University of Cambridge Department of Physics Computational Physics Self-study guide 2 Programming in Fortran 95 Dr. Rachael Padman Michaelmas 2007 Contents 1. THE BASICS 3 1.1 A very simple program 3 1.2 Running the program 4 1.3 Variables and expressions 5 1.4 Other variable types: integer, complex and character 8 1.5 Intrinsic functions 11 1.6 Logical controls 13 1.7 Advanced use of if and logical comparisons 15 1.8 Repeating ourselves with loops: do 16 1.9 The stop statement 17 1.10 Arrays 17 1.11 Array arithmetic 19 2 GOOD PROGRAMMING STYLE 21 2.1 Readability 21 2.2 Self-checking code 22 2.3 Write clear code that relates to the physics 22 3. INPUT TO AND OUTPUT FROM A F95 PROGRAM 24 3.1 F95 statements for I/O 24 4 GRAPHICS AND VISUALISATION 27 4.1 Plotting a data file 27 4.2 Getting help 28 4.3 Further examples 28 4.4 Printing graphs into PostScript files 29 SUGGESTED EXERCISE 1 30 5. PROGRAM ORGANISATION: FUNCTIONS AND SUBROUTINES 31 5.1 Functions 31 5.2 Formal definition 33 5.3 Subroutines 34 5.4 Local and global variables 34 5.5 Passing arrays to subroutines and functions 35 5.5.1 Size and shape of array known 35 5.5.2 Arrays of unknown shape and size 35 5.6 The intent and save attributes 36 6. USING MODULES 38 6.1 Modules 38 6.2 public and private attributes 41 7 NUMERICAL PRECISION AND MORE ABOUT VARIABLES 42 7.1 Entering numerical values 42 7.2 Numerical Accuracy 42 1 8 USE OF NUMERICAL LIBRARIES: NAG 44 8.1 A simple NAG example 44 8.2 A non-trivial NAG example: matrix determinant 44 9 SOME MORE TOPICS 47 9.1 The case statement and more about if 47 9.2 Other forms of do loops 48 SUGGESTED EXERCISE 2 49 Acknowledgements: This handout was originally prepared by Dr. Paul Alexander, and has been updated and maintained by Dr Peter Haynes of the TCM group. 2 1. The Basics In this section we will look at the basics of what a program is and how to make the program run or execute. The non-trivial example programs can be found in the directory: $PHYTEACH/part_2/examples with the name of the file the same as that of the program discussed in this guide. Some sections are more advanced and are indicated clearly indicated by a thick black line to the right of the text. These can be skipped certainly on a first reading and indeed you will be able to tackle the problems without using the material they discuss. 1.1 A very simple program A program is a set of instructions to the computer to perform a series of operations. Those operations will often be mathematical calculations, decisions based on equalities and inequalities, or special instructions to say write output to the screen. The program consists of “source code” which is “stored” in a text file. This code contains the instructions in a highly structured form. Each computer language has a different set of rules (or syntax) for specifying these operations. Here we will only consider the Fortran 90/95 (F95 for short) programming language and syntax. • Using emacs enter the following text into a file called ex1.f90, the .f90 part of the file name is the extension indicating that this is program source code written in the Fortran 90/95 language program ex1 ! ! My first program ! write(*,*) ’Hello there’ end program ex1 This is a complete F95 program. The first and last lines introduce the start of the program and show where it ends. Between the first and last lines are the program “statements”. The lines beginning with an exclamation mark are special statements called comments. They are not instructions to the computer, but instead are there to enable us (the programmer) to improve the readability of the program and help explain what the program is doing. The line beginning write is a statement giving a specific instruction to print to the screen. 3 Note that except within quotes: ⇒ Upper and lower case are NOT significant (different from Unix commands and files) ⇒ Blank lines and spaces are not significant. 1.2 Running the program Before we can run the program we must get the computer to convert this symbolic language (F95) into instructions it can understand directly. This process is called “compilation”. At the same time the computer will check our program source for errors in the syntax, but not for errors in our logic! In general programs will be assembled from source in many files; bringing all of these instructions together is called “linking”. We perform both of these tasks using the Unix command f95. • Type the following, the -o is an option saying where to place the output which in this case is a program which is ready to run, we call this an executable. (The default executable name is a.out). f95 -o ex1 ex1.f90 ¾ If you haven’t made any typing errors there should be no output to the screen from this command, but the file ex1 should have been created. By convention executable programs under Unix do not normally have a file extension (i.e. no “.xxx” in the file name). • To run the program type: ./ex1 ¾ Most Unix commands are files which are executed. The shell has a list of directories to search for such files, but for security reasons this list does not contain the current directory. The ‘./’ (dot slash) before ex1 tells the shell explicitly to look in the current directory for this file. ¾ The output should be the words “ Hello there”. • What happens if you make an error in the program? To see this let’s make a deliberate error. Modify the line beginning write to read: write(*,*) ’Hello there’ ’OK’ ¾ Save the file, and compile again : f95 -o ex1 ex1.f90 ¾ This time you get errors indicating that the syntax was wrong; i.e. you have not followed the rules of the F95 language! Correct the error by changing the source back to the original, recompile and make sure the program is working again. 4 1.3 Variables and expressions The most important concept in a program is the concept of a variable. Variables in a program are much like variables in an algebraic expression, we can use them to hold values and write mathematical expressions using them. As we will see later F95 allows us to have variables of different types, but for now we will consider only variables of type real. Variables should be declared before they are used at the start of the program. Let us use another example to illustrate the use of variables. • Enter the following program and save it to the file ex2.f90 program vertical ! ! Vertical motion under gravity ! real :: g ! acceleration due to gravity real :: s ! displacement real :: t ! time real :: u ! initial speed ( m / s) ! set values of variables g = 9.8 t = 6.0 u = 60 ! calculate displacement s = u * t - g * (t**2) / 2 ! output results write(*,*) ’Time = ’,t,’ Displacement = ’,s end program vertical • Compile and run the program and check the output is what you expect f95 -o ex2 ex2.f90 ./ex2 This program uses four variables and has many more statements than our first example. The variables are “declared” at the start of the program before any executable statements by the four lines: real :: g ! acceleration due to gravity real :: s ! displacement real :: t ! time real :: u ! initial speed ( m / s) After the declarations come the executable statements. Each statement is acted upon sequentially by the computer. Note how values are assigned to three of the variables and then an expression is used to calculate a value for the fourth (s). 5 Unlike in an algebraic expression it would be an error if, when the statement calculating the displacement was reached, the variables g, t and u had not already been assigned values. Some other things to note: 1. Comments are used after the declarations of the variables to explain what each variable represents. 2. The ‘*’ represents multiplication 3. The ‘**’ is the operator meaning “raise to the power of”, it is called technically exponentiation. 4. In this program we have used single letters to represent variables. You may (and should if it helps you to understand the program) use longer names. The variable names should start with a character (A-Z) and may contain any character (A-Z), digit (0-9), or the underscore (_) character. 5. Upper and lower case are not distinguished. For example therefore the variables T and t, and the program names vertical and Vertical are identical. The usefulness of variables is that we can change their value as the program runs. All the standard operators are available in expressions. An important question is if we have the expression g * t **2 what gets evaluated first? Is it g*t raised to the power of 2 or t raised to the power 2 then multiplied by g? This is resolved by assigning to each operator a precedence; the highest precedence operations are evaluated first and so on. A full table of numeric operators is (in decreasing precedence) Operator Precedence Meaning ** 1 Raise to the power of * 2 Multiplication / 2 Division + 3 Addition or unary plus - 3 Subtraction or unary minus You can change the precedence by using brackets; sub-expressions within brackets are evaluated first. Let’s look at ways of improving this program. An important idea behind writing a good program is to do it in such a way so as to avoid errors that you may introduce yourself! Programming languages have ways of helping you not make mistakes. So let’s identify some possible problems. • The acceleration due to gravity is a constant, not a variable. We do not wish its value to change. 