Present tutorial: This tutorial was prepared for our freshman engineering students using the student version of MATLAB, because symbolic computations are covered in almost no introductory textbook. We are pleased to make it available to the international community, and would appreciate suggestions for its improvement. MATLAB’s symbolic engine permits the user to do things easily that are difficult or impossible to do numerically.

The following topics are covered:

Differentiation

Indefinite integrals

Definite integrals

Conversion to numeric (“double”) variables for subsequent numeric calculations

Substitution

Plotting of equations and functions

Solution of equations: graphical approximations, single equations, multiple simultaneous equations, non-linear equations, equations with infinitely many solutions.

Limits

Taylor series

Variable precision arithmetic

Powerpoint lectures and homework assignments

First do the demos in MATLAB’s help. Click on Help, Demos tab at the top, Toolboxes, Symbolic Math.

In the following instructions, material in bold is to be entered in the MATLAB command line and executed. This can be done by cutting and pasting using Ctrl c to cut and Ctrl v to paste. (Use the small finger on the left hand to press Ctrl while pressing c or v with the index finger of the left hand at the same time. Ctrl x cuts the highlighted material.)

The first step is to tell MATLAB that the variables to be used are symbolic. This is most easily done using the syms command (see >> help syms). For example, in the following we will use x, y, a, b and c as symbolic quantities. So enter >> clear, syms x y a b c f g . In the Workspace window (or use >> whos) note that these variables are symbolic, rather than the usual double precision numeric variables. It’s a good idea to start with “clear” to make certain MATLAB doesn’t use previous values for variables.

Differentiation: The MATLAB symbolic toolbox is very useful for checking calculus problems. For the syntax for symbolic differentiation, see >> help sym/diff . For example, to find diff_f = where f = e-axx3bsin(cx) and a, b and c are unspecified constants, we do the following1:

>> clear, syms a b c x; f=exp(-a*x)*x^(3*b)*sin(c*x), diff_f = diff(f,x)

Notice that the result is quite complex. To ask MATLAB to try to simplify it, type:

>> diff_f_simp = simple(diff_f)

(see >> help sym/simple). MATLAB uses several algorithms for algebraic simplification, and chooses as a final answer the shortest of the results. If you prefer some other form listed, select the command producing it.

MATLAB can take higher order differentials. For example, to find diff2_f =we do

>> diff2_f=diff(f,x,2) using the already defined expression for f.

Indefinite integrals: For the syntax for symbolic integration, see >> help sym/int. Thus, for example, to find the indefinite integral int_g =, where g = e-axsin(cx), we do

>> clear, syms a c x; g=exp(-a*x)*sin(c*x), int_g = int(g,x)

Check this by taking >>diff_int = diff(int_g,x). This should give back g, but it doesn’t look the same because MATLAB doesn’t always give things in the simplest format. In this case, we again use the simple command >> diff_int_simp = simple(diff_int).

Note that MATLAB does not add the constant of integration (“C”) customarily shown for indefinite integrals -- this you must do by adding C after the int command.

Sometimes MATLAB is not able to integrate an expression symbolically. For example, try to find (where, as before, f = e-axx3bsin(cx) and a, b and c are unspecified constants).

>> clear, syms a b c x; f = exp(-a*x)*x^(3*b)*sin(c*x), int(f,x)

Notice that when MATLAB cannot find a solution, it gives a warning message and repeats the command. When this happens, you’re out of luck.

Definite integrals: For the definite integral, int_def_g =do

>> clear, syms a c x; g = exp(-a*x)*sin(c*x), int_def_g = int(g,x,-pi,pi)

When MATLAB is unable to find an analytical (symbolic) integral, such as with >>int(exp(sin(x)),x,0,1), a numerical solution can nevertheless be found using the “quad” command, e.g. >> quad('exp(sin)',0,1) . Generally speaking, it is best to follow the steps below:

1.Create the function to be integrated in the MATLAB editor and save it to your Current Directory, which must also be in the path (File / Set Path). For example, for our exp(sin(x)) function:

function out = func1(x)

out = exp(sin(x));

end

2.Use either one of the following formats: >> quad('func1',0,1) or >> quad(@func1,0,1)

Conversion to a double precision numeric variable: Often MATLAB's symbolic engine produces a symbolic answer, as in:

>> clear, syms x; h=exp(-x)*sin(x), int_def_h=int(h,x,-pi,pi)

where is retained in symbolic form (pi). Irrational numbers may also result, e.g. 4/3 rather than 1.3333. For use in subsequent numerical calculations, you must convert symbolic results into numeric “double” variables by the “double” command:

>> int_def_h_doub=double(int_def_h)

Substitution: Sometimes it is desired to substitute one value or parameter for another in an expression. This can be done using the “subs” command (see >> help sym/subs). For example, imagine we want a=2 and c=4 in int_def_g. This is done by:

>> clear, syms a c x; g = exp(-a*x)*sin(c*x), int_def_g = int(g,x,-pi,pi)

>> int_sub=subs(int_def_g,{a,c},{2,4})

Do not try to assign the values to the variables before the integration or differentiation is performed. (Try it in the above example and see what happens.)

