Physics 326G Winter Class 6

Transcription

1 Physics 36G Winter 008 Class 6 Today we will learn about functions, and also about some basic programming that allows you to control the execution of commands in the programs you write. You have already used several of Matlab s built-in functions, so you know roughly how they work. Every function is really a program written in an M-file. A function takes a number of arguments, which are input values that you give it, does something useful, then, usually, gives an output or return value. For example, when you do a = sin(x); x is the argument of the function SIN. The function calculates the sine of x and returns the result of that calculation, which here we have assigned to the variable a. Matlab functions can have many arguments and many return values. One big difference between functions and script files is that the variables used in a function are not accessible outside of that function. The exceptions to this are the variables you pass to the function as arguments, and the variables which the function gives back to you as output. (There is another exception, which we won t worry about here you can explicitly define variables to be global. This is useful sometimes but can also be dangerous.) Let s write a simple function. Open a new M-file in the Editor. The first line of a function must identify it as a function (rather than a script): function pressure = ideal_gas(density,temperature) Our function s name is ideal_gas. It takes two input arguments: variables called density and temperature. It has one output value, namely pressure. Note that in general any of these arguments could be arrays or strings; they don t have to be single numbers. The rest of the function is the series of Matlab commands we want to execute. The only thing we have to remember is to assign a value to the output variable, pressure. It is useful to make the first lines of a function after the definition line a series of comments that tells what the function does. Then you can use HELP with your own functions! Here is the complete function: function pressure = ideal_gas(moles,temperature) % This function takes the number of moles in a % litre of gas and % the temperature (in Kelvin) as inputs % and calculates the pressure using the ideal gas law. % It has been written to work with arrays, but moles and temperature % have to have the same length % check to see that the input arrays are the same length

2 if (length(moles) ~=length(temperature)), disp('error: input arrays must be the same length!') pressure = 0; return R = 8.057e-; % gas constant, liter atm/mol K % the ideal gas law is PV = nrt % where n is the number of moles of gas and R is the gas constant. % n/v is the number of moles per liter % so P = R*moles*temperature % note the dot-times in the line below, since we are working with arrays pressure = R*moles.*temperature; There are a few things to note here. The return statement s execution of the function. You don t need one at the of the function, since if Matlab gets to the of the function file it will return (stop) automatically anyhow. Early in the program we have an if statement. This allows you to do different things, deping on the value of a particular variable. In this case, the program checks the length of the input arrays. If they are not the same, it prints an error message and stops. If they are the same, it continues with the calculation. IF is an example of a control statement that allows you to control the execution of the program. More on this very soon. Type in this program and save it in a file ideal_gas.m. (You should always save functions in a file with a name that is the same as the function name; this avoids much confusion.) Try it out. Check that the HELP command works with your program. One thing you will need to do often is compare two quantities. Matlab has several relational operators that let you do this. They include > greater than < less than <= less than or equal to >= greater than or equal to = = equal to ~= not equal to The first four are obvious. Note that one equals sign alone is used to assign a value to a variable. To compare two quantities, you use two equals signs. Remember this, because it is very easy to make very confusing mistakes. The tilde ~ means not, so ~= is not equal to. The result of a relational operation is either 1 (meaning true) or 0 (meaning false). These operators work element-by-element on arrays of equal size. Type a =3;

3 b=4; then a<b which should give 1 (the relationship is true) a>b which should give 0 (the relationship is false). Now try it with arrays: c = [ ]; d = [ ]; Try all of the relational operators on these two arrays, and make sure you understand what happens. Matlab has a number of logical operators: & and or ~ not a&b (a AND b) returns a 1 (true) if both a and b are true, that is, if a and b are non-zero. a b (a OR b) returns a 1 (true) if either a or b or both are true, that is, if at least one of a and b is non-zero. ~a returns 1 if a is 0 (false), and returns 0 if a is non-zero (true). Logical operators also work on arrays. All of these operations can be used in conjunction with an IF statement to control program flow. IF statements allow Matlab to make decisions about what to deping on the result of a conditional test. For example if a>b, disp( a is greater than b ) The comma at the of the IF statement is optional; sometimes it helps to make the statement more readable. Every IF statement has to have an END statement associated with it. If the condition being tested is true, the statements after the IF up to the END are executed. If not, they are skipped. You can also use the ELSEIF statement to choose from among several possible outcomes: if a>b, disp( a is greater than b )