6 • We want to avoid using a variable which is not given a value; this could happen if we mistyped the name of a variable in one of the expressions. Consider the following modified form of our program: program vertical ! ! Vertical motion under gravity ! implicit none ! acceleration due to gravity real, parameter :: g = 9.8 ! variables real :: s ! displacement real :: t ! time real :: u ! initial speed ( m / s) ! set values of variables t = 6.0 u = 60 ! calculate displacement s = u * t - g * (t**2) / 2 ! output results write(*,*) ’Time = ’,t,’ Displacement = ’,s end program vertical We have changed three lines and some of the comments. The line: implicit none is an important statement which says that all variables must be defined before use. You should always include this line in all programs.1 The second change is to the line: real, parameter :: g = 9.8 This in fact defines g to be a constant equal to the value 9.8; an attempt to reassign g via a statement like the one in the original version (g = 9.8 on a line by itself) will now lead to an error. The syntax of this statement is as follows: After the definition of the variable type real we give a series of options separated by commas up until the ‘::’ after which we give the variable name with an optional assignment. 1 It is an unfortunate legacy of older versions of Fortran that you could use variables without defining them, and in that case Fortran supplied rules to determine what the variable type was. 7 We will meet more options later. Try out these new ideas: • Make these changes and make sure the program compiles. • Now make some deliberate errors and see what happens. Firstly add back in the line g = 9.8 but retain the line containing the parameter statement. • Compile and observe the error message. • Now change one of the variables in the expression calculating s, say change u to v. Again try compiling. • Fix the program. 1.4 Other variable types: integer, complex and character As we have hinted at, there are other sorts of variables as well as real variables. Important other types are integer, complex and character. Let’s first consider integer variables; such variables can only hold integer values. This is important (and very useful) when we perform calculations. It is also worth pointing out now that F95 also distinguishes the type of values you include in your program. For example a values of ‘3.0’ is a real value, whereas a value of ‘3’ without the ‘.0’ is an integer value. Some examples will illustrate this. Enter the following program: program arithmetic implicit none ! Define real and integer variables real :: d, r, rres integer :: i, j, ires ! Assign some values d = 2.0 ; r = 3.0 i = 2 ; j = 3 ! Now the examples rres = r / d ! Print the result, both text and a value. ! Note how the text and value are separated by ! a comma write(*,*) ’rres = r / d : ’,rres ! now some more examples ires = j / i; write(*,*) ’ires = j / i : ’,ires ires = r / i; write(*,*) ’ires = r / i : ’,ires rres = r / i; write(*,*) ’rres = r / i : ’,rres end program arithmetic 8 First some things to note about the program: 1. We can declare more than one variable of the same type at a time by separating the variable names with commas: real :: d, r, rres 2. We can place more than one statement on a line if we separate them with a semicolon: d = 2.0 ; r = 3.0 • Compile and run the program. Note the different output. The rule is that for integer division the result is truncated towards zero. Note that the same rules apply to expressions containing a constant. Hence: ires = 10.0 / 3 ! value of ires is 3 rres = 10 / 3 ! value of rres is 3.0 rres = 10.0 / 3.0 ! value of rres is 3.333333 • Make sure you are happy with these rules; alter the program and try other types of expression. Some expressions look a little odd at first. Consider the following expression: n = n + 1 where n is an integer. The equivalent algebraic expression is meaningless, but in a program this is a perfectly sensible expression. We should interpret as: “Evaluate the right hand side of the expression and set the variable on the left hand side to the value evaluated for the right hand side”. The effect of the above expression is therefore to increment the value of n by 1. Note the role played by the ‘=’ sign here: it should be thought of not as an equality but instead as an “assignment”. The complex type represents complex numbers. You can do all the basic numerical expressions discussed above with complex numbers and mix complex and other data types in the same expression. The following program illustrates their use. • Enter the program, compile and run it. Make sure you understand the output. program complex1 implicit none ! Define variables and constants complex, parameter :: i = (0, 1) ! sqrt(-1) complex :: x, y x = (1, 1); y = (1, -1) write(*,*) i * x * y end program complex1 9 The character data type is used to store strings of characters. To hold a string of characters we need to know how many characters in the string. The form of the definition of characters is as follows: character (len = 10) :: word ! word can hold 10 characters We will meet character variables again later. Rules for evaluating expressions The type of the result of evaluating an expression depends on the types of the variables. If an expression of the form a operator b is evaluated, where operator is one of the arithmetic operations above (+, -, *, /, **) the type of the result is given as follows with the obvious symmetric completion: Type of a Type of b Type of result integer integer integer integer real real integer complex complex real real real real complex complex complex complex complex N.B. The result of evaluating an integer expression is an integer, truncating as necessary. It is worth emphasising this again, although we met it above, since a very common error is to write ‘1 / 2’ for example, which by the above rules evaluates to zero. This can lead to non-obvious errors if hidden in the middle of a calculation. When a complex value is raised to a complex power, the principal value (argument in the range -π, π) is taken. Assignments take the form variable = expression, where variable has been declared and therefore has a type. If the type of the two do not agree, the following table determines the result Variable Expression Value assigned integer real truncated value integer complex truncated real part real integer convert to real real complex real part complex integer real part assigned value, imaginary part zero complex real real part assigned value, imaginary part zero 10 1.5 Intrinsic functions So far we have seen how to perform simple arithmetic expressions on variables. Real problems will involve more complicated mathematical expressions. As we shall see later, F95 enables you to define your own functions which return values. However, some functions are so common and important that they are provided for us as part of the language; these are called intrinsic functions. Let us consider a program to compute projectile motion. The program computes the horizontal, x, and vertical, y, position of the projectile after a time, t: x = u t cos a y = u t sin a - g t2 / 2 program projectile implicit none ! define constants real, parameter :: g = 9.8 real, parameter :: pi = 3.1415927 real :: a, t, u, x, y real :: theta, v, vx, vy ! Read values for a, t, and u from terminal read(*,*) a, t, u ! convert angle to radians a = a * pi / 180.0 x = u * cos(a) * t y = u * sin(a) * t – 0.5 * g * t * t vx = u * cos(a) vy = u * sin(a) - g * t v = sqrt(vx * vx + vy * vy) theta = atan(vy / vx) * 180.0 / pi write(*,*) ’x: ’,x,’ y: ’,y write(*,*) ’v: ’,v,’ theta: ’,theta end program projectile • Compile and run the program. It will wait. The statement “read(*,*)…” is requesting input from you. Enter three values for a, t and u. You should now get some output. • Examine this program carefully and make sure you understand how it works. • Note especially how we use the functions cos, sin, atan and sqrt much as you would use them in algebraic expressions. As always upper and lower case are equivalent. 