Graphing equations and functions: MATLAB’s symbolic engine has commands for plotting symbolic equations and functions, appropriately starting with “ez.” For two-dimensional graphs, see >> help ezplot. As an example here we use the equation for a circle of radius 2, x2+y2 = 4.

Step 1: Move everything to the left-hand side of the equation, i.e. x2 + y2 – 4 = 0

Step 2: Clear and define x and y as symbolic.

Step 3: Define a variable equal to the left-hand side of the equation, inside single

straight2 quotation marks.

Step 4: Use ezplot.

>> clear; syms x y; lhs = 'x^2+y^2 - 4'; ezplot(lhs,[-3,3,-2.5,2.5])

The [-3,3,-2,2] tells MATLAB that the x scale is to go from -3 to +3, and the y scale is to go from -2.5 to +2.5. Note that we have assigned x2 + y2 - 4 to the character variable “eq” (see the Workspace window or type >> whos).

You can edit the figure in the Figure window. First save it as “circle.fig” using File/Save As so you can retrieve it later from the Command window, without having to recreate it. Here are some things you can do in the Figure window:

Click on the arrow icon and then double-click on any portion of the figure you’d like to edit. In this way you can change the color and thickness of the curve, add points, change the axes, edit the title.

Edit / Copy Figure to place it on the Windows Clipboard. Then you can paste it into a Word document, for example.

While in the Figure window, press the Alt and Print Screen keys at the same time, to place the entire figure window on the clipboard. This can also be pasted into a Word document.

Experiment with other equations.

Three-dimensional plots can be created using the ezmesh and ezsurf commands. For these commands an explicit form must be used, i.e. z = f(x,y). It is the f(x,y) that is entered into the command. The code below illustrates some possibilities. The green lines with % are comments and are not executed by MATLAB.

% symplot3D.m

% Some 3d plots, where z = the expressions shown

% In the figure window, click on Tools, Rotate 3D.

% Then put cursor on figure, hold down left mouse,

% move cursor to rotate figure. Notice that this

% gives you a better feeling for its 3D shape.

% Have some fun and create your own shapes.

clear, clc

syms x y

ex1='sqrt(4-x^2-y^2)';

figure(1); ezsurf(ex1,[-2,2,-2,2]);

ex2='x^2-y^2';

figure(2); ezsurf(ex2);

ex3='x^2+2';

figure(3); ezmesh(ex3);

ex4='x + y';

figure(4); ezmesh(ex4);

Open the excellent MATLAB editor by >> edit, cut and paste the above code into your editor, save it on your Current Directory as symplot3D.m, make certain the Current Directory is in MATLAB’s path (File/Set Path), and then execute it in the Command window by >>symplot3D.

Select each figure in turn, and in the Figure window Tools / Rotate 3D. Put the cursor anywhere on the figure, hold down the left mouse button, and move the mouse to rotate the figure.

Solution of equations: The symbolic toolbox in MATLAB is capable of solving many types of equations, including non-linear ones and several simultaneous equations. See: >> help sym/solve. The steps in solving one or more equations using “solve” are:

Step 1: Define the variables in the equations as symbolic using the “syms” command.

Step 2: Define the equations (see the examples below).

Step 3: Solve the equations using the “solve” command.

Step 4: If there is more than one solution, decide which is physically reasonable and select it.

Step 5: Check the solution by substituting it back into the original equation(s).

We now look at some examples.

Example 1: Find the two solutions to the classic quadratic equation, ax2 + bx + c = 0.

>> clear, syms x y z a b c; eq = 'a*x^2+b*x+c = 0', [x]=solve(eq,x)

(Note that single quotation marks must be placed around an equation that is assigned to a variable, here “eq”.) As expected, we obtain two solutions. Suppose we want only the first. This is extracted by >> x1 = x(1) . This should be done manually or using a logical operator (“if” command), because MATLAB's symbolic engine doesn't always give multiple solutions in the same order. What is x(1) now could be x(2) in a subsequent trial.

