrix of approximated candidate solutions
#===================================
local n, m, fn, Sol2, kn, RowDel, i, j, SolList, SolListi, err, delCount;
n := nops(x);
m := nops(f);
fn := [seq(f[i], i = 1 .. n)];
kn := LinearAlgebra[RowDimension](Sol);
RowDel := proc (A::Matrix, r) A[[1 .. r-1, r+1 .. op(A)[1]], () .. ()] end proc; #prodecude deletes the row r of matrix A
Sol2 := Sol;
print(Sol2);
delCount := 0;
for i to kn do
SolList := LinearAlgebra[Row](Sol, 1 .. kn);
for j from n+1 to m do
SolListi := convert(SolList[i], list);
err := subsv(f[j], x, SolListi);
if tol < evalf(abs(err)) then
print(evalf(abs(err)));
Sol2 := RowDel(Sol2, i-delCount);
delCount := delCount + 1;
end if;
end do;
end do;
return(Sol2);
end proc:
#debug(ApproxSolMat);
Constructing the minimal polynomial:Getq := proc (Sol,u,x)
#===================================
# Sol : Solution Matrix
# x : list of variables
# u : primitive element
# Output : the minimal polynomial q(T)
#===================================
local i, j, d, q, uxi;
q := 1;
d :=LinearAlgebra[RowDimension](Sol);
for i to d do
uxi[i]:=subsv(u,x,Sol[i,..]);
q := expand(q*(T-uxi[i]));
end do;
q := expand(q);
end proc:
#debug(Getq);Constructing RUR:Getr:=proc( Sol, u, x, tol,it)
#===================================
# Sol : Solution Matrix
# x : list of variables
# u : primitive element in T
# tol : tolerance
# Output : list of rational univariate polynomials
#===================================
local d,n,l,uxi,P,i,m,r,j,List;
d,n := LinearAlgebra[Dimension](Sol);
for l from 1 to d do
uxi[l]:=subsv(u,x,Sol[l,..]);
end do;
for j from 1 to d do
P[j]:=product(T-uxi[k],k=1..j-1)*product(T-uxi[k],k=j+1..d);
end do;
List:=[];
for i from 1 to n do
r:=0;
for m from 1 to d do
r:=r+(Sol[m,i]*P[m]);
end do;
List:=[op(List),PolRat(simplify(r),tol,it)];
end do;
end proc:
#debug(Getr):
Local Newton:NewtonStep := proc( f, x, SolMat)
#===================================
# f : a list of polynomials in x
# x : a list of variables
# SolMat : Matrix of approximate solutions
# Output : Matrix of iterated approximate solutions
#===================================
local m, n, d, N, i, J, Jinv, JinvSol, fSol, Sol, SolList, SolM,s,j, S, fn;
n := nops(x);
N := nops(f);
d := LinearAlgebra[RowDimension](SolMat); #number of solutions
fn := [seq(f[i], i = 1 .. n)]; #A square subsystem (takes first n polynomials)
J := VectorCalculus[Jacobian](fn, x);
Jinv := LinearAlgebra[MatrixInverse](J);
for i from 1 to d do
JinvSol:= simplify(subsv(Jinv,x,SolMat[i]));
fSol := subsv(fn,x,SolMat[i]);
Sol[i]:= Vector[row](Vector[column](SolMat[i])-JinvSol.Vector[column](fSol));
end do;
#Updated Solution Matrix:
S:=<Sol[1]>;
for j from 2 to d do
S:=<S,Sol[j]>;
end do;
end proc:
#debug(NewtonStep):Checking the 2nd condition:Checkf := proc(f,x,r,qq,Ratq)
local k,n,v,Check2;
n:=nops(x);
for k from 1 to n do
v[k]:=simplify(r[k]/qq);
end do;
Check2:= map(proc (X) options operator, arrow; rem(X, Ratq, T) end proc, simplify(numer(subsv(f,x,v))));
end proc:Main Loop:Certify := proc(f, x, u, SolMat, tol, it_max)
#================================================
# f : list of polynomials in x
# x : list of variables
# u : a primitive element
# SolMat : Solution Matrix (rows are the coordinates of a solutions)
# tol : tolerance
# it_max : upper bound on the number of iterations
# Output : exact rational univariate representation (RUR)
#================================================
local i, j, n, N, Sol, q, r, qq, Ratq, Check1, Check2, CheckList, good, k,v;
n := nops(x); #number of varibles
N := nops(f); #number of equations
Sol:=SolMat;
#if n = N then
#Sol:=SolMat; #This part can be used, if the solutions are not known.