11 Common Intrinsic Functions Name Action ABS(A) absolute value of any A ACOS(X) inverse cosine in the range (0,π) in radians AIMAG(Z) imaginary part of Z AINT(X [,KIND]) truncates fractional part towards zero, returning real ANINT(X [,KIND]) nearest integer, returning real ASIN(X) inverse sine in the range (-π/2,π/2) in radians ATAN(X) inverse tangent in the range (-π/2,π/2) in radians ATAN2(Y,X) inverse tangent of Y/X in the range (-π,π) in radians CMPLX(X [,Y][,KIND] converts to complex X+iY; if Y is absent, 0 is used CONJG(Z) complex conjugate of Z COS(W) cosine of argument in radians COSH(X) hyperbolic cosine EXP(W) exponential function FLOOR(X) greatest integer less than X INT(A [,KIND]) converts to integer, truncating (real part) towards zero KIND(A) integer function, returns the KIND of the argument LOG(W) natural logarithm: if W is real it must be positive, if W is complex, imaginary part of result lies in (-π,π) LOG10(X) logarithm to base 10 MAX(R1,R2...) maximum of arguments, all of the same type MIN(R1,R2...) minimum of arguments, all of the same type MOD(R1,R2) remainder of R1 on division by R2, both arguments being of the same type (R1-INT(R1/R2)*R2) MODULO(R1,R2) R1 modulo R2: (R1-FLOOR(R1/R2)*R2) NINT(X [,KIND]) nearest integer REAL(A [,KIND]) converts to real SIGN(R1,R2) absolute value of R1 multiplied by the sign of R2 SIN(W) sine of argument in radians SINH(X) hyperbolic sine SQRT(W) square root function; for complex argument the result is in the right half-plane; a real argument must be positive TAN(X) tangent of argument in radians TANH(X) hyperbolic tangent • F95 has a set of over a hundred intrinsic functions, those in the list above are the most useful for scientific applications. • In this list A represents any type of numeric variable, R a real or integer variable, X and Y real variables, Z a complex variable, and W a real or complex variable. • Arguments in square brackets are optional. For an explanation of kind see section 7. 12 1.6 Logical controls So far all the programming statements we have met will simply enable us to produce efficient calculators. That is useful, but there is a lot more to programming. In this and Section 1.8 we introduce two crucial ideas. The first is the idea of taking an action conditional upon a certain criteria being met. An example will help to introduce this idea. For many years it was the case in Part IA of the Tripos that your maths mark was only included if it improved your overall result. Let us write a program to perform that simple sum. We read in four marks and output a final average. program tripos1 implicit none real :: p1, p2, p3, maths real :: av1, av2 ! read in the marks read(*,*) p1, p2, p3, maths ! work out two averages av1 = p1 + p2 + p3 av2 = av1 + maths av1 = av1 / 3.0 ; av2 = av2 / 4.0 ! use an if statement if (av2 > av1) then write(*,*) ’Final average = ’,av2 else write(*,*) ’Final average = ’,av1 end if end program tripos1 • Compile and run this program and make sure you understand how it works. • Note how the statements are indented. We use indenting to help show the logical structure of the program; indented statements are executed depending on the output of the test done by the if statement. The indenting is not essential, but it leads to a program which is much easier to follow. If you choose this style you can indent each level by any number of spaces as you wish. The if statement is the simplest, but most important, of a number of ways of changing what happens in a program depending on what has gone before. It has the general form: if (logical expression) action As another example we can use it to check for negative values: if (x < 0) x=0 ! replace negative x with zero 13 The if construct may also be used in more extended contexts (as above), such as: if (logical expression) then xxx else xxx end if Here if the condition is false the statements following the else are executed. We can also include additional tests which are treated sequentially; the statements following the first logical test to be reached which is true are executed: if (logical expression) then xxx else if (logical expression) then xxx else xxx end if Operators which may occur in logical expression are as follows: .lt. or < less than .le. or <= less than or equal .eq. or == equal .ge. or >= greater than or equal .gt. or > greater than .ne. or /= not equal .not. not .and. and .or. inclusive or and of course, brackets. Using brackets and the .not., .and. and .or. forms we can build up complicated logical expressions. • As an exercise consider the following. Suppose the rules for Part IA of the Tripos were changed so that: 1. The full maths course is always counted in the average 2. Quantitative biology mark is only counted if it improves the average 3. Elementary maths for biology is never counted. • Modify the program tripos1 to compute the average mark. One further piece of information is required which is an integer code indicating the type of maths paper taken. This integer code can be assumed to take the values: Full maths 1 Quantitative biology 2 Elementary maths 3 • One possible solution is available in the examples directory as tripos2.f90 14 if clauses may appear nested, that is one inside another. Suppose we wish to compute the expression ( ) adbx /= which fails if d < 0 or a is zero. If these were entered by a user then they could (incorrectly) take on these values. A good program should check this. Here is some code to do this which illustrates nested if clauses if (a /= 0.0) then if (d < 0.0) then write(*,*) ’Invalid input data d negative’ else x = b * sqrt(d) / a end if else write(*,*) ’Invalid input data a zero’ end if 1.7 Advanced use of if and logical comparisons In a large program it is likely that if clauses will be nested, i.e. appear one within another. This causes us no problems, but might make it less clear which end if goes with which if. To overcome this we can name the if clauses. An example illustrates the syntax. Let’s use the example we have just met: outer: if (a /= 0.0) then inner: if (d < 0.0) then write(*,*) ’Invalid input data d negative’ else inner x = b * sqrt(d) / a end if inner else outer write(*,*) ’Invalid input data a zero’ end if outer The names are outer and inner; note the syntax, especially the colon. Named if clauses are useful when you want to make your intention clear, but are not essential. The logical expressions we have met in if clauses can be used more generally with a logical variable. Logical variables take on the value of .true. or .false.. Here is a simple example which illustrates their use. logical :: l1, l2 l1 = x > 0.0 l2 = y /= 1.0 if (l1 .and. l2) then… This program fragment could equally well have been written if ((x > 0.0) .and. (y /= 1.0)) then Using logical variables may make some things easier to understand. 15 1.8 Repeating ourselves with loops: do Loops are the second very important concept needed in a program. If a set of instructions needs to be repeated, a loop can be used to do this repetition. As we shall see we have a lot of control over the loop and this makes them extremely powerful; this is especially true when combined with the if clauses we have just met. The general form of the do loop is: do var = start, stop [,step] xxx end do where as before the parts in square brackets are optional. ¾ var is an integer variable ¾ start is the initial value var is given ¾ stop is the final value ¾ step is the increment by which var is changed. If it is omitted, unity is assumed The loop works by setting var to start. If var ≤ stop the statements up to the end do are executed. Then var is incremented by step. The process then repeats testing var against stop each time around the loop. ¾ It is possible for the included statements never to be executed, for instance if start > stop and step is 1. This program is an example which computes factorials: program factorial implicit none ! define variables, some with initial values integer :: nfact = 1 integer :: n ! compute factorials do n = 1, 10 nfact = nfact * n write(*,*) n, nfact end do end program factorial • Modify the factorial program as follows. Change 10 to 100 and insert the following line before the end do. if (n > 10) exit ¾ What output do you get? Why? The exit command terminates the loop. • Write a program to calculate the binomial coefficient nCr. The program should read in values from the user for n and r and write out the answer. 