Solving Algebraic Equations S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn Math Review with Matlab: Single Variable Transcendental Equations

Solving Algebraic Equations:Transcendental Equations 2 Solving Transcendental Equations n Solve Command Solve Command n Polynomial Transcendental Example Polynomial Transcendental Example n Poly2sym Command Poly2sym Command n Exponential Transcendental Example Exponential Transcendental Example n Complex Solution Example Complex Solution Example

Solving Algebraic Equations:Transcendental Equations 3 Solve Command The Symbolic Toolbox’s solve command is used to solve both transcendental and algebraic equations solve(f) solves f in terms of the independent symbolic variable closest to x. f must be a symbolic expression or a entered as a text string enclosed by single quotes solve(f1,f2,...,fn) is used to solve multiple equations by separating the symbolic equations by commas as follows

Solving Algebraic Equations:Transcendental Equations 4 Polynomial Transcendental Equation Example Use Matlab’s Symbolic Toolbox solve command to solve the transcendental equation: » syms x » xs_sym=solve(x^2-3*x+2) xs_sym = [ 1] [ 2] » xs_dbl=double(xs) xs_dbl = 1 2

Solving Algebraic Equations:Transcendental Equations 5 Poly2Sym Command The sym2poly command converts a symbolic polynomial expression to a polynomial coefficient vector The poly2sym(f) command converts a polynomial coefficient vector to symbolic polynomial expression in terms of the default symbolic variable x The poly2sym(f, ' v ' ) command converts a polynomial coefficient vector to symbolic polynomial expression in terms of the symbolic variable v

Solving Algebraic Equations:Transcendental Equations 6 » p=[5,2,-1,1,10]; Poly2sym Example n Example: » sym_exp=poly2sym(p) sym_exp = 5*x^4+2*x^3-x^2+x+10 Polynomial Vector Symbolic Expression » poly_vec=sym2poly(sym_exp) poly_vec = Polynomial Vector

Solving Algebraic Equations:Transcendental Equations 7 Exponential Transcendental Equation Example » xs_sym=solve(exp(2*x)+3*exp(x)-54) xs_sym = [ log(-9)] [ log(6)] Complex Solution Real Solution Use the solve command to solve the transcendental equation: » xs_dbl=double(xs) xs_dbl = i

Solving Algebraic Equations:Transcendental Equations 8 Verify Log e (-9) In this case -  = -9, so: In Matlab, log is the natural log of base e: Verify ln(-  ) is a complex number :

Solving Algebraic Equations:Transcendental Equations 9 Complex Solution Example n Solve the nonlinear equation: » syms x » rt = solve('x^2+exp(x)+5') rt = *i n Verify the result » z=subs( ' x^2+exp(x)+5 ',rt) z =.2e-30

Solving Algebraic Equations:Transcendental Equations 10 Summary The solve command can be used to solve symbolic transcendental equations with real or complex solutions poly2sym and sym2poly commands can be used to convert polynomial vectors to/from symbolic expressions