Root

Root[f,k]

represents the exact k root of the polynomial equation f[x]0.

Root[{f1,f2,},{k1,k2,}]

represents the last coordinate of the exact vector {a1,a2,} such that ai is the ki root of the polynomial equation fi[a1,,ai-1,x]0.

Root[{f,x0}]

represents the exact root of the general equation f[x]0 near x=x0.

Root[{f,x0,n}]

represents n roots of the equation f[x]0 near x=x0.

Details and Options  • f must be a Function object such as (#^5-2#+1)&.
• Root[f,k] is automatically reduced so that f has the smallest possible degree and smallest integer coefficients.
• The ordering used by Root[f,k] takes real roots to come before complex ones, and takes complex conjugate pairs of roots to be adjacent.
• The coefficients in the polynomial f[x] can involve symbolic parameters.
• For linear and quadratic polynomials f[x], Root[f,k] is automatically reduced to explicit rational or radical form.
• For other polynomials, ToRadicals can be used to convert to explicit radicals.
• In Root[{f1,f2,},{k1,k2,}], fi must be a Function object with i formal parameters, and fi[a1,,ai-1,x] should be a polynomial in x of degree at least ki.
• If for all i, fi[x1,,xi] is a polynomial in x1,,xi with rational number coefficients, then RootReduce can be used to represent Root[{f1,f2,},{k1,k2,}] in the Root[f,k] form.
• Root[{f,x0}] represents an exact root of the general equation f[x]0, which can be transcendental.
• In Root[{f,x0}], x0 must be an approximate real or complex number such that exactly one root of f[x] lies within the numerical region defined by its precision.
• Root[{f,x0,n}] represents n roots, counting multiplicity, that lie within the numerical region defined by the precision of x0.
• N finds the approximate numerical value of a Root object.
• Operations such as Abs, Re, Round, and Less can be used on Root objects.
• Root[f,k] is treated as a numeric quantity if f contains no symbolic parameters.
• Root by default isolates the complex roots of a polynomial using validated numerical methods. SetOptions[Root,ExactRootIsolation->True] will make Root use symbolic methods that are usually much slower.

Examples

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Basic Examples(3)

Solution to a quintic:

 In:= Out= Numerical values:

 In:= Out= Real solutions to an exp-log equation:

 In:= Out= Real solution to a system of equations:

 In:= Out= Neat Examples(1)

Introduced in 1996
(3.0)
|
Updated in 2012
(9.0)