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Symbolic Math Toolbox

This tutorial will utilize the symbolic math toolbox. This short tutorial will explain some of the basic functionalities that will be used in future tutorials.

The Symbolic Math Toolbox is a powerful tool for calculations, as it lets us use variables instead of numbers. This can be useful to reuse complex matrices or equations and substitute values when needed.

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Create a Symbolic Variable

to create a symbolic variable you can use the syms command followed by your desired variables. 

syms var

If you only work with real numbers, as will be the case for this tutorial, extend the syms with a real to reduce symbolic computations for imaginary numbers. 

syms var real

You can create multiple variables at once, if you extend it with a number, it will be shown as an indices on display

syms var1 var2 var3 real real
MyVars = [var1 var2 var3]
MyVars =

$$ \displaystyle \left(\begin{array}{ccc} {\textrm{var}}_1 & {\textrm{var}}_2 & {\textrm{var}}_3 \end{array}\right) $$

When spelling out greek letters as variables, they be converted into their symbols on display: 

syms alpha beta gamma Delta delta 
MyVars = [alpha, beta, gamma, Delta, delta]
MyVars =

$$ \displaystyle \left(\begin{array}{ccccc} \alpha & \beta & \gamma & \Delta & \delta \end{array}\right) $$

Substituting variables

using the subs() function we can substitute symbolic variables for values, making it easy to reuse e.g. equations :

syms alpha beta x y z real 
Equation1 = x^2 ;
Equation2 = 2 * x + y;
Value1 = subs(Equation1, x, 3)
Value1 = \(\displaystyle 9\)

Value2 = subs(Equation2, [x, y], [3, 1])
Value2 = \(\displaystyle 7\)

We can combine functions like: 

Equation3 = Equation1 + Equation2
Equation3 = \(\displaystyle x^2 +2\,x+y\)

Equation4 = Equation1 * Equation2
Equation4 = \(\displaystyle x^2 \,{\left(2\,x+y\right)}\)

Equation5 = subs(Equation4, [x, y], [alpha, beta])
Equation5 = \(\displaystyle \alpha^2 \,{\left(2\,\alpha +\beta \right)}\)

You can also substitute vectors for variables resulting in a vector output where each row relates to the corresponding input row. 

timevec= linspace(0,10,5)' %this creates an equally spaced row vector from 0 to 10 in 5 steps. 

timevec = 5x1
         0
    2.5000
    5.0000
    7.5000
   10.0000


EquationVector = subs(Equation4,x,timevec)
EquationVector =

$$ \displaystyle \left(\begin{array}{c} 0\newline \frac{25\,y}{4}+\frac{125}{4}\newline 25\,y+250\newline \frac{225\,y}{4}+\frac{3375}{4}\newline 100\,y+2000 \end{array}\right) $$

Converting symbolic variables

Sometimes you get error messages when attempting to combining numerical and symbolic variables. Avoid this by converting one of them:

After substituting the Workspace variable is still considered a symbolic variable. Even if your new variable only contains numbers, MATLAB will consider them a symbolic variable, which may lead to difficulties. You can convert it to a numeric by: 

ValueDouble = double(Value2)

ValueDouble = 7

If you try to insert symbolic variables into numeric variables you need to convert the numeric matrix into a symbolic matrix: 

Matrix_1 = ones(3)

Matrix_1 = 3x3
     1     1     1
     1     1     1
     1     1     1


Matrix_symb = [alpha, beta; 
               gamma, delta]
Matrix_symb =

$$ \displaystyle \left(\begin{array}{cc} \alpha & \beta \newline \gamma & \delta \end{array}\right) $$ 

Matrix_combined = sym(Matrix_1)
Matrix_combined =

$$ \displaystyle \left(\begin{array}{ccc} 1 & 1 & 1\newline 1 & 1 & 1\newline 1 & 1 & 1 \end{array}\right) $$ 

Matrix_combined(1:2,1:2) = Matrix_symb
Matrix_combined =

$$ \displaystyle \left(\begin{array}{ccc} \alpha & \beta & 1\newline \gamma & \delta & 1\newline 1 & 1 & 1 \end{array}\right) $$

Working with Symbolic Variables

The Symbolic Math Toolbox gives us some powerful tools when doing calculus.

Differentiating with Symbolic Variables

We can differentiate an expression like: 

Equation = x^2 + x * y^2 + y^3 + 5
Equation = \(\displaystyle x^2 +x\,y^2 +y^3 +5\)

diff_Eq1 = diff(Equation, x)
diff_Eq1 = \(\displaystyle y^2 +2\,x\)

You can also differentiate w.r.t. multiple variables at once: 

diff_Eq2 = diff(Equation, x, y)
diff_Eq2 = \(\displaystyle 2\,y\)

diff_Eq3 = diff(Equation, x, x)
diff_Eq3 = \(\displaystyle 2\)

diff_Eq4 = diff(Equation, y, y)
diff_Eq4 = \(\displaystyle 2\,x+6\,y\)

This can also be used in a matrix, where each element 

MatrixEquation = [x^2,      x*y,       x*y*z; 
                  x*y,      y^2,       y*z  ; 
                  x*y*z,    y*z,       z^2  ]
MatrixEquation =

$$ \displaystyle \left(\begin{array}{ccc} x^2 & x\,y & x\,y\,z\newline x\,y & y^2 & y\,z\newline x\,y\,z & y\,z & z^2 \end{array}\right) $$ 

diff_Matrix1 = diff(MatrixEquation, x)
diff_Matrix1 =

$$ \displaystyle \left(\begin{array}{ccc} 2\,x & y & y\,z\newline y & 0 & 0\newline y\,z & 0 & 0 \end{array}\right) $$ 

diff_Matrix2 = diff(MatrixEquation, x,y)
diff_Matrix2 =

$$ \displaystyle \left(\begin{array}{ccc} 0 & 1 & z\newline 1 & 0 & 0\newline z & 0 & 0 \end{array}\right) $$ 

diff_Matrix3 = diff(MatrixEquation, x,y,z)
diff_Matrix3 =

$$ \displaystyle \left(\begin{array}{ccc} 0 & 0 & 1\newline 0 & 0 & 0\newline 1 & 0 & 0 \end{array}\right) $$

Rewriting Equations

We can rewrite expressions using the symbolic toolbox: 

Eq = (x + 1) * (x + y)
Eq = \(\displaystyle {\left(x+y\right)}\,{\left(x+1\right)}\)

The collect() function will return an equation where all expressions are pooled based on the desired variable and its identical powers 

Eq_collect_x = collect(Eq, x)
Eq_collect_x = \(\displaystyle x^2 +{\left(y+1\right)}\,x+y\)

The simplify() function lets us factorize an expression and applying cancellation rules

Simple_Eq = simplify(Eq_collect_x)
Simple_Eq = \(\displaystyle {\left(x+y\right)}\,{\left(x+1\right)}\)

Solving Equations

The symbolic toolbox also allows us to solve for variables: 

syms a b real
Eq1 = a * 5 + b == 0
Eq1 = \(\displaystyle 5\,a+b=0\)

Eq2 = a * 15 + b == 4
Eq2 = \(\displaystyle 15\,a+b=4\)

solutions = solve([Eq1, Eq2], [a, b])

solutions = struct with fields:
    a: 2/5
    b: -2

We can also let it solve for a variable that depends on others: 

parameter_solution_a = solve(Eq1, a)
parameter_solution_a = \(\displaystyle -\frac{b}{5}\)