16 1.9 The stop statement We have just seen how the exit command can be used to terminate a do loop. If you wish execution of your program to cease, you can insert a stop statement; this can incorporate some text, which is output when your program halts and identifies where this happened, e.g. stop ’this is where it all ends up’ 1.10 Arrays A great deal of scientific computation involves the manipulation of vectors, matrices and more general arrays of numbers. In F95 we can have an array of variables set up in the declarations statements. How do we specify arrays? The simplest way is to give the dimension in parentheses. real :: a(3) ! a is an array of 3 values: a vector real :: m(3,3) ! m is a rank 2 array: a matrix We call the part in parentheses a shape. Each element of the array can be addressed in the program using a similar notation. Here is a simple example: program vector implicit none real :: v(3) real :: x integer :: i v(1) = 0.25 v(2) = 1.2 v(3) = 0.2 ! compute the modulus squared of the vector x = 0.0 do i=1,3 x = x + v(i)*v(i) end do write(*,*) ’Modulus squared = ’,x end program vector Notice how we use a loop to compute the sum over all elements of the vector. A second example will show us how we can implement simple vector and matrix operations: 17 program linalg implicit none real :: v1(3), v2(3), m(3,3) integer :: i,j v1(1) = 0.25 v1(2) = 1.2 v1(3) = 0.2 ! use nested do loops to initialise the matrix ! to the unit matrix do i=1,3 do j=1,3 m(j,i) = 0.0 end do m(i,i) = 1.0 end do ! do a matrix multiplication of a vector ! equivalent to v2i = mij v1j do i=1,3 v2(i) = 0.0 do j = 1,3 v2(i) = v2(i) + m(i,j)*v1(j) end do end do write(*,*) ’v2 = ’,v2 end program linalg • Enter this program, compile and run it. Make sure you understand the output. • Try modifying the program to multiply two matrices. We can also have arrays of integer, complex or any other data types declared in analogous ways. Arrays may be declared using other forms than those given above which can be useful for different situations. The dimension option to the declaration may be used to set a shape for all declared variables which do not have a particular shape specified for them. The dimension statement serves several purposes in a F95 declaration. In the following, note the critical nature of the punctuation, particularly ‘,’, ‘:’ and ‘::’. An example of the simplest form of an array declaration for a matrix might be: real, dimension(10,11) :: a ! a is a rank 2 array 18 The array subscripts are assumed to start at unity, but this can be altered by using the explicit form of the declaration in which the range of the array subscripts is given separated by a colon: real, dimension(0:9) :: a ! vector of 10 elements ! starting at 0 We can also declare a variable to be an array of unknown size. We do this as follows real, dimension(:), allocatable :: a and at an appropriate time, when it is known how big this array needs to be, we use the following (say to create an array of 10 elements): m=10 allocate(a(m)) where m is an integer variable. When the use of the array in the program is finished, the space can be released by using deallocate(a) 1.11 Array arithmetic One very useful feature of F95 is the ability to work with whole arrays. In most programming languages one can, say, add two arrays using a loop. For example real :: a(10), b(10), c(10) integer :: i [some statements to setup the arrays] do i=1,10 c(i) = a(i) + b(i) end do F95 allows you to perform whole array operations in a natural way. Most of the normal arithmetic operations can be carried out on arrays, where they apply in an element by element fashion. The above example can be written as: real :: a(10), b(10), c(10) [some statements to setup the arrays] c = a + b 19 • Here are some more examples which illustrate array arithmetic: real, dimension(3,3) :: a, b real, dimension(3) :: x, y, z integer, dimension(10) :: idx idx = 1 ! set all elements of idx to 1 a=b ! copies the array b into a x=y+1 ! x(i) = y(i)+1 for i=1,2,3 z=atan2(y,x) ! z(i) = atan2(y(i),x(i)) for i=1,2,3 • You can refer to a subset of an array and treat it as another array: ¾ z( (/1,3,6/) ) a length 3 array with elements set toz(1), z(3), z(6) ¾ z(m:n) is an array of length (n-m+1) formed from the elements of z starting at m and ending at n ¾ z(m:n:c) is an array of length (n-m+1)/c formed from the elements of z starting at m and ending at n incremented by c ¾ x(1:5) = y(2:6) copies elements 2 to 6 of y into elements 1 to 5 of x ¾ z(1:3) = y(1:5:2) copies elements 1,3,5 of y into elements 1,2,3 of z ¾ a(2,:) = z copies the vector z into the second row of a • There is a conditional, where, which operates on an array for simple function forms e.g. to replace negative elements of an array z with their absolute values: where (z < 0.0) z=-z ¾ More generally it has the form: where (logical array test) [statements if test true] elsewhere [statements if test false] end where ¾ For example to take the logarithm of the positive elements of an array: real, dimension(1000) :: a where (a > 0.0) a = log(a) elsewhere a = 0.0 end where • There are a number of intrinsic procedures taking array arguments e.g. dot_product takes 2 arguments of rank 1 and the same size and returns their inner product matmul performs matrix multiplication on 2 array arguments with compatible size and rank maxval returns maximum element of an integer or real array minval returns minimum element of an integer or real array product returns the product of the elements of an array sum returns the sum of the elements of an array 20 2 Good Programming Style In section 1 we have covered some of the basics of programming. We will return to programming later when we look in even more detail at F95. In this section we will briefly consider some rules for good practice in developing programs. When you come to tackle the computing exercise we will be looking for how you have tackled some of the issues we shall now discuss. 2.1 Readability Your program should be as easy to follow in terms of its logical structure as possible. There are a number of ways we have already met that help us do this. Let us recap some of them. First use comments. A general rule for comments is that you should use a comment when the F95 statements you write are not self explanatory. There is no need, for example, to add comments to obvious computational expressions. However you may want to add comments to the top of a block of expressions explaining how the following code relates to the physical problem. • Similarly if you have a loop, a comment of the form below is of no help: ! loop from 1 to 10 do i=1,10 ¾ But a comment of the following form, say in a program calculating a binomial might be very useful: ! loop to calculate nCr do k=1,r So use comments sensibly to make the code understandable. • What we don’t want to hear is: “I have written my program, now I just need to comment it before handing it in” This is bad practice because comments are part of the program and should be there as much to help you follow your own intentions in programming as for the head of class to follow it. • Another aspect of readability is indenting code blocks between do loops and in if clauses. This is very good practice. It uses the layout of the program to show at a glance the logical structure. Our strong advice is to make good use of indenting. Again it helps as much in program development as it does in presenting the final program. 21 2.2 Self-checking code We have already seen in some of the examples how we can use checks to avoid numerical errors. There are a number of numerical operations which are “poorly defined”. These include, among many others: a) division by zero b) taking the square root or logarithm of a negative real number Alternatively we may know the range of possible allowed values for a variable and can include checks to make sure this is not violated. Sometimes we can be sure that variables cannot take on illegal values, other times we cannot. For example values may be supplied to the program by a user and the values may be wrong. Alternatively we may know that under certain conditions a variable may, for example, become negative and all this really means is that it should be set equal to zero; in fact the formula we are computing may explicitly state something like: ⎩ ⎨ ⎧ >= otherwise0 0... xz . In either case you must be careful to check arguments to make sure they are “in range”. We have seen examples of this already and you should go back now and revise these methods. Once again it is essential in program design to be sensible. Do not check a variable if it cannot be out of range; this just slows your code down. For example the following would be bad programming style: real :: x [some statements] x = sin(y) + 1.0 if (x >= 0.0) z = sqrt(x) Here x can never be less than zero; the test is not wrong, but clearly unnecessary and indicates a poor appreciation of the logic of the program. 2.3 Write clear code that relates to the physics We are not aiming in this course to develop ultra-efficient programs or the shortest possible program etc. Our aim is for you to learn the basics of computational physics. Therefore you should aim to write your code so that it relates as clearly as possible to the physics and computational physics algorithms you are using as possible. You can split long expressions over many lines, for example, by using the continuation marker. 22 If the last character of a line is an ampersand ‘&’, then it is as if the next line was joined onto the current one (with the ‘&’ removed). Use this to lay out long expressions as clearly as possible. • Another technique is to split long expressions using intermediate calculations. A simple example would be replacing something like: res = sqrt(a + b*x + c*x*x + d*x*x*x) + & log(e * f / (2.345*h + b*x)) with t1 = a + b*x + c*x*x + d*x*x*x t2 = E * F / (2.345*h + b*x) res = sqrt(t1) + log(t2) • Think about the choice of variable names. You can make the variable names very clear with names such as energy or momentum. This can be very helpful, but also cumbersome in long expressions. A useful rule of thumb is that if there is an accepted symbol for a physical quantity consider using that (e.g. E and p); use longer more descriptive names if one does not exist. We will return to the topic of programming style later when we consider how the program can be broken up into smaller units. This will be the main job in the next section of the course. 