symbolic math toolbox

S. Awad, Ph.D.

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

E.C.E. Department

University of Michigan

Math Review with Matlab:

Simplification

Symbolic Math Toolbox

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Symbolic Toolbox: Simplifications and Substitutions

Symbolic Simplifications Pretty Command Factor Command Collect Command Expand Command Simplify Command Simple Command


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Pretty Command

The pretty command can be used to display symbolic expression in a format that resembles type-set mathematics

Pretty

pretty(s) prints the symbolic expression s

pretty(s,n) prints s using screen width n instead of the default 79


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Pretty Examples» syms x» f=x^3 - 6*x^2 + 21*x -6;

» g=(x-1)*(x-2)*(x-3);» pretty(g) (x - 1) (x - 2) (x - 3)

» h=x*(x*(x-6)+11)-6;» pretty(h) x (x (x - 6) + 11) - 6

Product Polynomial

Nested Products

Polynomial» pretty(f) 3 2 x - 6 x + 21 x - 6


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Factor Command The factor(f) command factors f into polynomial

products

» f = x^3 -6*x^2 +11*x -6;» y = factor(f)y =(x-1)*(x-2)*(x-3)

» y = factor(x^5-1)y =(x-1)*(x^4+x^3+x^2+x+1)

)3)(2)(1(116 623 xxxxxx

)1)(1(1 2345 xxxxxx


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Collect Command The collect command collects coefficients of a

symbolic expression and rewrites it as powers of a polynomial

collect(s,v) s is a symbolic expression matrix

v is the independent polynomial variable

If v is omitted, collect uses rules to determine a default variable


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Collect Examples Create symbolic

expression f(x,t)=(1+x)t+xt

» syms x t» f=(1+x)*t+x*tf =(1+x)*t+x*t

Specify collecting the x terms

Specify collecting the t terms

Unspecified independent variable collects the x variable

» f_col_x = collect(f,x)f_col_x =2*x*t+t

» f_col_t = collect(f,t)f_col_t =(1+2*x)*t

» f_col = collect(f)f_col =2*x*t+t


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Expand Command The expand(s) command writes each element of the

symbolic expression s as a product of its factors

Types of expandable expressions include:

Polynomial expressions Trigonometric expressions Exponential expressions Logorithmetic expressions

This is the inverse of the collect command


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Expand Examples

» syms a x y» expand(a*(x+y)) ans = a*x+a*y

» expand(exp(x+y))ans =exp(x)*exp(y)

Polynomial Expansion

Exponential Expansion

ayaxyxa

yxyx eee )(

Polynomial and exponential expansion examples


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Involved Expand Example Given the following function of x:

xxf 1cos3cos)(

1) Expand f(x) by hand to get a polynomial function of x

2) Verify the result using the symbolic expand command


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Expansion Approach To expand f(x) by hand, represent the inverse cosine

portion as a new function z

xxf 1cos3cos)(

xz 1cos

Expand cos(3z) in terms of z Once cos(3z) is expanded, substitute back in z=cos-

1(x)

Let:

zxf 3cos)( Thus:


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Expand cos(3z) Term

zzxf

zzzz

zzzz

zzzz

zzzz

zzzzz

zzzz

zz

zxf

cos3cos4)(

cos2cos2coscos2

cos2cos2coscos2

coscos12coscos2

cossincoscos2

cossin2sincos1cos2

2sinsincos2cos

2cos

3cos)(

3

33

33

23

23

2

Begin by expanding f(x) in terms of z


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Substitute and Simplify From the previous work:

zzxf cos3cos4)( 3 )(cos 1 xz

Substitute:

Simplify:

xxxf 113 coscos3coscos4)(

xxxf 34)( 3


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Expand Verification

This is easily verified in Matlab

» expand( cos(3*acos(x)) )ans =4*x^3-3*x

xxxxf 34cos3cos)( 31


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» syms x» f1=sin(x)^2 + cos(x)^2 + log(x);» f1_smplfy = simplify(f1) f1_smplfy = 1+log(x)

Simplify Command The simplify(s) command performs algebraic,

trigonometric, and logarithmic identities and relationships to simplify each element of the the symbolic matrix s

Trigonometric Identity:

1)(cos)(sin 22 xx


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» syms a b» f=exp(a*log(b));» f_smplfy=simplify(f)f_smplfy =b^a » f_expnd = expand(f)f_expnd = b^a

Simplify Example Simplify the expression:

a

b

bbaf

ebafa

),(

),( ln

baebaf ln),(

Expand gives the same result


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Example Methods for Simplification: Collect Similar Terms Trigonometric Identities Log/Complex Number Relations

Simple Command r = simple(s) tries different algebraic simplifications

and looks for the shortest form of the entire symbolic matrix s. If the result r is not specified, all intermediate steps are displayed to the screen.

[r,how] = simple(s) does not display intermediate simplifications, but returns the shortest form, as well as a string describing the simplification method used


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Simplify Example Use the simple command to simplify the function f from the

previous example and show intermediate steps

» f=exp(a*log(b));» f_smpl=simple(f)simplify:bâ

radsimp:exp(a*log(b))

combine(trig):exp(a*log(b))

convert(sincos):exp(a*log(b))

convert(tan):exp(a*log(b))

collect(b):exp(a*log(b)) f_smpl =bâ

abxf )(

factor:exp(a*log(b))

expand:bâ

combine:exp(a*log(b))

convert(exp):exp(a*log(b))


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Best Simplify Method Perform the

simplification again but show only the result

» [f_smpl]=simple(f) f_smpl =b^a

Recall from a previous example that the expand and simplify methods gave the same results

Also show which simplification was used » [f_smpl,how]=simple(f)

f_smpl =b^a how =expand


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Simple Example

» f=sym( '(1+1/2*2^(1/2))^2+1+1/2*2^(1/2)') f =(1+1/2*2^(1/2))^2+1+1/2*2^(1/2) » f_smpl=simple(f)f_smpl =5/2+3/2*2^(1/2)

The simple command can also be used to simplify symbolic mathematical expressions without dependent variables


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Summary The pretty command can be used to display symbolic

expressions in mathematical type-set form

The factor, collect, expand, and simplify commands can be used to reduce a symbolic expression to shorter forms

The simple command implements multiple simplification methods to simplify a symbolic expression to its shortest form

The simple command can also return the best simplification method used to reduce the symbolic expression

symbolic math toolbox

Documents

prettyh x x x

x termsspecify

prettyg x

x variable f

x y ans

x y expanda

following function of

xt xt syms x t f