symbolic math toolbox

S. Awad, Ph.D.

M. Corless, M.S.E.E.

E.C.E. Department

University of Michigan

Math Review with Matlab:

Fundamentals

Symbolic Math Toolbox

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Symbolic Toolbox: Fundamentals

Fundamentals of Matlab’s 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 Use sym to create a

symbolic variable x:

» x=sym('x');

Use syms to create several symbolic variables at one time

Use who to view al variables in the workspace

» who

Your variables are:

a b x y

» syms y a b


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

Viewing Workspace Variables

» 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 Expressions Symbolic Expressions:

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

» f = 2*x^2 + x + 1;» g = a*x^2 + b*x + 5g =a*x^2+b*x+5

» 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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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

Numerical Representation

'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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» rho=(1+sqrt(5)/2)

» rho_float = sym(rho,'f')rho_float = '1.0f1bbcdcbfa54'*2^(1)

» rho_decimal = sym(rho,'d')rho_decimal =2.1180339887498949025257388711907

Symbolic Representation Example

Symbolic

Variables

rho = 2.1180

Double-Precision

Floating Point

Variable


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digits, by itself, displays the current accuracy (default = 32 digits)

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


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Digits Example» digitsDigits = 32» rho=(1+sqrt(5)/2);» rho_decimal = sym(rho,'d')

Default Precision (32 Digits)

Adjusted Precision (7 Digits)

» digits(7)» rho_decimal_7=sym(rho,'d')

rho_decimal_7 =2.118034

rho_decimal =2.1180339887498949025257388711907


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» x=sym(3);y=sym(4);» z_sym = x/yz_sym =3/4 » z_float = double(z_sym)z_float = 0.7500

Double Command The double command coverts a symbolic variable to a

general Matlab double floating point number

Symbolic Variable

Double Float Variable


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Declaring Real Variables To declare real symbolic variables:

Or use shorthand notation:

» x = sym('x','real');» y = sym('y','real');

» syms x y real» whoYour variables are:x y


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Declaring Complex Variables

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

Use real to find the real part

» syms x y

» z=x+i*y; % or z=x+j*yz=x+i*y

Use imag to find the imaginary part

» z_real = real(z)z_real =x

» 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

» x=sym('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

» x=sym('x','unreal');» conj(x)ans =conj(x)


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Abstract functions are useful for solving algebraic and differential equations

Abstract Functions A symbolic variable can represent an abstract

function: f=sym('f(x)')where the input argument is a string

» f=sym('2*x+2') f =2*x+2


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Abstract Function Example Find the

determinant and inverse of the matrix z:

c

b

a

z

00

00

00

» syms a b c» z=[ a 0 0; 0 b 0; 0 0 c]z =[ 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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c

b

a

00

00

00

Matrix Manipulation Example

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

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

Symbolic variables can be declared as real, complex, or converted to the default unreal state

Abstract functions can be created and manipulated symbolically

The double command converts symbolic variables into Matlab double precision floating point variables

symbolic math toolbox

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