Symbolic Math Toolbox

Introduction

This example shows how to create and use symbolic variables and expressions using Symbolic Math Toolbox™.

For example, the statement

x = sym('x');

produces a symbolic variable named x.

You can combine the statements

a = sym('a'); t = sym('t'); x = sym('x'); y = sym('y');

into one statement involving the "syms" function.

syms a t x y

You can use symbolic variables in expressions and as arguments to many different functions.

r = x^2 + y^2

theta = atan(y/x)

e = exp(i*pi*t)
 
r =
 
x^2 + y^2
 
 
theta =
 
atan(y/x)
 
 
e =
 
exp(pi*t*i)
 

It is sometimes desirable to use the "simplify" function to transform expressions into more convenient forms.

f = cos(x)^2 + sin(x)^2

f = simplify(f)
 
f =
 
cos(x)^2 + sin(x)^2
 
 
f =
 
1
 

Derivatives and integrals are computed by the "diff" and "int" functions.

diff(x^3)

int(x^3)

int(exp(-t^2))
 
ans =
 
3*x^2
 
 
ans =
 
x^4/4
 
 
ans =
 
(pi^(1/2)*erf(t))/2
 

If an expression involves more than one variable, differentiation and integration use the variable which is closest to 'x' alphabetically, unless some other variable is specified as a second argument. In the following vector, the first two elements involve integration with respect to 'x', while the second two are with respect to 'a'.

[int(x^a), int(a^x), int(x^a,a), int(a^x,a)]
 
ans =
 
[ piecewise([a == -1, log(x)], [a ~= -1, x^(a + 1)/(a + 1)]), a^x/log(a), x^a/log(x), piecewise([x == -1, log(a)], [x ~= -1, a^(x + 1)/(x + 1)])]
 

You can also create symbolic constants with the sym function. The argument can be a string representing a numerical value. Statements like pi = sym('pi') and delta = sym('1/10') create symbolic numbers which avoid the floating point approximations inherent in the values of pi and 1/10. The pi created in this way temporarily replaces the built-in numeric function with the same name.

pi = sym('pi')

delta = sym('1/10')

s = sym('sqrt(2)')
 
pi =
 
pi
 
 
delta =
 
1/10
 
 
s =
 
2^(1/2)
 

Conversion of floating point values to symbolic constants involves some consideration of roundoff error. For example, with either of the following statements, the value assigned to t is not exactly one-tenth.

t = 1/10, t = 0.1
t =

    0.1000


t =

    0.1000

The technique for converting floating point numbers is specified by an optional second argument to the sym function. The possible values of the argument are 'f', 'r', 'e' or 'd'. The default is 'r'.

'f' stands for 'floating point'. All values are represented in the form (2^e+N*2^(e-52)) or -(2^e+N*2^(e-52)) where N and e are integers. This captures the floating point values exactly.

sym(t,'f')
 
ans =
 
3602879701896397/36028797018963968
 

'r' stands for 'rational'. Floating point numbers obtained by evaluating expressions of the form p/q, p*pi/q, sqrt(p), 2^q and 10^q for modest sized integers p and q are converted to the corresponding symbolic form. This effectively compensates for the roundoff error involved in the original evaluation, but may not represent the floating point value precisely.

sym(t,'r')
 
ans =
 
1/10
 

If no simple rational approximation can be found, an expression of the form p*2^q with large integers p and q reproduces the floating point value exactly.

sym(1+sqrt(5),'r')
 
ans =
 
910872158600853/281474976710656
 

'e' stands for 'estimate error'. The 'r' form is supplemented by a term involving the variable 'eps' which estimates the difference between the theoretical rational expression and its actual floating point value.

sym(t,'e')
 
ans =
 
eps/40 + 1/10
 

'd' stands for 'decimal'. The number of digits is taken from the current setting of DIGITS used by VPA. Fewer than 16 digits loses some accuracy, while more than 16 digits may not be warranted.

digits(15)
sym(t,'d')

digits(25)
sym(t,'d')
 
ans =
 
0.1
 
 
ans =
 
0.1000000000000000055511151
 

The 25 digit result does not end in a string of 0's, but is an accurate decimal representation of the floating point number nearest to 1/10.

MATLAB® language vector and matrix notation extends to symbolic variables.

n = 4;

A = x.^((0:n)'*(0:n))

D = diff(log(A))
 
A =
 
[ 1,   1,   1,    1,    1]
[ 1,   x, x^2,  x^3,  x^4]
[ 1, x^2, x^4,  x^6,  x^8]
[ 1, x^3, x^6,  x^9, x^12]
[ 1, x^4, x^8, x^12, x^16]
 
 
D =
 
[ 0,   0,   0,    0,    0]
[ 0, 1/x, 2/x,  3/x,  4/x]
[ 0, 2/x, 4/x,  6/x,  8/x]
[ 0, 3/x, 6/x,  9/x, 12/x]
[ 0, 4/x, 8/x, 12/x, 16/x]