23 3. Input to and output from a F95 program We have already seen some examples of outputting information from a program (write) and reading information from the terminal (read). In this section we will look in detail at input and output and explain those strange ‘*’s. To save some writing let’s introduce some jargon: we will call input and output I/O. 3.1 F95 statements for I/O Input and output to a F95 program are controlled by the read and write statements. The manner in which this is done is controlled by format descriptors, which may be given as character variables or provided in a format statement. For economy of effort we will only outline the latter method, together with the default mechanism. The form of the I/O statements is as follows: read(stream, label [, end=end][, err=err]) list and write(stream, label) list where • stream is a number previously linked to a file, or a character variable, or *, where * here indicates the default value, usually the screen of a terminal session. If stream is a character variable, the result of the write is stored in that variable, and can be manipulated as such within the program. • label is the number of a format statement, or * for free format. • list is a list of items to be transferred, separated by commas, possibly including text strings enclosed in quotation marks. • The optional items end and err are so that you can provide statement labels end and err to which control moves in the event that the end of data is reached prematurely (end) , or some error is encountered (err). The precise details of how the output should look are governed by the format definition. This takes the form: label format (format descriptors) • label is an integer, corresponding to the label appearing in the read or write statement. More than one read or write can refer to the same label. • format descriptors is a comma-separated list of items describing how the output is to be presented, possibly including text items. The latter should be enclosed in single quotation marks as in character strings. 24 Possible formats for numeric items are nIw to output integers nFw.d to output real or complex in fixed-point form nEw.d to output real or complex in floating-point form n is an optional repeat count (how many of each item is in the I/O list). w is the total number of characters per number, including signs and spaces: the field width. d is the number of decimal digits to be output within w. For non-numeric items the descriptor Aw is available, which causes the next w characters to be output To access a file for input or output you can use the open statement: open([unit=]stream, err=escape, action=action, file=name) There are further possible arguments which should not be needed for this course, but which you may look up in a text book. • stream is the identifier which appears in the read or write statement • escape is a statement label, to which control is transferred if there is a problem opening the file. • action is one of ‘read’, ‘write’ or ‘readwrite’, depending how you intend to use the file. • name is the name of the file (in quotes) and may be held in a character variable. Having opened a file, linking it to a stream, and read through it, you can move back to the beginning using: rewind(stream) When you have completed I/O to a particular file, you can use the close instruction to close the file and tidy things up: close(stream) • Try entering the following example, compile and run it and examine the output program format1 implicit none integer :: i do i=1,20 write(*,1) i, i*i, i**3 end do 1 format(i4,i6,i8) end program format1 25 • Modify the program to use “free-format” output and compare the results (you could output both in one program for comparison of course). • The next example illustrates how to send the output from your program to a file program fileio implicit none integer :: i open(20, file=’cubes.dat’) do i=1,100 write(20,1) i, i*i, i**3 end do close(20) 1 format(i4,i6,i8) end program fileio • Modify the program to send a copy of the output to two files simultaneously. 26 4 Graphics and Visualisation There are many ways in which to plot data. It is very difficult to write graphics applications in F95, so generally it is easier (and better) to use an application. One such application is gnuplot, which is a free plotting program that can plot data files and user-defined functions. It can’t do everything you might possibly want, but it is very easy to use. We introduce gnuplot here – there is documentation available via ‘help’ within the program and on the course web site. gnuplot is a command-line driven program. Typing gnuplot at the terminal you will see that the prompt changes. You will want to use the help command to find out more information. You will also be able to output graphs from gnuplot in a form you can import into, say, Microsoft Word, when you produce your report. 4.1 Plotting a data file gnuplot expects data to be arranged in columns in an ordinary text file, e.g. # Gnu population in Antarctica since 1965 1965 103 1970 55 1975 34 1980 24 1985 10 You can have as many columns as you like. Comments are indicated by ‘#’. The recommended way you use gnuplot to produce results from your F95 program is therefore to write out results to a file and use gnuplot to plot them. Let’s look at a simple F95 program to do just that. program outputdata implicit none real, dimension(100) :: x, y integer :: i ! setup x and y with some data do i=1,100 x(i) = i * 0.1 y(i) = sin(x(i)) * (1-cos(x(i)/3.0)) end do ! output data to a file open(1, file=’data1.dat’, status=’new’) do i=1,100 write(1,*) x(i), y(i) end do close(1) end program outputdata 27 The file data1.dat should contain the two columns of numbers, exactly the format needed by gnuplot. To plot the data is very easy: • Enter this program, compile and run it and produce the data file data1.dat. • Start up gnuplot. • Within gnuplot give the command: plot ’data1.dat’ 4.2 Getting help You can get help by typing ‘?’ or ‘help’ within gnuplot. The on-line help is very good. You can also abbreviate commands to save typing. 4.3 Further examples As well as plotting data we can plot functions in gnuplot. • For example to plot one of the trigonometric functions type the following: plot sin(x)*cos(x) • In fact, gnuplot lets us define a function: pop(x) = sin(x)*(1-cos(x/3.0)) • Then we can plot this function, for 0 ≤ x ≤ 10, say, as follows: plot [0:10] pop(x) • To plot the data file created using the F95 program of the previous section we can use: plot ’data1.dat’ • And you can also plot both the function and the data together: plot [0:10] ’data1.dat’, pop(x) • By default, data files are plotted point by point. If you want lines joining the points: plot ’data1.dat’ with linesp • If you want lines only: plot ’data1.dat’ w lines • To control which colour each set of lines and points comes out, see help plot. For example, to make the data come out with colour 2 (dotted lines), and pop(x) with colour 1, plot [0:10] ’data1.dat’ w lines 2 2, pop(x) \ w lines 1 1 28 • The backslash enables you to continue on to another line if the commands become long. • To plot column 4 of ‘flib.dat’ against column 2 of the same file: plot ’flib.dat’ u 2:4 w linesp • this gives column 2 on the x axis and 4 on the y axis. You can also plot points with error bars. The following command plots column 4 versus column 2, with columns 5 and 6 defining the upper and lower error bars: plot ’flib.dat’ u 2:4:5:6 w errorbars 4.4 Printing graphs into PostScript files • The following sequence changes the terminal type to PostScript and replots the most recent plot to a file called file.ps: set term post set output ’file.ps’ replot • Don’t forget to set the terminal type back to X11 when you are done plotting to the file. set term X • In order to get graphs that are readable when included in papers, I recommend: set size 0.6,0.6 before plotting to the file. This reduces the size of the graph while keeping the font size and point size constant. There are other output types from gnuplot. In particular you may want to use the CGM terminal type instead of the PostScript terminal type as above (use cgm instead of post). This produces a file which can be read directly into Microsoft Word and converted to a Word drawing. 