S. Awad, Ph.D.

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Symbolic Math Toolbox
Math Review with Matlab
Fundamentals

• S. Awad, Ph.D.
• M. Corless, M.S.E.E.
• E.C.E. Department
• University of Michigan

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Fundamentals of Matlabs Symbolic Toolbox

• 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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Defining Symbolic Variables
xsym('x')

• Use sym to create a symbolic variable x

syms y a b

• Use syms to create several symbolic variables at
  one time

who Your variables are a b x y

• Use who to view al variables in the workspace

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Viewing Workspace Variables

• Use whos to view all workspace variables with
  their associated size, bytes, and class
  information

n1.0t1.1 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 Expressions

• Symbolic Expressions

f 2x2 x 1 g ax2 bx 5 g
ax2bx5

• Symbolic and Numerical Conversions to perform a
  mathematical operation and create a new symbolic
  variable delta

delta sym('1sqrt(2)/2') f delta2
delta f (11/22(1/2))211/22(1/2)
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Numerical Representation

• 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 Representation Example
rho(1sqrt(5)/2)

• Double-Precision
• Floating Point
• Variable

rho 2.1180
rho_float sym(rho,'f') rho_float
'1.0f1bbcdcbfa54'2(1)
rho_decimal sym(rho,'d') rho_decimal
2.1180339887498949025257388711907
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Digits Command

• 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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Digits Example
digits Digits 32 rho(1sqrt(5)/2)
rho_decimal sym(rho,'d')

• Default Precision (32 Digits)

rho_decimal 2.1180339887498949025257388711907
digits(7) rho_decimal_7sym(rho,'d')
rho_decimal_7 2.118034
Adjusted Precision (7 Digits)
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Double Command

• The double command coverts a symbolic variable to
  a general Matlab double floating point number

xsym(3)ysym(4) z_sym x/y z_sym 3/4

Symbolic Variable
z_float double(z_sym) z_float 0.7500
Double Float Variable
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Declaring Real Variables

• To declare real symbolic variables

x sym('x','real') y sym('y','real')

• Or use shorthand notation

syms x y real who Your variables are x
y
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Declaring Complex Variables
syms x y zxiy or zxjy z xiy

• To construct a complex number use i or j to
  represent the imaginary part

z_real real(z) z_real x

• Use real to find the real part

• Use imag to find the imaginary part

z_imag imag(z) z_imag y
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Unreal

• The 'unreal' argument to sym can be used to
  convert a real variable to a purely formal
  variable with no additional properties

xsym('x','real') conj(x) ans x

• If x is real, the complex conjugate of x will be x

• If x is unreal, the complex conjugate of can not
  be further simplified

xsym('x','unreal') conj(x) ans conj(x)
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Abstract Functions

• A symbolic variable can represent an abstract
  function fsym('f(x)')where the input argument
  is a string

• Abstract functions are useful for solving
  algebraic and differential equations

fsym('2x2') f 2x2
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Abstract Function Example
syms a b c z a 0 0 0 b 0 0 0 c z a,
0, 0 0, b, 0 0, 0, c

• Find the determinant and inverse of the matrix z

determinant det(z) determinant abc
inverse inv(z) inverse 1/a, 0, 0
0, 1/b, 0 0, 0, 1/c
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Matrix Manipulation Example

• Change the first element of the matrix from a to
  g

z(1,1)'g' z g, 0, 0 0, b, 0 0,
0, c
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Summary

• 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