#else
#Sol := ApproxSolMat(f, x, SolMat, tol) ;
#end if;
### The Main Loop ###
for i from 1 to it_max do
Sol:=NewtonStep(f,x,Sol);
q := Getq(Sol, u, x); #The minimal polynomial
Ratq:=PolRat(q,tol,i); #The minimal polynomial with rationalized coefficients
r := Getr(Sol, u, x, tol,i);#rational univariate polynomials
qq:=diff(Ratq,T);
Check1:= rem(simplify(subsv(u,x,r)-T*qq), Ratq, T);
for k from 1 to n do
v[k]:=simplify(r[k]/qq);
end do;
Check2:= map(proc (X) options operator, arrow; rem(X, Ratq, T) end proc, simplify(numer(subsv(f,x,v))));
### Checking 3 statements:###
good := true;
if gcd(Ratq,qq)<>1 then good := false; end if;
if Check1<>0 then good := false; end if;
for j from 1 to N do
if Check2[j] <> 0 then good := false; end if;
end do;
if good = true then
return(printf("\134n Rational Representation is: \134n %a with minimal polynomial %a \134n number of iterations: %a",r,Ratq,i));
end if;
end do;
print("number of maximum iteration exceeded, no RUR exists.");
end proc:
#debug(Certify):Main Loop: (with Randomized square subsytem input)Certify_R := proc(f, Randf, x, u, SolMat, tol, it_max)
#================================================
# f : list of polynomials in x
# Randf : randomized square subsystem (with full rank Jacobian)
# x : list of variables
# u : a primitive element
# SolMat : Solution Matrix (rows are the coordinates of the solutions)
# tol : tolerance
# it_max : upper bound on the number of iterations
# Output : exact rational univariate representation (RUR)
#================================================
local i, j, n, N, q, r, qq, Ratq, Check1, Check2, CheckList, good, k,up_f,Sol;
n := nops(x); #number of variables
N := nops(f); #number of equations
Sol:=SolMat;
### The Main Loop ###
for i from 1 to it_max do
#Sol:=NewtonStep(Randf,x,Sol);
q := Getq(Sol, u, x);
Ratq:=PolRat(q,tol,i);
r := Getr(Sol, u, x, tol,i);#rational univariate polynomials
qq:=diff(Ratq,T);
Check1:= rem(simplify(subsv(u,x,r)-T*qq), Ratq, T);
Check2:= Checkf(f,x,r,qq,Ratq);
### Checking 3 statements:###
good := true;
if gcd(Ratq,qq)<>1 then good := false; end if;
if Check1<>0 then good := false; end if;
for j from 1 to N do
if Check2[j] <> 0 then good := false; end if;
end do;
if good = true then
return(printf("\134n Rational Representation is: \134n %a with minimal polynomial %a \134n number of iterations: %a",r,Ratq,i));
end if;
Sol:=NewtonStep(Randf,x,Sol);
end do;
print("number of maximum iteration exceeded, no RUR exists.");
end proc:
#debug(Certify_R):4.1 An illustrative example: p.25it:=10; #set an upper bound on number of iterations