29 Suggested Exercise 1 Modify the projectile program to output the path of a projectile launched from the ground. The program should read in the initial speed, the angle to the horizontal and a time interval at which to output a series of x,y values (as two columns) suitable for plotting with gnuplot. Points should be output until the projectile reaches the horizontal plane in which it was launched. It is a good idea to show this program to a demonstrator or head of class and discuss with them how it might be improved. You can find a solution to this problem in the examples directory $PHYTEACH/part_2/examples/exercise1.f90 30 5. Program Organisation: Functions and Subroutines In this section we will consider ways to structure your program, and in particular the use of functions and subroutines. We shall start off by assuming that the functions are to be defined in the same file as the main program. However this is rather cumbersome for a large program, and in the next section we will consider how to split the program between multiple files. Splitting the program in this way results in different program segments. Subroutines and functions are the main way in which you can structure your F95 program. The main difference between them is that a function returns a value through its name, while the subroutine does not (a function must have at least one argument, while a subroutine may have none). The type of the function name must be declared both where it is defined and in the segment from which it is called. The function is used like any of the intrinsic functions while the subroutine is accessed via a call statement as we shall see below. We will use the term routine to refer to both subroutines and functions. Direct communication between one program segment and a routine is through the arguments. Operations coded in the routine act directly on the variables referred to via the arguments in the calling statement. Other variables declared within the routine are local to that routine and not in general accessible elsewhere, or indeed in a subsequent call to the routine. In what follows we will introduce functions and subroutines and then discuss how grouping functions and subroutines in different files can be an additional way of structuring your program. 5.1 Functions We have already met the idea of functions in the form of intrinsic functions, i.e. functions which are part of F95 and which are used to calculate standard mathematical functions. We can also define our own functions for use in a program; this is a very powerful feature of all programming languages. Using functions has a number of advantages: 1. The code to implement the calculation can be written just once, but used many times. 2. The functions can be tested separately from the rest of the program to make sure they give the correct result; we can then be confident when using them in the larger program. Let’s consider an example where we define two functions; in fact this is a simple program to find the root of an equation using Newton’s method. The functions we define are for the mathematical function we are seeking the root of and its first derivative. 31 program newton ! ! Solves f(x) = 0 by Newton’s method ! implicit none integer :: its = 0 ! iteration counter integer :: maxits = 20 ! maximum iterations integer :: converged = 0 ! convergence flag real :: eps = 1.0e-6 ! maximum error real :: x = 2 ! starting guess ! introduce a new form of the do loop do while (converged == 0 .and. its < maxits) x = x - f(x) / df(x) write(*,*) x, f(x) its = its + 1 if (abs(f(x)) <= eps) converged = 1 end do if (converged == 1) then write(*,*) ’Newton converged’ else write(*,*) ’Newton did not converge’ end if contains function f(x) real :: f, x f = x**3 + x - 3.0 end function f function df(x) ! first derivative of f(x) real :: df, x df = 3 * x**2 + 1 end function df end program newton There are lots of things to notice about this example. • We have introduced a modified form of the do statement. Instead of looping a fixed number of times we loop while a condition is true. We shall return to this later. • The functions f and df are defined. They appear in the program after the word contains. contains marks the end of the main code of the program and indicates the start of the definition of functions and, as we shall see in a moment, subroutines. 32 • Each function starts with the keyword function and ends with end function and the function name; this is quite like the syntax we have already met introducing the program itself. • Notice how within the function the function name is treated like a variable; we define its type and assign it a value; this is the value returned from the function and used when the function is called. • The functions defined in this way are called internal functions since they are defined within the program itself. We shall see the nature of this in a little while. • Values are supplied to the function via its arguments, i.e. the variables specified in parentheses (the x in this example); note each argument must have its type specified. ¾ Enter this program, compile and run it. Try to understand the output. Modify the functions to some other form to find the roots of different functions using Newton’s method. 5.2 Formal definition The structure of a routine is the same as for a main program, except that the first line defines the routine name. Thus the form for a function is: function name(arg1, arg2, ....) [declarations, including those for the arguments] [executable statements] end function [name] and for a subroutine: subroutine name(arg1, arg2, ....) [declarations, including those for the arguments] [executable statements] end subroutine [name] Additionally for functions name must occur in a declaration statement in the function segment itself. The value the function takes is defined by a statement where name is assigned a value just like any other variable in the function segment. The arguments must appear in the declarations as will any local variables; if you change the value of one of the arguments then the corresponding variable in the program “calling” the function or subroutine will also change. For this reason it is imperative that the type of dummy arguments agree with that of the variables in the call of the routine. If you wish to terminate execution of a function or subroutine before the last statement in the subroutine (e.g. you may test some condition with an if clause and decide there is no more to do) then you can use the return statement to terminate the routine. 33 5.3 Subroutines Subroutines are very similar to functions except that they do not return a value. Instead they have an effect on the program in which they are invoked by modifying the arguments to the subroutine. Here is a simple example which defines a subroutine called swap which swaps the values of its two arguments: program swapmain implicit none real :: a, b ! Read in two values read(*,*) a, b call swap(a,b) write(*,*) a,b contains subroutine swap(x, y) real :: x, y, temp temp = x x = y y = temp end subroutine swap end program swapmain • Compile this program and see what happens to the values typed in. You should find the values are interchanged. • Note how the subroutine is called and that its action is to modify its arguments. • One important difference between a function and subroutine is that a function must always have at least one argument, whereas a subroutine need not have any. 5.4 Local and global variables Variables that are defined within a given routine (and are not arguments to the routine) are local to that routine, that is to say they are only defined within the routine itself. If a variable is defined in the main program then it is also available within any internal routine (i.e. one defined after the contains statement). • We say that these variables are global. Here is a simple example: 34 program set implicit none real :: a, b ! Read in value of a read(*,*) a call setval(b) write(*,*) b contains subroutine setval(x) real :: x x = a ! value of a is from main program end subroutine setval end program set This program is very contrived, it simply sets the value of the argument of the call to setval to the value of the variable a in the main program. In this example a is a variable global to the main program and the subroutine setval. 5.5 Passing arrays to subroutines and functions Arrays can of course be used as arguments to subroutines and functions, however there are one or two special considerations. 5.5.1 Size and shape of array known If the size and shape of the array are known, one can simply repeat the definition in the subroutine when the arguments are declared. For example, if v and r are vectors of length 3 a subroutine declaration in which they are passed would look like: subroutine arr1(v, r) real :: v(3) real :: r(3) and of course you could have used the alternative form using dimension instead: subroutine arr1(v, r) real, dimension(3) :: v, r 5.5.2 Arrays of unknown shape and size The problem is how to deal with arrays of unknown length. This is important when for example we wish to write a subroutine which will work with arrays of many different lengths. For example, we may wish to calculate various statistics of an array. We can in this case use the following definition: 35 subroutine stats(y, ybar) real, dimension(:) :: y real :: ybar ! local variables integer :: i, n n = size(y) ybar = 0.0 do i=1,n ybar = ybar + y(i) end do ybar = ybar / n end subroutine stats • Here we have declared the array y to be of unknown size, but of an assumed shape; that is it is a vector and not a matrix. The intrinsic function size returns the number of elements in the array i.e. the length of the vector. • We can do the same for a two-dimensional array, a matrix. The corresponding definition would be: real, dimension(:,:) :: a2d If you use this form for passing arrays and declare the subroutines in a different file from the calling program, it is essential that you use the concept of modules discussed in the section 6. 