Intro to Scientific Computing PHYS 27/193 Physics Department University of the Pacific

User Defined Functions

As you know we can plot all kinds of mathematical functions (see Builtin Functions) with gnuplot. However, an important class of functions are ones that we can make up ourselves out of other functions.

If gnuplot is not already running, start gnuplot

We already learned how to use gnuplot as a calculator to define our own variables, as you did in HW 5 where you calculated the power output of the sun. There, you defined constants made of numbers and builtin constants, such as

gnuplot> T = 5.5e3 # set temp. of sun's surface gnuplot> r = 7e8 # solar radius gnuplot> A = 4 * pi * r**2 # surface area gnuplot> print A 6.15752160103599e+018 # in m**2 gnuplot> P = flux * T**4 * A # solar power output gnuplot> print P 3.27/19391028086952e+026 # in watts

You can also define functions.

Try this:

gnuplot> f(x) = 3*x**2 - 2*x + 4 gnuplot> plot f(x)

You should have gotten a parabola.

You can also define a function of several variables:

gnuplot> f(x,a,b) = a*x - b gnuplot> plot f(x,1,2), f(x,3,-2)

You can use functions in other functions

gnuplot> set hidden3d gnuplot> set isosamples 30 gnuplot> r(x,y) = sqrt(x*x + y*y) gnuplot> splot [-2:2][-2:2] exp(-r(x,y))*cos(pi*r(x,y))

You can build quite complicated functions with gnuplot. Remember however, that the variable that is uses for plotting in a 2-D plot is always x.

Actually what I said above about x is not 100% true; you can do: set polar which will make the default plot variable t (for "theta"). This makes gnuplot plot functions as distance from the origin (radius) vs. angle t. The default range is 2*pi radians).

The Conditional Function

One very useful construct that I sometimes use is the C programming if/then short hand notation which gnuplot recognizes:

( conditional phrase ) ? ( if TRUE value ) : ( if FALSE value )

The conditional phrase is a question such as

x == 0            # is x equal to zero?
x != 0            # is x NOT equal to zero?
X > 4 || x < 3    # is x greater than 4 OR less than 3?
x >= 0 && x <= y  # is x greater to or equal to zero AND less than or equal to y?

If the question part is TRUE, the result is the thing between the ? and : If the question is FALSE, the result is the thing after the :

Here's an example. Plot a function whose value is f(x) = 50 if x is between 0 and 1, but f(x) = 0 otherwise.

gnuplot> f(x) = (x>0 && x<1) ? 50 : 0 gnuplot> set xzeroaxis gnuplot> plot [-1:2][-10:60] f(x)

Try it! Such constructs could be handy in Digital Signal Processing classes.

You can even define functions recursively, such as the factorial function, n! = n*(n-1)*(n-2)...3*2*1.

gnuplot> fac(n) = (n != 0) ? n*fac(n-1) : 1 gnuplot> print fac(3) 6 gnuplot> print fac(5) 120 gnuplot> print fac(8) 40320 gnuplot> print fac(10) 3628800

Functions with Constants

One last thing. Notice that I can define a function with a variable, c, in it:

gnuplot> f(x) = c/sqrt(1 + x) gnuplot> plot f(x) undefined variable: c

However when I try to plot the function, gnuplot complains that the variable c is not defined, hence it, rightly, doesn't know how to plot the function.

If I go ahead and define c

gnuplot> c = 5 gnuplot> plot f(x)

gnuplot can now plot the function, since it knows how to evaluate it. I can update the value of c by typing c = 10, and replot, and I'll get a new function f(x) = 10/sqrt(1 + x).

Notice too in your plot that gnuplot stopped plotting for x <= 1, since the function is complex there (in fact it's singular at x = -1). It will however, still tell you the complex value if you ask:

gnuplot> print f(-2) {0.0, -5.0}

Next, we'll use the ability to define functions with constants to let gnuplot find the "Best" function that fits our data.

Next: Fitting Functions to Data