root

Represent the roots of the polynomial ${\mathit{x}}^3+1$ using root. The root function returns a column vector. The elements of this vector represent the three roots of the polynomial.

syms x
p = x^3 + 1;
root(p,x)

ans = 
$$\left(\begin{array}{c}\mathrm{root}\left(x^3+1,x,1\right)\\ \mathrm{root}\left(x^3+1,x,2\right)\\ \mathrm{root}\left(x^3+1,x,3\right)\end{array}\right)$$

$\mathrm{root}\left({\mathit{x}}^3+1,\mathit{x},1\right)$ represents the first root of p, while $\mathrm{root}\left({\mathit{x}}^3+1,\mathit{x},2\right)$ represents the second root, and so on. Use this syntax to represent roots of high-degree polynomials.

Find the roots of the quadratic polynomial $$x^2-x-1$$. You can use the root function to represent these roots.

syms x
p = x^2 - x - 1;
r = root(p,x)

r = 
$$\left(\begin{array}{c}\mathrm{root}\left(x^2-x-1,x,1\right)\\ \mathrm{root}\left(x^2-x-1,x,2\right)\end{array}\right)$$

To convert these roots to high-precision floating point numbers, you can use vpa.

rVpa = vpa(r)

rVpa = 
$$\left(\begin{array}{c}-0.61803398874989484820458683436564\\ 1.6180339887498948482045868343656\end{array}\right)$$

When solving a high-degree polynomial, solve represents the roots by using root. Alternatively, you can either return an explicit solution by using the MaxDegree option or return a numerical result by using vpa.

Find the roots of x^3 + 3*x - 16.

syms x
p = x^3 + 3*x - 16;
R = solve(p,x)

R = 
$$\left(\begin{array}{c}\mathrm{root}\left(z^3+3\hspace{0.17em}z-16,z,1\right)\\ \mathrm{root}\left(z^3+3\hspace{0.17em}z-16,z,2\right)\\ \mathrm{root}\left(z^3+3\hspace{0.17em}z-16,z,3\right)\end{array}\right)$$

Find the roots explicitly by setting the MaxDegree option to the degree of the polynomial. Polynomials with a degree greater than 4 do not have explicit solutions.

Rexplicit = solve(p,x,"MaxDegree",3)

Rexplicit = 
$$\begin{array}{l}\left(\begin{array}{c}{\sigma }_1-\frac{1}{{\sigma }_1}\\ \frac{1}{2\hspace{0.17em}{\sigma }_1}-\frac{{\sigma }_1}{2}-\frac{\sqrt{3}\hspace{0.17em}\left(\frac{1}{{\sigma }_1}+{\sigma }_1\right)\hspace{0.17em}\mathrm{i}}{2}\\ \frac{1}{2\hspace{0.17em}{\sigma }_1}-\frac{{\sigma }_1}{2}+\frac{\sqrt{3}\hspace{0.17em}\left(\frac{1}{{\sigma }_1}+{\sigma }_1\right)\hspace{0.17em}\mathrm{i}}{2}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1={\left(\sqrt{65}+8\right)}^{1/3}\end{array}$$

Calculate the roots numerically by using vpa to convert R to high-precision floating point.

Rnumeric = vpa(R)

Rnumeric = 
$$\left(\begin{array}{c}2.1267693318103912337456401562601\\ -1.0633846659051956168728200781301-2.5283118563671914055545884653776\hspace{0.17em}\mathrm{i}\\ -1.0633846659051956168728200781301+2.5283118563671914055545884653776\hspace{0.17em}\mathrm{i}\end{array}\right)$$

If the call to root contains parameters, substitute the parameters with numbers by using subs before calling vpa.

You can use the root function as input to Symbolic Math Toolbox functions such as simplify, subs, and diff.

Simplify an expression containing root using the simplify function.

syms x
r = root(x^6 + x, x, 1);
simplify(sin(r)^2 + cos(r)^2)

ans = $$1$$

Substitute for parameters in root with numbers using subs.

syms b
subs(root(x^2 + b*x, x, 1), b, 5)

ans = $$\mathrm{root}\left(x^2+5\hspace{0.17em}x,x,1\right)$$

Substituting for parameters using subs is necessary before converting root to numeric form using vpa.

Differentiate an expression containing root with respect to a parameter using diff.

diff(root(x^2 + b*x, x, 1), b)

ans = 
$$\frac{\mathrm{root}\left(x^2+b\hspace{0.17em}x,x,1\right)}{b}$$

Find the inverse Laplace transform of a ratio of two polynomials using ilaplace. The inverse Laplace transform is returned in terms of root.

syms s
G = (s^3 + 1)/(s^6 + s^5 + s^2);
H = ilaplace(G)

H = 
$$t-\left({\sum }_{k=1}^4\frac{{\mathrm{e}}^{t\hspace{0.17em}\mathrm{root}\left(z^4+z^3+1,z,k\right)}}{4\hspace{0.17em}\mathrm{root}\left(z^4+z^3+1,z,k\right)+3}\right)$$

When you get the root function in output, you can use the root function as input in subsequent symbolic calculations. However, if a numerical result is required, convert the root function to a high-precision numeric result using vpa.

Convert the inverse Laplace transform to numeric form using vpa.

H_vpa = simplify(vpa(H))

H_vpa = $$t+0.30881178580997278695808136329347\hspace{0.17em}{\mathrm{e}}^{-1.0189127943851558447865795886366\hspace{0.17em}t}\hspace{0.17em}\cos \left(0.60256541999859902604398442197193\hspace{0.17em}t\right)-0.30881178580997278695808136329347\hspace{0.17em}{\mathrm{e}}^{0.5189127943851558447865795886366\hspace{0.17em}t}\hspace{0.17em}\cos \left(0.666609844932018579153758800733\hspace{0.17em}t\right)-0.6919689479355443779463355813596\hspace{0.17em}{\mathrm{e}}^{-1.0189127943851558447865795886366\hspace{0.17em}t}\hspace{0.17em}\sin \left(0.60256541999859902604398442197193\hspace{0.17em}t\right)-0.16223098826244593894459034019473\hspace{0.17em}{\mathrm{e}}^{0.5189127943851558447865795886366\hspace{0.17em}t}\hspace{0.17em}\sin \left(0.666609844932018579153758800733\hspace{0.17em}t\right)$$