320
Introduction MCS to Symbolic Computation
Fall 2005
Maple Lecture 15. Dening Mathematical Functions
We rst consider how to turn formulas into functions. As many functions are not continuous and cannot be dened by one single formula, we will see how to create piecewise functions. This lecture note corresponds to [1, Sections 8.1, 8.2, and 8.3]. Chapter 6 of [2] is recommended reading.
15.1 Mathematical Functions
Maple oers a vast library of mathematical functions. Here we see how to add our own functions. [> expT := 30+40*exp(-0.2*t); # cooling off
Suppose we wish to evaluate this expression for several values of t. There are three ways for doing this: 1) Evaluation as a formula. Suppose we wish to know the value at t = 1.5: [> eva := subs(t=1.5,expT); [> evalf(eva,30); This is not such a good way to evaluate, although we may make it better with a macro: [> macro(evatemp = evalf @ subs); The @ symbol stands for function composition (we will see more applications of this in lecture 17). Here is means: apply evalf after subs, i.e.: at the result of subs. (Note that the macro is an abbreviation mechanism like an alias, except that the alias also aects the output.) We can use this macro then in a similar fashion as the substitute: [> evatemp(t=1.5,expT); evatemp(t=2.3,expT); 2) Evaluation with a procedure. We can also use the formula in a procedure. > proctemp := proc(x) # define function > description cooling temperature; > global expT; # expression is global > return evalf(subs(t=x,expT)); > end proc; [> proctemp(2.4); proctemp(2); # evaluate function A shorter way to make a procedure from the formula is via [> f := codegen[makeproc](expT,t); Notice we must tell makeproc the name of the independent variable. 3) Evaluation using unapply. A fast and convenient way to turn a formula into a function goes like this: [> funtemp := unapply(expT,t); [> funtemp(1.5); It also works for functions in several variables: [> monkey_saddle := x^3 - 3*x*y^2; [> mky := unapply(monkey_saddle,x,y); [> mky(2,3); # polynomial in two variables # function of two variables # sanity check # verify
To see why we call this polynomial a monkey saddle, execute the following: [> plot3d(mk,-2..2,-2..2); Jan Verschelde, September 26, 2005 UIC, Dept of Math, Stat & CS Lecture 15, page 1
MCS 320
Introduction to Symbolic Computation
Fall 2005
15.2 Arrow Operators
This is the fourth way to dene functions from formulas. Arrow operators are close to the notation used in mathematics to denote functions. [> arrtemp := t -> 30 + 40*exp(-0.2*t); [> arrtemp(1.7); Note that we have a problem if we wish to use the formula expT: [> arrtemp2 := t -> expT; [> arrtemp2(1.9); This situation is particularly annoying if expT is so huge we have no courage to retype it. The solution is with substitute: [> arrtemp3 := subs(body = expT, t -> body); [> arrtemp3(2.1); Also multiple arguments work with the arrow operator. Suppose we wish to give the accuracy, say we only want two decimal places instead of 10: [> newtemp := subs(body = expT, (t,n) -> evalf(body,n)); [> newtemp(3,2); newtemp(3,3); newtemp(3,4); With subs we have a very exible mechanism to extend functions.
15.3 Piecewise Dened Functions
Not all functions are as smooth and continuous as the one we used as an example so far. For example, consider a step function. > step := proc(x) > description definition of a step function; > if x < 0 then > return 0 > elif x < 1 > return 1; > else > return > 2; end if > end proc; Let us check the function at some points: [> step(0); step(1/2); step(1); step(1.5); While the above denition worked ne for evaluation, we have a problem when it comes to dierentiating : [> diff(step,x); But that is incorrect, as the derivative does not exist at 0 and 1. [> int(step,x=0..3); This does not make much sense either. We better dene the step function dierently: [> step2 := piecewise(x<0,0,x<1,1,2); Now you see that we can dierentiate and integrate: [> diff(step2,x); [> int(step2,x=0..3); Jan Verschelde, September 26, 2005 UIC, Dept of Math, Stat & CS Lecture 15, page 2 # same as procedure step
MCS 320
Introduction to Symbolic Computation
Fall 2005
15.4 Assignments
1. Dene a function midpoint, which returns the average of two arguments given on input. For example, midpoint(2,3) returns 3/2, midpoint(a,b) returns a/2 + b/2. 2. For two positive natural numbers a and b we can dene the least common multiple lcm(a, b) via their ab greatest common divisor gcd(a, b), i.e.: lcm(a, b) = gcd(a,b) . When a and/or b are zero, lcm(a, b) = 0. Use the arrow operator and piecewise to dene the function my lcm which returns the least common multiple of two natural numbers. To test your least common multiple, take random numbers and compare with Maples lcm procedure. Also test the cases when zeroes are given on input. 3. Consider the general quadratic polynomial p = ax2 +bx+c, with the coecients as parameters. Suppose we wish to create a general quadratic function of x, for example f (1) returns a + b + c, f (2) returns 4a + 2b + c, etc... Type p := a*x2 + b*x + c and, without retyping the formula for p, create f in four dierent ways. Each time, test your function on some values for x. 4. The quadratic formula allows to write the roots of p = ax2 + bx + c explicitly in terms of a, b, and c. (a) Use the output of the solve command (i.e., the output of solve(p,x)) to create a function which returns the rst root, in function of the parameters a, b, and c. (b) From the output of solve, select the discriminant b2 4ac (use the op command) and create a function which re...