Finding Roots

root(f(var1, var2, ...), var1, [a, b]) Returns the value of var1 to make the function f equal to zero. If a and b are specified, root finds var1 on the interval [a,b]. Otherwise, var1 must be defined with a guess value before root is called. When a guess value is used, root uses the Secant or Mueller method; in the case where root bracketing is used, root uses the Ridder or Brent method.

polyroots(v) Returns a vector containing the roots of the polynomial whose coefficients are in v. By default, polyroots uses a LaGuerre method. If you want to use the Companion Matrix method instead, right-click on the polyroots function and select the desired method from the menu. LaGuerre's method is iterative and searches for solutions in the complex plane. The Companion Matrix method converts the equations into an eigenvalue problem.

Arguments:

• f is a scalar-valued function of any number of variables.
• var1 is a scalar variable found in f, the variable with respect to which the root is found. Define a complex guess to solve for a complex root.
• a, b (optional) are real numbers, a < b, bounds of the interval on which a root is to be found. Since these values bound a root, f(a) and f(b) must be of opposite sign.
• v is a vector containing coefficients of a polynomial in which the first element is the constant term, and 2 ≤ length(v) ≤ 99.

The root function can only solve one equation in one unknown. To solve several equations simultaneously, use Find or Minerr.

Notes:

• The root function is dependent on TOL, but does not respond to a TOL that is larger than 10^-5. This is the maximum convergence criteria. In addition, values of TOL smaller than 10^-12 do not likely produce superior results, and may cause the algorithm not to converge.
• For functions with multiple roots, the root returned depends on the guess value. If the guess value is very close to a minimum or maximum of f, the root function may fail to converge or converge to a root far away from the guess value. In choosing an appropriate guess value or brackets, it helps to graph the function.

• Other problems that may cause the root function to fail to converge, or to converge to an unexpected root are:

  • no roots to the expression
  • the roots are too far from the initial guess
  • local maxima and minima between the guess value and the root
  • discontinuities between the initial guess and the roots
  • the expression has a complex root but the initial guess was real (or vice versa)
  • multiple closely-spaced roots (try reducing the value of TOL to distinguish between them)
  • roots in flat function regions (try reducing the value of TOL, or finding the root of f(x) divided by its derivative)
• Functions with rapid variation can cause the root solver to return tiny complex parts even when a real result is expected.
• If you wish to solve an equation of the form f(x) = g(x), use an expression like
  x0 := root(f(x) − g(x),x).
• If two roots are close together, you may have to reduce TOL to distinguish between them.
• For an expression f(x) with a known root r, solving for additional roots of f(x) is equivalent to solving for roots of h(x) = (f(x))/ (x – r). Dividing out known roots like this is useful for resolving two roots that may be close together. It's often easier to solve for roots of h(x) as defined here than it is to try to find other roots for f(x) with different guesses.
• If f(x) has a small slope near its root, then root(f(x),x) may converge to a value r that is relatively far from the actual root. In such cases, even though |f(r)|< TOL, r may be far from the point where f(r) = 0. To find a more accurate root, decrease the value of TOL. Or, try finding root(g(x),x), where g(x) = [f(x))]/[(d/dx)*f(x)]).
• You can also solve equations in one unknown symbolically, using the solve keyword.

QuickSheet: root

QuickSheet: polyroots