Example 2: Find the solution to e2x=3y:

>> clear, syms x y; eq = 'exp(2*x) = 3*y', [x] = solve(eq,x)

(Is the log in this solution the natural log or the log base 10? See

http://people.clarkson.edu/~wilcox/ES100/formcomp.doc)

We use a similar procedure to solve simultaneous equations. First, let us look at a set of linear equations, and then non-linear equations.

Example 3: Find the solution to the following set of linear equations:

2x-3y+4z = 5

y+4z+x = 10

-2z+3x+4y = 0

>> clear, syms x y z;

>> eq1 = '2*x-3*y+4*z = 5'

>> eq2 = 'y+4*z+x = 10',

>> eq3 = '-2*z+3*x+4*y = 0'

>> [x,y,z] = solve(eq1,eq2,eq3,x,y,z)

Notice that we did not have to enter x, y and z in any particular order in defining the equation variables (unlike the numerical MATLAB method using matrices with \ or inv). However, the variables in the output are in alphabetical order, regardless of how they are written in the command. So take care to list the variables in the brackets in alphabetical order. (For example, if you use [z,x,y] rather than [x,y,z] in the above example the result produced for z is actually x, that for x is actually y, and that for y is actually z.) Again, the results may be symbolic and require conversion to numeric form for subsequent calculations:

>> x_d = double(x), y_d = double(y), z_d = double(z)

Example 4: We can also solve simultaneous non-linear equations, e.g. y=2ex and y=3-x2:

>> clear, syms x y; eq1='y=2*exp(x) ', eq2='y=3-x^2', [x,y]=solve(eq1,eq2,x,y)

Note that this gives only one solution; there are two as we see below. Again, to convert x and y from symbolic variables to numeric double variables, use the “double” command. (Use “whos” or look in the Workspace window to follow this transformation.)

We can obtain an approximate solution by plotting these two equations and looking at their intersections. As above, we first move everything to the left hand side of the equations.

>> clear, syms x y; eq1 = 'y - 2*exp(x)'; eq2 = 'y-3+x^2';

>> ezplot(eq1), hold on; ezplot(eq2); hold off

The "hold on" command tells MATLAB to write following plots on top of the first one, rather than replacing it. Approximate values of the solutions (intersections) can be obtained in the Figure window by clicking on Tools / Zoom In, clicking on a desired intersection to enlarge that area of the graph, clicking on Tools / Data Cursor, and then clicking on that intersection again to give the approximate values of x and y there. Compare these results with those found above using the solve command.

Example 5: There are equations with an infinite number of solutions, for example sin(x) = 1/2.

It is helpful to see some of these solutions by plotting y = sin(x)-1/2 and seeing approximately where it has values of zero:

>> clear, syms x; eq='y - sin(x) + 1/2'; ezplot(eq,[-6,6,-2,1])

>> hold on; eq0='0'; ezplot(eq0); hold off

The "hold on" command tells MATLAB to write following plots on top of the first one, rather than replacing it. Plotting 0 puts a horizontal line on the graph. Intersections of the sine wave with this line represent solutions of the first equation. Approximate values can be obtained in the Figure window by clicking on Tools / Zoom In, clicking on a desired intersection to enlarge that area of the graph, clicking on Tools / Data Cursor, and then clicking on that intersection again to give the approximate values of x and y there. Try this on the first two intersections to the right of the origin (x > 0). Write down your results to compare with those found below using solve commands.

The symbolic solve command finds only one of these solutions and then stops:

>> clear, syms x; eq = 'sin(x)-1/2=0'; [x] = solve(eq,x); x_doub=double(x)

But suppose we want the solution that lies between x = 2 and x = 3. There are two ways to do this, one using MATLAB's symbolic engine and the (easier) using its numeric engine. First we use the symbolic fsolve command (see >> mhelp fsolve), for which the syntax is

>> clear, x = maple('fsolve(sin(x)=1/2,x,2..3)')

While not immediately apparent, >> whos shows that the result is a 1x33 character array (made up of the ASCII decimal codes for each number and the period.) Try >> x_d = double(x). To convert to a number that can be used in subsequent calculations, we must use the “str2num” (convert string to number) command:

>> x_d = str2num(x)

(Unfortunately, there are many symbolic commands that are not included in the Student version of MATLAB. These can be recognized by the use of “mhelp” and “maple.” For these, you’ll have to use the full professional version of MATLAB that is installed on Clarkson’s computers.)

Now for an easier numerical method to solve the above problem. See >> help fzero. Here are the steps involved:

1.Convert the equation to a function consisting of everything moved to the left-hand side, e.g. func2(x) = sin(x)-1/2.

