double

Convert symbolic numbers to double precision by using double. Symbolic numbers are exact, while double-precision numbers have round-off errors.

Convert $\pi$ and $\frac{1}{3}$ from symbolic form to double precision.

symN = sym([pi 1/3])

symN = 
$$\left(\begin{array}{cc}\pi & \frac{1}{3}\end{array}\right)$$

doubleN = double(symN)

doubleN = 1×2

    3.1416    0.3333

For information on round-off errors, see Recognize and Avoid Round-Off Errors.

Variable-precision numbers created by vpa are symbolic values. When a MATLAB function does not accept symbolic values, convert variable precision to double precision by using double.

Convert $\pi$ and $\frac{1}{3}$ from variable-precision form to double precision.

vpaN = vpa([pi 1/3])

vpaN = $$\left(\begin{array}{cc}3.1415926535897932384626433832795& 0.33333333333333333333333333333333\end{array}\right)$$

doubleN = double(vpaN)

doubleN = 1×2

    3.1416    0.3333

Convert the symbolic numbers in matrix symM to double-precision numbers by using double.

a = sym(sqrt(2));
b = sym(2/3);
symM = [a b; a*b b/a]

symM = 
$$\left(\begin{array}{cc}\sqrt{2}& \frac{2}{3}\\ \frac{2\hspace{0.17em}\sqrt{2}}{3}& \frac{\sqrt{2}}{3}\end{array}\right)$$

doubleM = double(symM)

doubleM = 2×2

    1.4142    0.6667
    0.9428    0.4714

When converting symbolic expressions that suffer from internal cancellation or round-off errors, increase the working precision by using digits before converting the number.

Convert a numerically unstable expression Y with double. Then, increase precision to 100 digits by using digits and convert Y again. This high-precision conversion is accurate, while the low-precision conversion is not.

Y = ((exp(sym(200)) + 1)/(exp(sym(200)) - 1)) - 1;
lowPrecisionY = double(Y)

lowPrecisionY = 
0

digitsOld = digits(100);
highPrecisionY = double(Y) 

highPrecisionY = 
2.7678e-87

Restore the old precision used by digits for further calculations.

digits(digitsOld)

Solve the trigonometric equation $\sin \left(2\mathit{x}\right)+\cos \left(\mathit{x}\right)=0$ by using solve. Set the ReturnConditions option to true to return the complete solution, parameters used in the solution, and conditions on those parameters.

syms x
eqn = sin(2*x) + cos(x) == 0;
[solx,params,conds] = solve(eqn,x,ReturnConditions=true)

solx = 
$$\left(\begin{array}{c}\frac{\pi }{2}+\pi \hspace{0.17em}k\\ 2\hspace{0.17em}\pi \hspace{0.17em}k-\frac{\pi }{6}\\ \frac{7\hspace{0.17em}\pi }{6}+2\hspace{0.17em}\pi \hspace{0.17em}k\end{array}\right)$$

params = $$k$$

conds = 
$$\left(\begin{array}{c}k\in \mathbb{Z}\\ k\in \mathbb{Z}\\ k\in \mathbb{Z}\end{array}\right)$$

The solver does not create the variable $\mathit{k}$ for the parameters in the MATLAB® workspace. Create this variable. Find the solutions for $\mathit{k}=2$ by using subs.

syms k
sols_k2 = subs(solx,k,2)

sols_k2 = 
$$\left(\begin{array}{c}\frac{5\hspace{0.17em}\pi }{2}\\ \frac{23\hspace{0.17em}\pi }{6}\\ \frac{31\hspace{0.17em}\pi }{6}\end{array}\right)$$

The solutions are exact symbolic numbers. Convert these numbers to double-precision numbers.

doublesols_k2 = double(sols_k2)

doublesols_k2 = 3×1

    7.8540
   12.0428
   16.2316

Create a symbolic expression S that represents ${\mathit{A}}^2-2\mathit{A}+{\mathit{I}}_2$, where $\mathit{A}$ is a 2-by-2 symbolic matrix variable.

syms A 2 matrix
S = A*A - 2*A + eye(2)

S = $${\mathrm{I}}_2-2\hspace{0.17em}A+A^2$$

Substitute $\mathit{A}$ with symbolic numbers $\left[\begin{array}{cc}\cos \left(\frac{\pi }{5}\right)& \sin \left(\frac{\pi }{4}\right)\\ -1& 0\end{array}\right]$.

Aval = [cos(sym(pi)/5) sin(pi/4); -1 0];
symS = subs(S,A,Aval)

symS = 
$$\begin{array}{l}-2\hspace{0.17em}{\Sigma }_1+{{\Sigma }_1}^2+{\mathrm{I}}_2\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\Sigma }_1=\left(\begin{array}{cc}\frac{\sqrt{5}}{4}+\frac{1}{4}& \frac{\sqrt{2}}{2}\\ -1& 0\end{array}\right)\end{array}$$

Convert the result to a double-precision matrix.

doubleS = double(symS)

doubleS = 2×2

   -0.6706   -0.8422
    1.1910    0.2929