MATLAB Basics Symbolic math

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MATLAB Basics - Symbolic math

• Engineering mathematics, Week 4

Neural Signal Processing Laboratory at Yonsei BME
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Algebra

• Symbolic Math ToolboxUsing MATLAB's Symbolic
  Math Toolbox, you can carry out algebraic or
  symbolic calculation, such as factoring
  polynomials or solving algebraic equations
• To perform symbolic computations, you must use
  syms to declare the variables you plan to use to
  be symbolic variables

syms x y (x - y) (x - y)2 ans (x -
y)3 expand(ans) ans x3-3x2y3xy2-y3
factor(ans) ans (x-y)3
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Simplify, Simple

• MATLAB has a command called simplify, which you
  can sometimes use to express a formula as simply
  as possible

syms x y simplify((x3 - y3) / (x - y)) ans
x2xyy2

• MATLAB has a more robust command, called simple,
  that sometimes does a better job than simplify.

syms x y sin(x) cos(y) cos(x) sin(y)
simple(ans) ans sin(x y)
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Symbolic expressions, variable precision, and
exact arithmetic (i)

• MATLAB uses floating point arithmetic for its
  calculations. Using the Symbolic Math Toolbox,
  you can also do exact arithmetic with symbolic
  expressions.

cos(pi / 2) ans 6.1232e-017

• The answer is written in floating point format
  and means 6.1232 x 10-17. However, we know that
  cos(p/2) is really equal to 0. The inaccuracy is
  due to the fact that typing pi in MATLAB gives an
  approximation to p accurate to about 15 digits,
  not its exact value.

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Symbolic expressions, variable precision, and
exact arithmetic (ii)

• To compute an exact answer, instead of an
  approximate answer, we must create an exact
  symbolic representation of p/2 by typing
  sym('pi/2')

cos(sym('pi / 2')) ans 0 This is the
expected answer !
Another example
1/2 1/3 ans 0.8333
sym('1/2') sym('1/3') ans 5/6
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Symbolic expressions, variable precision, and
exact arithmetic (iii)

• Finally, you can also do variable-precision
  arithmetic with vpa. For example, to print 50
  digits of , type

vpa('sqrt(2)', 50) ans 1.41421356237309504880
16887242096980785696718753769
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Solving equations (i)

• You can solve equations involving variables with
  solve or fzero. For example, to find the
  solutions of the quadratic equation
  , type

solve('x2 - 2x - 4 0') ans 5(1/2)1
1-5(1/2)

• Note that the equation to be solved is specified
  as a string. The answer consists of the exact
  (symbolic) solutions . To get
  numerical solutions, type double(ans) or vpa(ans)
  to display more digits

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Solving equations (ii)

• The command solve can solve higher degree-degree
  polynomial equations, as well as many other types
  of equations.

x, y solve('x2 - y 2', 'y - 2x 5') x
122(1/2) 1-22(1/2) y
742(1/2) 7-42(1/2) Note that the answer
is symbolic representation.
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Solving equations (iii)

• Some equations cannot be solved symbolically, and
  in these cases solve tries to find a numerical
  answer.

solve('sin(x) 2 - x') ans
1.1060601577062719106167372970301

• Sometimes there is more than one solution, and
  you may not get what you expected.

solve('exp(-x) sin(x)') ans
-2.0127756629315111633360706990971 2.703074511
5909622139316148044265i The answer is a
complex number
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Solving equations (iv)

• Though it is a valid solution of the equation,
  there are also real number solutions (red arrows).

fzero(inline('exp(-x) - sin(x)'), 0.5) ans
0.5885 fzero(inline('exp(-x) - sin(x)'), 3) ans
3.0964