Sturm

Sturm is a MATLAB toolbox implementing two classes

• Poly, which implements a polynomial object
• Sturm, which implements the Sturm algorithm for real roots computations.

Features

Poly class store and manipulate a polynomial:

• easy initialization
• +,-,* operations implemented on the objects
• division of polynomial with remainder
• derivative and integral of a polynomial

Sturm class store and manipilate Sturm sequence for real roots separation and computation:

• build Sturm sequence
• build intervals separating roots
• compute all the real roots in an interval

Class Poly

Build a polynomial

build empy polynomial

p = Poly();
p.print();

build a polynomial

q = Poly([1,2,3,4,5]);
q.print();

build a monomial x+3

q.set_monomial(3);
q.print();

setup a polynomial

q.set_by_coeffs([5,4,3,2,1]);
q.print();

scale a polynomial in such a way max absolute value of polynomial coefficients is 1

q.normalize();
q.print();

Evaluate polynomial

evaluate polynomial on sampled values

y = q.eval([1,2,3,4,5]);
disp(y);

evaluate polynomial derivative

y = q.eval_D([1,2,3,4,5]);
disp(y);

Perform some basic operations

Build

p = Poly([1,2,3]);     % build a polynomial
q = Poly([1,2,3,4,5]); % build a polynomial
fprintf('p(x) = %s\n',p.to_string);
fprintf('q(x) = %s\n',q.to_string);

addition

res = p+q;
fprintf('p(x)+q(x) = %s\n',res.to_string);

scalar addition

res = 1+p;
fprintf('p(x)   = %s\n1+p(x) = %s\n',p.to_string,res.to_string);

scalar addition and subtraction

res = p+1;
fprintf('1+p(x) = %s\n',res.to_string);

res = p-1;
fprintf('p(x)-1 = %s\n',res.to_string);

res = 1-p;
fprintf('1-p(x) = %s\n',res.to_string);

polynomial multiplications

res = p*q;
fprintf('p(x)*q(x) = %s\n',res.to_string);

% multiplications by a scalar
res = p*10;
fprintf('p(x)*10 = %s\n',res.to_string);

% multiplications by a scalar
res = 3*p;
fprintf('p(x)*10 = %s\n',res.to_string);

Integral and derivative

Integral

Iq = q.integral;
fprintf('q(x)        = %s\nint(q(x),x) = %s\n',q.to_string,Iq.to_string);

Derivative

Dq = q.derivative;
fprintf('q(x)  = %s\nq''(x) = %s\n',q.to_string,Dq.to_string);

Division with remainder

p.set_by_coeffs([1,0,-3,5,0,3,0,2]);
[s,r] = p.divide(q);
fprintf('p(x)  = %s\n',p.to_string);
fprintf('q(x)  = %s\n',q.to_string);
fprintf('p(x)/q(x) = %s\n',s.to_string);
fprintf('remainder = %s\n',r.to_string);

% check operation
res = q*s+r;
fprintf('q(x)*s(x)+r(x) = %s\n',res.to_string);
res = res - p;
fprintf('q(x)*s(x)+r(x)-p(x) = %s\n',res.to_string);

set to 0 coefficients less than epsi

epsi = 100*eps;
res.purge(epsi);
fprintf('q(x)*s(x)+r(x)-p(x) = %s\n',res.to_string);

Greater Common Divisor

set GCD a multiple of polynomial g = 1+2x+3x^2

% GCD
g   = Poly([1,2,3]);
q   = q*g;
p   = p*g;
res = p.GCD(q);
fprintf('p(x) = %s\n',p.to_string);
fprintf('q(x) = %s\n',q.to_string);
fprintf('GCD(p(x),q(x)) = %s\n',res.to_string);

Class Sturm

build a Sturm sequence from a polynomial

S = Sturm();
S.build(p);
S.print();

separate roots

S.separate_roots(-10,10);
S.print();

x = -2:0.01:2;
y = p.eval(x);
plot(x,y);

refine roots

S.refine_roots();
S.print();
p.eval(S.roots())

Reference

• en.wikipedia.org/wiki/Sturm%27s_theorem
• en.wikipedia.org/wiki/Jacques_Charles_François_Sturm
• lara.epfl.ch/w/_media/sar10/sturms_proof.pdf