f_41:=[64*x1*x2+16*x2,x1^2+x2^2-1,x1-x2^2-x3-1,8*x1-16*x2^2+17];
xxx:=[x1,x2,x3];
u:=x2;
tol:=10^(-5);
z1:=[0.250,0.968,-2.188]; z2:=[-0.250,-0.968,-2.188];
Sol:=Matrix([z1,z2]);
#A randomized square subsystem:
Randf := [(1/4)*f_41[1]+f_41[2], f_41[2]+f_41[3], f_41[2]+f_41[4]];
Certify_R(f_41, Randf, xxx, u, Sol, tol, it); #finds RUR with randomized subsytem input
Certify(f_41, xxx, u, Sol, tol, it); #without randomized square subsystem, the main proc gets the first n polynomial as a square system. With tol:=0.002tol:=0.002;
it:=5;
Certify_R(f_41, Randf, xxx, u, Sol, tol, it);
Certify(f_41, xxx, u, Sol, tol, it);4.2 Clustered Roots Example: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVElU29sN0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJmV2YWxmRidGL0YyLUko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This solution obtained via Bertiniz1:=[+I*1.154700538379822e+00,-I*5.773502691899748e-01, -I*1.154700538379477e+00, +I*5.773502691892773e-01]:z2:=[-0.200000000000000263266e1 , +I*0.173205080756887634555e1,-0.200000000000000007503e1 , -I*0.173205080756887217939e1]:z3:=[0.199999999999999355504658989633e1 ,I*0.173205080756887381138130474341e1,0.199999999999999083209528858593e1 , -I*0.173205080756887594433688804939e1]:z4:=[-0.199999999999999541539242592181e1 , -I*0.173205080756887624280889663887e1,-0.199999999999999862898008525028e1 ,+I*0.173205080756887254266420683042e1]:z5:=[ +I*0.115470053837925381913641368202e1, +I*0.577350269189624924621611036924e0, -I*0.115470053837925410586476726352e1, -I*0.577350269189625162576512437922e0]:z6:=[0.200000000000000071379568135220e1, -I*0.173205080756887583611942418313e1,0.200000000000000018973674060425e1 ,+I*0.173205080756887661083091934619e1]:z7:=[ -I*0.115470053837925344175464891871e1, +I*0.577350269189628009654995874356e0, +I*0.115470053837925276996165547253e1, -I*0.577350269189626860010422698762e0]:z8:=[ -I*0.115470053837925280782847098611e1, -I*0.577350269189626535575605147198e0, +I*0.115470053837925101166556082723e1, +I*0.577350269189625142347592586494e0]:
Sol:=Matrix([z1,z2,z3,z4,z5,z6,z7,z8]):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Solution: 9 Digits 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Solution: 5 Digits 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Solution: 2 Digits accuracySol2:=evalf(Sol,2);Solution: 4 Digits accuracySol4:=evalf(Sol,4);Solution: 3 Digits accuracySol3:=evalf(Sol,3);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RUR: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it_max := 