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

Symbolic Toolbox:Simplifications and Substitutions 2 Symbolic Simplifications n Pretty Command n Factor Command n Collect Command n Expand Command n Simplify Command n Simple Command

Symbolic Toolbox:Simplifications and Substitutions 3 Pretty Command The pretty command can be used to display symbolic expression in a format that resembles type-set mathematics PrettyPretty pretty(s) prints the symbolic expression s pretty(s,n) prints s using screen width n instead of the default 79

Symbolic Toolbox:Simplifications and Substitutions 4 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

Symbolic Toolbox:Simplifications and Substitutions 5 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)

Symbolic Toolbox:Simplifications and Substitutions 6 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

Symbolic Toolbox:Simplifications and Substitutions 7 Collect Examples n Create symbolic expression f(x,t)=(1+x)t+xt » syms x t » f=(1+x)*t+x*t f = (1+x)*t+x*t n Specify collecting the x terms n Specify collecting the t terms n 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

Symbolic Toolbox:Simplifications and Substitutions 8 Expand Command The expand(s) command writes each element of the symbolic expression s as a product of its factors n Types of expandable expressions include: u Polynomial expressions u Trigonometric expressions u Exponential expressions u Logorithmetic expressions This is the inverse of the collect command

Symbolic Toolbox:Simplifications and Substitutions 9 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 n Polynomial and exponential expansion examples

Symbolic Toolbox:Simplifications and Substitutions 10 Involved Expand Example n Given the following function of x: 1)Expand f(x) by hand to get a polynomial function of x 2)Verify the result using the symbolic expand command

Symbolic Toolbox:Simplifications and Substitutions 11 Expansion Approach n To expand f(x) by hand, represent the inverse cosine portion as a new function z n Expand cos(3z) in terms of z n Once cos(3z) is expanded, substitute back in z=cos -1 (x) Let: Thus:

Symbolic Toolbox:Simplifications and Substitutions 12 Expand cos(3z) Term n Begin by expanding f(x) in terms of z

Symbolic Toolbox:Simplifications and Substitutions 13 Substitute and Simplify n From the previous work: n Substitute: n Simplify:

Symbolic Toolbox:Simplifications and Substitutions 14 Expand Verification n This is easily verified in Matlab » expand( cos(3*acos(x)) ) ans = 4*x^3-3*x

Symbolic Toolbox:Simplifications and Substitutions 15 » 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:

Symbolic Toolbox:Simplifications and Substitutions 16 » 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 n Simplify the expression: Expand gives the same result

Symbolic Toolbox:Simplifications and Substitutions 17 n 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

Symbolic Toolbox:Simplifications and Substitutions 18 Simplify Example n 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: b^a 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 = b^a factor: exp(a*log(b)) expand: b^a combine: exp(a*log(b)) convert(exp): exp(a*log(b))

Symbolic Toolbox:Simplifications and Substitutions 19 Best Simplify Method n 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 n Also show which simplification was used » [f_smpl,how]=simple(f) f_smpl = b^a how = expand

Symbolic Toolbox:Simplifications and Substitutions 20 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) n The simple command can also be used to simplify symbolic mathematical expressions without dependent variables

Symbolic Toolbox:Simplifications and Substitutions 21 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