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Math Review with Matlab:
Symbolic
Math Toolbox
Fundamentals
S. Awad, Ph.D.
M. Corless, M.S.E.E.
E.C.E. Department
University of Michigan
Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Fundamentals of Matlab’s
Symbolic Toolbox
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Creating Symbolic Variables
Defining Symbolic Expressions
Defining Numerical Representation
Converting Symbolic Variables to Doubles
Creating Real Symbolic Variables
Creating Complex Symbolic Variables
Manipulating Abstract Functions
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Defining Symbolic Variables
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Use sym to create a
symbolic variable x:
Use syms to create
several symbolic
variables at one time
Use who to view al
variables in the
workspace
» x=sym('x');
» syms y a b
» who
Your variables are:
a
b
x
y
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Viewing Workspace Variables
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Use whos to view all workspace variables with their
associated size, bytes, and class information
» n=1.0;t=[1.1 2.2 3.3];
» whos
Name
Size
Bytes
Class
a
1x1
126
sym object
b
1x1
126
sym object
n
1x1
8
double array
t
1x3
24
double array
x
1x1
126
sym object
y
1x1
126
sym object
Grand total is 12 elements using 536 bytes
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Symbolic Expressions
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Symbolic Expressions:
» f = 2*x^2 + x + 1;
» g = a*x^2 + b*x + 5
g =
a*x^2+b*x+5
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Symbolic and Numerical Conversions to perform a mathematical
operation and create a new symbolic variable delta:
» delta = sym('1+sqrt(2)/2');
» f = delta^2 + delta;
f =
(1+1/2*2^(1/2))^2+1+1/2*2^(1/2)
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Numerical Representation
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The command sym(A,flag) converts a numeric
scalar or matrix, A, to symbolic form
The flag argument specifies the technique for
converting floating point numbers
'f' Exactly represents Floating Point values in the form
'1.F'*2^(e) or '-1.F'*2^(e) where F is a string of 13
hexadecimal digits and e is an integer. (This form may
not be convenient for subsequent manipulation)
'd'
Represents Decimal numbers where the number of
digits is taken from the current setting of DIGITS
(described later)
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Symbolic Representation
Example
» rho=(1+sqrt(5)/2)
rho =
2.1180
Double-Precision
Floating Point
Variable
» rho_float = sym(rho,'f')
rho_float =
'1.0f1bbcdcbfa54'*2^(1)
Symbolic
Variables
» rho_decimal = sym(rho,'d')
rho_decimal =
2.1180339887498949025257388711907
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Digits Command
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The digits command is used to set the number
of digits of accuracy used for future numeric
computations on symbolic variables
digits(n) sets accuracy to n digits for
subsequent calculations. Where n represents an
integer
digits, by itself, displays the current accuracy
(default = 32 digits)
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Digits Example
» digits
Default Precision (32 Digits)
Digits = 32
» rho=(1+sqrt(5)/2);
» rho_decimal = sym(rho,'d')
rho_decimal =
2.1180339887498949025257388711907
» digits(7)
» rho_decimal_7=sym(rho,'d')
rho_decimal_7 =
2.118034
Adjusted Precision (7 Digits)
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Double Command
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The double command coverts a symbolic variable to a
general Matlab double floating point number
» x=sym(3);y=sym(4);
» z_sym = x/y
z_sym =
Symbolic Variable
3/4
» z_float = double(z_sym)
z_float =
Double Float Variable
0.7500
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Declaring Real Variables
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To declare real symbolic variables:
» x = sym('x','real');
» y = sym('y','real');
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Or use shorthand notation:
» syms x y real
» who
Your variables are:
x
y
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Declaring Complex
Variables
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To construct a
complex number use
i or j to represent
the imaginary part
» syms x y
» z=x+i*y; % or z=x+j*y
z=
x+i*y
Use real to find the
real part
» z_real = real(z)
z_real =
x
Use imag to find the
imaginary part
» z_imag = imag(z)
z_imag =
y
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Unreal
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The 'unreal' argument to sym can be used to
convert a real variable to a purely formal variable with
no additional properties
If x is real, the
complex conjugate
of x will be x
» x=sym('x','real');
» conj(x)
ans =
x
If x is unreal, the
complex conjugate
of can not be further
simplified
» x=sym('x','unreal');
» conj(x)
ans =
conj(x)
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Abstract Functions
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A symbolic variable can represent an abstract
function: f=sym('f(x)')where the input
argument is a string
Abstract functions are useful for solving algebraic
and differential equations
» f=sym('2*x+2')
f =
2*x+2
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Abstract Function Example
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Find the
determinant and
inverse of the
matrix z:
 a 0 0


z   0 b 0
0 0 c 
»
»
z
[
[
[
syms a b c
z=[ a 0 0; 0 b 0; 0 0 c]
=
a, 0, 0]
0, b, 0]
0, 0, c]
» determinant = det(z)
determinant =
a*b*c
» inverse = inv(z)
inverse =
[ 1/a,
0,
0]
[
0, 1/b,
0]
[
0,
0, 1/c]
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Matrix Manipulation
Example
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Change the first element
of the matrix from a to g:
ag 0 0
 0 b 0


0 0 c 
» z(1,1)='g'
z =
[ g, 0, 0]
[ 0, b, 0]
[ 0, 0, c]
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Symbolic Toolbox: Fundamentals
Math Review with Matlab
U of M-Dearborn ECE Department
Summary
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Matlab can be used to create and manipulate symbolic
variables and expressions
Symbolic variables representing numbers can be
displayed with adjustable accuracy
The double command converts symbolic variables into
Matlab double precision floating point variables
Symbolic variables can be declared as real, complex, or
converted to the default unreal state
Abstract functions can be created and manipulated
symbolically
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