5; Certify_R(g_def, Randg, xxxx, u, Sol4, tol, it_max);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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv4.4 Cyclic Systems:Cyclic System with n variables:QyQ+SShDeWNsaWNmRzYiZio2I0kieEdGJTYnSSJuR0YlSSJnR0YlSSJpR0YlSSVsaXN0RyUqcHJvdGVjdGVkR0kjeHhHRiVGJUYlQyc+OCQtSSVub3BzR0YuNiM5JD44JzciPjgoNyQtSSNvcEdGLkY1Rj0/KDgmIiIiRkEsJkYyRkEhIiJGQUkldHJ1ZUdGLkMkPiY4JTYjRkAtSSRzdW1HRiU2JC1JKHByb2R1Y3RHRiU2JCZGOzYjLCZJImpHRiVGQUkia0dGJUZBL0ZUO0ZBRkAvRlM7RkFGMj5GODckLUY+NiNGOEZHPkY4NyRGZW4tSSdleHBhbmRHRi42IywmLUZONiQmRjs2I0ZTRldGQUZDRkFGJUYlRiVGQw==Construct Cyclic-4 sytem: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Solution Matrix: obtained via 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:Qyw+SSdpdF9tYXhHNiIiIiYiIiI+SSR0b2xHRiUtSSJeRyUqcHJvdGVjdGVkRzYkIiM1ISImRic+SSJ1R0YlLCpJI3gxR0YlRidJI3gyR0YlIiIjSSN4M0dGJSEiIkkjeDRHRiUiIiRGJz5JJXh4eHhHRiU3JkYzRjRGNkY4RictSSpDZXJ0aWZ5X1JHRiU2KUkkZzQ0R0YlSShSYW5kZzQ0R0YlRjtGMUklU29sNUdGJUYpRiRGJw==Certify_R(g44, Randg44, xxxx, u, Sol3, tol, it_max)Construct Cyclic-9 system:QzA+SSd2YXJfZjlHNiI3NEkjeDFHRiVJI3gyR0YlSSN4M0dGJUkjeDRHRiVJI3g1R0YlSSN4NkdGJUkjeDdHRiVJI3g4R0YlSSN4OUdGJUYnRihGKUYqRitGLEYtRi5GLyIiIj5JI2Y5R0YlLUkoQ3ljbGljZkdGJTYkRiQiIipGMD5JJHh4eEdGJTcrRidGKEYpRipGK0YsRi1GLkYvRjA+SSRmMTBHRiUsJkYnRjBGKiEiIkYwPkkkZjExR0YlLCZGJ0YwRi1GPUYwPkkjZzlHRiU3JS1JI29wRyUqcHJvdGVjdGVkRzYjRjJGO0Y/RjA+SSdSYW5kZzlHRiU3KyZGMjYjRjAmRjI2IyIiIyZGMjYjIiIkJkYyNiMiIiUmRjI2IyIiJiZGMjYjIiInJkYyNiNGNkY7Rj9GPQ==JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Capresse System PUR:#restart;
Cr:=[6240+1568*T^2-6176/3*T^4+160/3*T^6, 1560-3688*T^2-1256/3*T^4-40/3*T^6, -9984-13952*T^2-4096/3*T^4+128/3*T^6, -1560+3688*T^2+1256/3*T^4+40/3*T^6]; qC:=1521+5004*T^2+1302*T^4-188/3*T^6+T^8; qq:=diff(qC,T);simplify(Cr[1]/qq);
simplify(Cr[2]/qq);
simplify(Cr[3]/qq);
simplify(Cr[4]/qq);den:=expand(T*(T^6-47*T^4+651*T^2+1251));
gcdex(qC,den,T,'s','t');t;
rem(den*t,qC,T);
v1:=rem(4/3*(5*T^6-193*T^4+147*T^2+585)*t,qC,T);
v2:=rem(-1/3*(5*T^6+157*T^4+1383*T^2-585)*t,qC,T);
v3:=rem(16/3*(T^6-32*T^4-327*T^2-234)*t,qC,T);
v4:=rem(1/3*(5*T^6+157*T^4+1383*T^2-585)*t,qC,T);
v:=[v1,v2,v3,v4];
g := [x_1^3*x_3-4*x_1^2*x_2*x_4-4*x_1*x_2^2*x_3-2*x_2^3*x_4-4*x_1^2-4*x_1*x_3+10*x_2^2+10*x_2*x_4-2, x_1*x_3^3-4*x_1*x_3*x_4^2-4*x_2*x_3^2*x_4-2*x_2*x_4^3-4*x_1*x_3+10*x_2*x_4-4*x_3^2+10*x_4^2-2, 2*x_1*x_2*x_4+x_2^2*x_3-2*x_1-x_3, x_1*x_4^2+2*x_2*x_3*x_4-x_1-2*x_3]; xxxx := [x_1, x_2, x_3, x_4]check:=subsv(g,xxxx,v);
rem(check[1],qC,T);
rem(check[2],qC,T);
rem(check[3],qC,T);
rem(check[4],qC,T);
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