4 elseif a<b disp( a is less than b ) else disp( a is equal to b ) Write a script that takes two numbers from the keyboard, and then prints out which of the two is the biggest. Often you will want your program to execute a series of commands a given number of times. For example you might want to calculate something for a series of different values of a variable. You can do this with a FOR loop. Here is a silly example: for ii=1:10, fprintf( hello, this is step number ii=%d \n,ii) The statements between the FOR and the END are executed once for each element of the array specified in the FOR statement. The first time, the counter ii is set equal to the first element of the array, the second time, ii is set equal to the second element of the array, and so on. You can use any one-dimensional array; it does not have to be a sequence. For example, for ii=rand(1,10), fprintf( hello, is this step number ii=%f?\n,ii) It is worth pointing out that large loops in Matlab are usually very slow. Often you can eliminate FOR loops by taking advantage of Matlab s ability to do arithmetic with arrays all at one time, and you should do this whenever possible. For example if a aand b are arrays, it is better to write c = a+b; than for ii=1:length(a), c(ii) = a(ii)+b(ii); It takes some practice to use array arithmetic effectively but it is well worth it. In addition to gaining a few milliseconds because your program will run faster, your code will be shorter and easier to read, and you will make fewer mistakes. The Taylor series for sin(x) issin x = k k ( 1) x k = 0 ( k + ) + 1 1!. Write a function that takes as arguments the angle x in degrees and kmax, the number of terms to include in the sum. Your function should then convert x in to radians, then uses a FOR loop to calculate the sum up to k=kmax, and output

5 the result to the command window. Use a FOR loop to calculate the sum. The FACTORIAL function will be helpful. Matlab also has WHILE loops. These are useful when you know you want to iterate a loop, but you don t know at the outset how many iterations of the loop you will need. For example, you could modify the Taylor series program you wrote above so that it would keep going until the terms being added to the sum were smaller than some value epsilon. You could do something like: k = 0 total = 0; epsilon = 1e-5; term = 1 % start with a big number while term>epsilon, term =. total = total+term; k = k+1; In this example, the calculation will stop when the kth term is less than Instead of incrementing k automatically in the FOR loop, here you have to increment k in a separate line inside the WHILE loop because you don t know how many terms you will need. Each WHILE has to have an END associated with it, just as for FOR loops. The commands BREAK and CONTINUE are useful in loops. A BREAK command inside a loop will stop execution of that loop, the program will jump to the END statement of that loop and carry on from there. A CONTINUE statement will stop execution of the iteration of the loop that you are executing right now, and go on to the beginning of the next pass through the loop. Some practice problems: 1. Write a function called quadroots that finds the real roots of a quadratic equation ax + bx + c = 0. The coefficients a, b, and c should be arguments of the function. Arrange it so that if b 4ac> 0 then the program will print out a message saying that there are two real roots, then give the roots. If b 4ac= 0 it should print out a message saying that there is one real root, then give the root. Finally, if it is less than zero it should say that there are no real roots. Use your function to find the real roots of the following equations: x + 8x 3= 0 15x + 10x+ 5 = 0 18x + 1x + = 0. Write a program using a loop that calculates the sum of the first m terms of the series

6 m ( ) n= 0 n 1 1, then calculates the difference between the result and the exact result of π/4. n Modify the program you wrote in () so that it will keep increasing n until the difference between the sum and the exact result is less than 10-4.