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Solving system of n equations
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- it can show up in root(), in the form root(polynomial in z, z) or root(polynomial in z, z, number) . These stand in for the set of values, z, such that the polynomial evaluated at z becomes 0 -- the roots of the polynomial. z is a placeholder variable in this situation, that does not necessarily correspond to any of your variables, but one of your variables involves an expression that involves roots of a polynomial. root() was added in R2015b. vpa() of root() will resolve down to specific numeric values only in the case where all of the coefficients of the polynomial are numeric or standard functions applied to a numeric value (e.g., z^5 - cos(pi/7)*z + exp(2))
- it can show up in RootOf(), in the form RootOf(expression in z, z) or RootOf(expression in z, z, number). This is similar to root() but the expression does not have to be a polynomial. These stand in for the set of values, z, such that the expression evalulated at z becomes 0 -- the roots of the expression. z is a placeholder variable in this situation, that does not necessarily correspond to any of your variables, but one of your variables involves an expression that involves roots of an expression. RootOf are more likely to show up in older releases of MATLAB, but can still show up in cases where the expression is not a polynomial. vpa() of RootOf() is only certain to resolve down to specific numeric values only in cases similar to what root() can handle. Because RootOf() might involve more complicated functions such as erfc() or ilaplace or integrals, even when there are no other unbound symbolic variables, MATLAB might not be able to find a software floating point approximation, so vpa() of a RootOf() sometimes returns a RootOf() rather than a software floating point number.
- it can show up in the results of solve() outside of a root() or RootOf() . When this happens, you probably need to add the 'ReturnConditions', true option to solve() and assign the output of solve to a single variable, after which you should look at the Parameters field of the solution. If Parameters is non-empty, it will be a list of variables that the Symbolic Toolbox introduced to make the solution easier to write. For example if x and y both involve the result of calculating an expression, such as x = 3*expression^2 + 7, and y = expression/19, then instead of putting the full value of the expression into x and y, the Symbolic Toolbox might introduce a temporary variable, such as z, and then say x = 3*z^2 + 7, y = z/19 . You then need to look in the Conditions property of the solution to see how the temporary variable is defined or constrained. Typically the Symbolic Toolbox will only define a temporary variable if the variable can have multiple values, but it also sometimes defines a temporary variable for complicated conditions, such as "z has to be such that arccos(sqrt(z-2))/Pi is a positive integer and z = sin(a^3+log(b))" . Sometimes it is possible, with some work, to show that the conditions cannot be met. When a Parameter shows up in the result of solve(), then vpa() will never get rid of the parameter: you have to find solutions for the Conditions and subs() those into the solutions.
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