259 Lecture 18 The Symbolic Toolbox

2  MATLAB has a set of built-in commands that allow us to work with functions in a fashion similar to Mathematica.  Octave can also perform some of the same functionality via the “symbolic” package – we will look at this after the MATLAB features!  For more on the commands available, type “help Symbolic Toolbox”.  The commands we will look at are:  sym, syms, diff, int, simplify, pretty, subs, double, and ezplot.  For help on these commands, use the help file.  If a symbolic command has the same name as a numerical command, such as “diff”, typing “help sym/commandname” will give help on the symbolic command.  Try “help diff” and “help sym/diff”.

3 sym  To define symbolic variables in MATLAB, use the “sym” command.  Example 1a: Here’s how to define variables x, y, and a, and functions  f(x) = x 3 + 2x 2 +x-1  g(x) = sin(x)  h(y) =  (a 2 - y 2 )/(y+1)  x = sym(‘x’)  y = sym(‘y’)  a = sym(‘a’)  f = x^3+2*x^2+x-1  g = sin(x)  h = sqrt(a^2-y^2)/(y+1)  Typing “pretty(h)” will make h(y) look nicer!  Note that in MATLAB, “syms x y a” also works to define symbolic variables!

4 diff  To find the derivative of a function defined symbolically, we use the “diff” command.  Example 2a: Find the following derivatives of the functions from Example 1a: f’(x), f’’(x), g’(x), h’’’(y).  Also find dy/dx if y = x + x 2 – x 4 or  y = tan(x) sin(x) -ln(x)/e x  diff(f)  fprime = diff(f)  f2prime = diff(f,2)  gprime = diff(g)  h3prime = diff(h,3)  pretty(h3prime)  simplify(h3prime)  pretty(simplify(h3prime))  diff('x + x^2 - x^4')  diff('tan(x)^sin(x) - log(x)/exp(x)')  pretty(ans)

5 subs  To evaluate a symbolic function we use the command “subs”.  Example 3a: For the functions defined in Example 1a, find each of the following:  f(-3)  f(v) where v = [1 2 4]  g’(pi/4)  h(1)  h(1) with a = 2  h(y) with a = 2  subs(f,-3)  v = [1 2 4]  subs(f,v)  subs(gprime,pi/4)  subs(h,1)  b = subs(h,1)  subs(b,2)  subs(h,'a',2)

6 ezplot  We can plot symbolic functions with the command “ezplot”!  Example 4a: Use ezplot to graph the functions f(x), g(x), g’(x), f’(x), and f’’(x) defined in Example 1a.  Note that the default settings for ezplot can be changed with title, xlabel, and ylabel.  The default x-interval of [-2, 2] can also be changed.  ezplot(f)  ezplot(f,[-1,1])  ezplot(g)  ezplot(gprime,[0,2*pi])  One way to plot multiple graphs via ezplot: ezplot(f) hold on ezplot(fprime) ezplot(f2prime) title('Plot of f and it''s derivatives.') hold off

7 int  To find indefinite or definite integrals in MATLAB, we use “int”.  Example 5: Find each integral:  int(f)  int(g,0,pi)  int(h,'y',0.5,1)  pretty(ans)  sym(a,'positive')  int(h,'y',0.5,1)  pretty(ans)  int(h,'a',0.5,1)  int('x + x^2 - x^4')

8 funtool  Finally, here’s a way to work with functions that is more “user friendly”.  Typing “funtool” brings up a function calculator in MATLAB.  funtool is a visual function calculator that manipulates and displays functions of one variable.  At startup, funtool displays graphs of a pair of functions, f(x) = x and g(x) = 1.  The graphs plot the functions over the domain [-2*pi, 2*pi].  funtool also displays a control panel that lets you save, retrieve, redefine, combine, and transform f and g.

Symbolic Package in Octave  To perform symbolic manipulation in Octave, first we need to load the “symbolic” package.  The symbolic package, as well as other add-ons can be found here: Octave downloads from Source Forge (both Windows and Mac OS X installers can be found here: Octave Homepage:  A reference for the “symbolic” package can be found here (choose “Function Reference”): 9

Symbolic Functions in Octave  Once the “symbolic” package is installed, turn on Octave and type “pkg load all” to load all installed packages.  To enable the symbolic features in Octave, type “symbols”. 10

11 sym in Octave  To define symbolic variables in Octave, use the “sym” command.  Example 1b: Here’s how to define variables x, y, and a, and functions  f(x) = x 3 + 2x 2 +x-1  g(x) = sin(x)  h(y) =  (a 2 - y 2 )/(y+1)  x = sym(‘x’)  y = sym(‘y’)  a = sym(‘a’)  f = x^3+2*x^2+x-1  g = Sin(x)  h = Sqrt(a^2-y^2)/(y+1)

12 diff in Octave  To find the derivative of a function defined symbolically in Octave, we use the “differentiate” command.  Example 2b: Find the following derivatives of the functions from Example 1b: f’(x), f’’(x), g’(x), h’’’(y).  Also find dy/dx if y = x + x 2 – x 4 or  y = tan(x) sin(x) -ln(x)/e x  differentiate(f,x)  fprime = differentiate(f,x)  f2prime = differentiate(f,x,2)  gprime = differentiate(g,x)  h3prime = differentiate(h,y,3)  differentiate(x + x^2 - x^4,x)  differentiate(Tan(x)^Sin(x) -Log(x)/Exp(x),x)

13 subs in Octave  To evaluate a symbolic function in Octave, we use the command “subs”.  Example 3b: For the functions defined in Example 1b, find each of the following:  f(-3)  g’(pi/4)  h(1)  h(1) with a = 2  h(y) with a = 2  subs(f,x,-3)  subs(gprime,pi/4)  subs(h,y,1)  b = subs(h,y,1)  subs(b,a,2)  subs(h,a,2)

14 ezplot in Octave  We can plot symbolic functions with the command “ezplot”!  Example 4b: Use ezplot to graph the functions f(x), g(x), g’(x), f’(x), and f’’(x) defined in Example 1b.  Note that the default settings for ezplot can be changed with title, xlabel, and ylabel.  The default x-interval of [-2, 2] can also be changed.  Recall that f = x^3+2*x^2+x-1 g = Sin(x)  ezplot(‘x^3+2*x^2+x-1’)  ezplot(‘x^3+2*x^2+x- 1’,[-1,1])  ezplot(‘sin(x)’)  ezplot(‘cos(x)’,[0,2*pi])  One way to plot multiple graphs via ezplot: ezplot(‘x^3+2*x^2+x-1’) hold on ezplot(‘3*x^2+4*x+1’) ezplot(‘6x+4’) title('Plot of f and it''s derivatives.') hold off

15 References  Using MATLAB in Calculus by Gary Jenson  MATLAB Help File