Solving Algebraic Equations

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Solving AlgebraicEquations
Math Review with Matlab
Single Variable Transcendental Equations

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

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Solving TranscendentalEquations

• Solve Command
• Polynomial Transcendental Example
• Poly2sym Command
• Exponential Transcendental Example
• Complex Solution Example

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Solve Command

• The Symbolic Toolboxs 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

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Polynomial Transcendental Equation Example

• Use Matlabs Symbolic Toolbox solve command to
  solve the transcendental equation

syms x xs_symsolve(x2-3x2) xs_sym
1 2
xs_dbldouble(xs) xs_dbl 1 2
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Poly2Sym Command

• 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

• The sym2poly command converts a symbolic
  polynomial expression to a polynomial coefficient
  vector

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Poly2sym Example

• Example

p5,2,-1,1,10
Polynomial Vector
sym_exppoly2sym(p) sym_exp 5x42x3-x2x
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Symbolic Expression
poly_vecsym2poly(sym_exp) poly_vec 5 2
-1 1 10
Polynomial Vector
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Exponential Transcendental Equation Example

• Use the solve command to solve the transcendental
  equation

xs_symsolve(exp(2x)3exp(x)-54) xs_sym
log(-9) log(6)
xs_dbldouble(xs) xs_dbl 2.1972 3.1416i
1.7918
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Verify Loge(-9)

• In Matlab, log is the natural log of base e

• Verify ln(-a) is a complex number

• In this case -a -9, so

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Complex Solution Example

• Solve the nonlinear equation

• Verify the result

syms x rt solve('x2exp(x)5') rt
3.5668799472626199482146544957277
4.8873680198538466074569067078051i
zsubs('x2exp(x)5',rt) z .2e-30
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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