2.Create this function in the MATLAB editor, save to the Current Directory, and make certain the current directory is in the path (File / Set Path). For example:

function out = func2(x)

out = sin(x) - 1/2;

end

3.Plot this function over the range of interest to see where the solutions are, e.g.:

>> clear, x = - 4:0.01:4; plot(x,func2(x))

4.Use one of the following formats to find the solution:

>> fzero('func2',3) or >> fzero(@func2,3) or >> fzero('func2', [2,3])

MATLAB finds the value of x near 3 that gives func2 = 0. Notice (“whos” or Workspace) that the result is double, so no conversions are necessary for subsequent numeric calculations.

Limits: The symbolic toolbox also contains a command for finding limits of expressions, e.g.:

>> clear, syms x a; value = limit(sin(a*x)/x,x,0)

There are cases where the limit depends on the direction from which it is approached. Consider the plot of tan(x) versus x generated by:

>> ezplot('tan(x)')

Suppose we want the value of tan(x) at x = p/2. We could use MATLAB's numeric engine as follows:

>> tan(pi/2)

Does this give a correct result? Recall that tan(p/2) can be either + or -, depending whether we approach from the left or the right. The numeric solution is a very large number, which might by okay for some applications but not for others. We can use the symbolic engine to find the two correct solutions, as follows:

>> tanhalfpiright = limit(tan(x),x,pi/2,'right')

>> tanhalfpileft = limit(tan(x),x,pi/2,'left')

As an exercise, find . The result may surprise you!

Taylor series: Taylor series are polynomial approximations for functions that are valid near a particular point. Differentials of increasingly higher order are used to generate this polynomial (e.g., see your calculus test). The syntax is taylor(expression, order, variable, point)

Example: Expand ex about x = 3 to order 5:

>> clear, syms x; Tay_expx = taylor(exp(x),5,x,3)

This is exactly correct only at x = 3. Compare this approximation at x = 2 with the exact value:

>> approx=subs(Tay_expx,x,2), exact=exp(2), frac_err=abs(1-approx/exact)

The approximation improves as one adds more terms to the polynomial (order), or moves closer to the point about which the expansion was made. This can be illustrated by:

>> ezplot(Tay_expx), hold on, ezplot(exp(x)), hold off

Summations: MATLAB’s symbolic engine is also capable of doing symbolic summations, even those with infinitely many terms! The “symsum” command is used (see help symsum) Let us consider, where f(k) is a function of an integer k. The syntax to calculate this using

MATLAB is syms k, symsum(f,k,m,n). If a factorial (!) is desired, since ! is also a numeric command, we must tell MATLAB to use the symbolic form by including the term involving in the factorial, say (1+2*k)!, in sym, i.e. sym('(1+2*k)!'). Infinite sums are particularly interesting, e.g.: , the solution for which is found by:

>> clear, syms k x; symsum((-1)^k*x^(2*k+1)/sym('(2*k+1)!'),k,0,Inf)

As an exercise, find and. The results may surprise you!

Variable Precision Arithmetic (VPA): We can control the precision with which MATLAB performs calculations involving irrational numbers. We illustrate this using the golden ratio:

Read >> help vpa and >> help sym/vpa to learn about the two ways of setting the precision of calculating an irrational number. What is the current (default) number of digits?

>> clear, digits

This is the default accuracy of MATLAB's numeric engine. The VPA syntax for calculating R from S is: R = vpa(S) & R = vpa('S'), depending on whether you want the answer accurate to about 16 significant figures or to full precision in symbolic form. Find the difference between the results of calculating the golden ratio by these two methods.

>> phi1 = vpa((1+sqrt(5))/2), phi2 = vpa('(1+sqrt(5))/2'), difference = phi1 - phi2

Which is closer to the correct irrational value? When "difference" is evaluated, MATLAB takes phi2 and expresses it to the current number of digits. The number of digits to which the answer is rounded off can be controlled by the optional argument D, i.e. vpa(S,D) or vpa('S',D). Compare the following:

>> clear, phi1 = vpa((1+sqrt(5))/2, 4), phi2 = vpa('(1+sqrt(5))/2'), difference = phi1 - phi2

>> clear, phi1 = vpa((1+sqrt(5))/2), phi2 = vpa('(1+sqrt(5))/2',4), difference = phi1 - phi2

>> clear, phi1 = vpa((1+sqrt(5))/2,100), phi2 = vpa('(1+sqrt(5))/2',100), difference = phi1 - phi2

Thus, increasing D beyond 32 does not improve the precision using the numerical format, but does add more significant figures using the symbolic format (with the quotation marks). So if you want to know an irrational number to more significant figures, use symbolic methods.

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