# CASA Function: realroot_a

Isolate real roots of a polynomial with algebraic number coefficients in strictly seperated intervals using de-recursive algorithms and norm.

### Calling Sequence:

• r := realroot_a(minimal,interval,poly)
• r := realroot_a(minimal,interval,poly,widthgoal)

### Parameters:

minimal : polynom
• the defining polynomial of an algebraic number alpha
interval : list
• an isolating interval for alpha
poly : polynom
• a square-free univariate polynomial with algebraic number coefficients ( alpha).
widthgoal : numeric
• maximal size of each isolating interval.

### Result:

R : list
• a sorted list of strictly separated isolating intervals for all real roots of the polynomial poly

### Description:

• The function returns a sorted list of strictly separated isolating intervals for all real roots of the polynomial poly.
• The width of the interval is less than or equal to the optional parameter widthgoal (a positive number). If widthgoal is omitted, the most convenient width is used for each interval returned. The polynomial with algebraic number coefficients has to be square-free.

### Examples:

> eq1 := expand(2*x^4 - 3*x^2*y + y^2*(y-1)^2);

> Ay := 8*y^2 - 16*y - 1:

> a := RootOf(Ay):

> eq2 := subs(y=a, eq1):

> # Algebraic sqrfree

> Geq2 := evala(Gcd(eq2, diff(eq2, x))):

> evala(Divide(eq2, Geq2, evaln(eq2))):

> eq2 := primpart(subs(a=y, eq2));

> realroot_a(Ay, [-10,10], eq2);