5.6 The intent and save attributes We have seen how subroutines can modify their arguments, but often we will want some of the arguments to be modifiable and others just values supplied to the routine. Recall how we discussed the use of parameters as a way of protecting us from misusing something which should be a constant. In a similar way F95 provides a way of stating which arguments to a routine may be modified and which not. By default all arguments can be modified. The extra syntactic protection offered is by the intent attribute, which can be added to the declaration. It can have the values out, inout or in depending on whether you wish to modify, or not, the value of the argument in the routine call. This is another example of an option to a declaration statement (again c.f. the parameter definition earlier). Here is the specification: subroutine name(arg1, arg2, .....) real, intent(in) :: arg1 real, intent(out) :: arg2 in the following example any attempt to change the variable x would result in a compiler error. 36 subroutine setval(x, y) ! set y to value of x real, intent(in) :: x real, intent(out) :: y y = x end subroutine setval • Modify the setval example above to use this subroutine. • Deliberately introduce an error and within the subroutine attempt to change the value of the first argument; try compiling the program and see what happens. Local variables which are declared within a routine will in general not retain their values between successive calls to the routine. It is possible to preserve values after one return into the next call of the routine by specifying the save attribute in the declaration of that variable, e.g. real, save :: local This of course cannot be used for the arguments of the routine. 37 6. Using Modules As programs grow larger it becomes less convenient to use a single file for the source code. Also it is very often the case that the functions and subroutines you have written can be used in more than one program. We need a way to organise the source code to make this easy to do. In this section we will look at modules which enable the program to be split between multiple files and make it easy to “re-use” the code you write in different programs. 6.1 Modules Modules are a very useful tool when we split programs between multiple files. Modules give us a number of advantages: 1. The module is useful for defining global data, and may be used as an alternative way of transmitting information to a routine. 2. The variables, etc., declared in a module may be made available within any routines at the choice of the programmer. 3. Modules can be imported for use into another program or subroutine. Functions and variables defined in the module become available for use. The compiler is also able to cross-check the calls to subroutines and functions and use of variables just as if they had been defined as internal routines. The definition of a module is module name [statement declarations] [contains [subroutine and function definitions] ] end module [name] The module is incorporated in the program segment or routine in which it is to be used by the use statement: use name • You can have as many different modules as you like. Each module will be contained in a separate file and may be compiled (and checked) separately. • Modules can be re-used between different programs and may be used multiple times in the same program. • The variables declared in the module before the contains statement are global to the module and become global variables in any routine in which the use statement appears. • The use statement can appear in the main program or any other subroutine or module which needs access to the routines or variables of another module. 38 We should now have a look at some examples. Let’s return to the swapmain program from section 5.3 and implement this in two files using modules. First edit the file swapmain.f90 and to contain: program swapmain use swapmod ! use statements must come first implicit none real :: a, b ! Read in two values read(*,*) a, b call swap(a,b) write(*,*) a, b end program swapmain Now in the second file, called swapmod.f90 the module is defined: module swapmod implicit none ! you need this in every module ! ! Global declarations would appear here if any ! contains ! routines provided by this module subroutine swap(x, y) real :: x, y, temp temp = x x = y y = temp end subroutine swap end module swapmod • Now compile the swapmod.f90 file: f95 -c swapmod.f90 ¾ The –c option is a request to compile only and not try to produce an executable program. ¾ Use ls to see what files you have. You should have swapmod.mod and swapmod.o files. 39 The swapmod.o file contains the compiled code for all the routines in the swapmod.f90 file. When you compile a module an extra file called name.mod is created. This file is needed for as long as you want to include the module in programs. • Now compile the complete program, making available the compiled code from the swapmod module so that the “linker” can bring together all the machine instructions necessary to define a complete working program. This is done by giving the name of the compiled routine on the f95 command line: f95 -o swap swapmain.f90 swapmod.o ¾ Run the program and check the output. Within the module you may define as many routines as you like; they are all included with the same use statement. Let us now look at an example which makes use of modules to provide global data. Again this is a rather contrived example designed to illustrate as simply as possible the use of modules. This time let’s have three files. • Let’s start with a file called gd.f90 which defines a module called gd; this defines some global data but no routines: module gd implicit none ! define global data real :: a_g, b_g end module gd • Note how we include an implicit none statement in the module; each program element (main program and each module) must have the implicit none statement. • Now the second file, setval.f90 defines a setv routine which is part of a module setval module setval ! make the global data available use gd implicit none contains subroutine setv(a, b) real :: a, b ! set arguments to global values a = a_g; b = b_g end subroutine setv end module setval 40 • Finally the main program testset.f90 program testset ! include global data use gd ! make available the setval routine use setval real :: x, y ! read values from terminal read(*,*) a_g, b_g ! set x and y and check output call setv(x, y) write(*,*) x, y end program testset • Compile the two modules: f95 -c gd.f90 f95 -c setval.f90 • Now the main program, creating an executable program: f95 -o testset testset.f90 gd.o setval.o • Run the program and check the output. These two examples, although simple, illustrate the usefulness of modules. 6.2 public and private attributes The examples of modules in the previous section were quite simple. In general you are likely to want a module which not only declares some global data, but also defines routines in the same module. However we may want to differentiate variables which are global and available in all routines in which the module is used, as we saw in the last section, and variables which are to be global only within the module (i.e. used by all the routines in the module, but not available in the routines or program segments which include the module via the use statement). We can in fact achieve this using the private and public attributes. For example the variable a_g in the following will be available in all program segments including the module in which this definition occurs, whereas the variable b is a global variable only within the current module real, public :: a_g real, private :: b The default is for variables is to be public. 41 7 Numerical precision and more about variables 7.1 Entering numerical values We have already seen how to enter simple numerical values into a program. What happens if the values need to be expressed in the form a × 10b? F95 provides a way of doing it. For example the numbers on the left of the following table are entered in the form indicated on the right: 1.345 1.345 2.34×107 2.34e7 or 2.34d7 2.34×10−7 2.34e-7 or 2.34d-7 7.2 Numerical Accuracy Variables in any programming language use a fixed amount of computer memory to store values. The real variables we have discussed so far have surprisingly little accuracy (about 6-7 significant figures at best). We will often require more accuracy than this, especially for any problem involving a complicated numerical routine (e.g. a NAG routine). In that case you will need to use what is called a “double precision variable”. Such variables have twice the accuracy of normal reals. F95 in fact provides a very flexible system for specifying the accuracy of all types of variables. To do this it uses the idea of kind; this really means the accuracy of a particular type of variable. You do not need to go into this in much detail, we will give a recipe for obtaining high precision, “double precision”, variables. Each variable type may have a number of different kinds which differ in the precision with which they hold a number for real and complex numbers, or in the largest number they can hold for integer variables. When declaring variables of a given type you may optionally specify the kind of that variable as we shall see below. A few intrinsic functions are useful: kind(x) Returns the kind (an integer variable) of the argument x. x may be any type. selected_int_kind(n) Return the kind necessary to hold an integer as large as 10n. selected_real_kind(p[,r]) Return the kind necessary to hold a real in the range 10-r to 10r with precision p (number of significant decimals). Here we want to specify variables of high precision and there is an easy way of doing this. The notation for a double precision constant is defined; so that for instance 1.0 is designated 1.0d0 if you wish it to be held to twice the accuracy as ordinary single precision. To define variables as double precision, you first need to establish the 42 appropriate kind parameter and then use it in the declaration of the variable. Here is a useful recipe: ! find the kind of a high precision variable, by finding ! the kind of 1.0d0 integer, parameter :: dp=kind(1.0d0) ! use dp to specify new real variables of high precision real(kind=dp) :: x,y ! and complex variables complex(kind=dp) :: z ! and arrays real(kind=dp), dimension(30) :: table The integer parameter dp can now also be used to qualify constants in the context of expressions, defining them as double precision, e.g. 0.6_dp 43 8 Use of numerical libraries: NAG There are many libraries of subroutines performing mathematical and graphics operations. The library you are encouraged to use here is the NAG library, which has an on-line documentation system. One important feature of the NAG library in particular is that all the variables are of double precision type, and all arguments must be declared as such when NAG routines are called. The on-line information is available via the link on the course web page. When using the NAG library you must tell the linker where to find the library code. This is done by adding some extra information to the f95 command line; e.g. let’s imagine we were compiling and linking (to form an executable program) a simple program in the file ex.f90. The command we need is: f95 -o ex ex.f90 -lnag The -lnag tells the linker to search the NAG library file for the code that it needs. 8.1 A simple NAG example • Enter this simple program and try compiling and running it; it should report some information about the NAG library we are going to use program naginfo use nag_f77_a_chapter implicit none write(*,*) ’Calling NAG identification routine’ write(*,*) call a00aaf end program naginfo 8.2 A non-trivial NAG example: matrix determinant Use the link on the course web page to bring up the NAG documentation. Find the documentation for the routine F03AAF. The first page of the documentation appears slightly cryptic: 44 F03AAF – NAG Fortran Library Routine Document Note. Before using this routine, please read the Users’ Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details. 1 Purpose F03AAF calculates the determinant of a real matrix using an LU factorization with partial pivoting. 2 Specification SUBROUTINE F03AAF(A, IA, N, DET, WKSPCE, IFAIL) INTEGER IA, N, IFAIL real A(IA,*), DET, WKSPCE(*) 3 Description This routine calculates the determinant of A using the LU factorization with partial pivoting, PA = LU, where P is a permutation matrix, L is lower triangular and U is unit upper triangular. The determinant of A is the product of the diagonal elements of L with the correct sign determined by the row interchanges. 4 References [1] Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer-Verlag 5 Parameters 1: A(IA,*) – real array Input/Output Note: the second dimension of A must be at least max(1,N). On entry: the n by n matrix A. On exit: A is overwritten by the factors L and U, except that the unit diagonal elements of U are not stored. 2: IA – INTEGER Input On entry: the first dimension of the array A as declared in the (sub)program from which F03AAF is called. Constraint: IA ≥ max(1,N). 3: N – INTEGER Input On entry: n, the order of the matrix A. Constraint: N ≥ 0. 4: DET – real Output On exit: the determinant of A. 5: WKSPCE(*) – real array Workspace Note: the dimension of WKSPCE must be at least max(1,N). 6: IFAIL – INTEGER Input/Output On entry: IFAIL must be set to 0, -1 or 1. On exit: IFAIL = 0 unless the routine detects an error (see Section 6). The note at the beginning draws attention to the bold italicised term real, which you should simply interpret as “double precision” or real(kind(1.0d0)) in all NAG documentation. 45 One would expect a subroutine which calculates a determinant to need arguments including the array in which the matrix is stored, the size of the matrix, and a variable into which to place the answer. Here there are just a couple extra. • Because the library is written in Fortran 77 (F77, an earlier version of Fortran than we are using), which does not support dynamic memory allocation, it is often necessary to pass “workspace” to a routine. The routine will use this space for its own internal temporary requirements. Hence WKSPCE(*). • Array dimensions of unknown size are denoted by a ‘*’, not a ‘:’. • Most NAG routines have the ifail argument which is used to control error handling and communicate error information back to the calling routine. In general you should set ifail = 0 before calling a NAG routine, and test it on return (zero indicates success). Here is the example, enter this program compile and test and make sure it gives the result you expect! program nagdet ! we make available the chapter f module of nag use nag_f77_f_chapter implicit none ! variables used in the program real(kind(1.0d0)) :: m(3,3), d, wrk(2) integer i, n, ifail ! assign values to only the upper 2x2 portion of ! the 3x3 array m(1,1)=2 ; m(1,2)=0 m(2,1)=0 ; m(2,2)=2 ! set up input values and call the NAG routine ! note the difference between i and n values i=3 ; n=2 ; ifail=0 call f03aaf(m,i,n,d,wrk,ifail) if (ifail == 0) then write(*,*) ’Determinant is ’,d else write(*,*) ’F03AAF error:’,ifail end if end program nagdet Further examples of F95 programs calling the NAG library can be found in the lecture handout on the Physics of Computational Physics. 46 9 Some more topics This section can be skipped at a first reading. 9.1 The case statement and more about if If you have several options to choose between in a program, you can use the case statement, which takes the form: select case (expression) case(value1,value2) . . case(value3) . . case default . . end select • expression must be an expression which evaluates to an integer, character or logical value. • value1, value2, … must be possible values for the expression. • if the expression matches either of value1 or value2 then the code following the case statement listing that value is executed. • If no match is found, the code following the optional case default statement is executed. A simple example is the use of a case statement to take action on user input to a program: program choice implicit none character(len=1) :: x write(*,*) ’Select a or b’ read(*,*) x select case(x) case(’A’,’a’) write(*,*) ’A selected’ case(’B’,’b’) write(*,*) ’B selected’ case default write(*,*) ’Error: unknown option’ end select end program choice 47 The same outcome could have been achieved using an if clause as follows: if (logical expression) then … else if (logical expression) then … else if (logical expression) then … else … end if • Try writing the above case statement using this form of the if clause. Ask a demonstrator to check that it is correct. 9.2 Other forms of do loops We have already met the do loop in which the number of times round the do loop we go is determined at the start: [name:] do var = start, stop [,step] xxx end do [name] There are two other forms of the do loop which are useful; one we have seen already in a example: [name:] do while (logical expression) xxx end do [name] Here the loop is repeated while the logical expression evaluates to .true.. The final form of the do loop requires that you use an exit statement somewhere within it, as the do loop is set up to loop indefinitely: [name:] do xxx need some test which will result in an exit end do [name] 48 Suggested Exercise 2 Write a program which initialises two arrays real(kind(1.0d0)) :: x(128), y(128) as the real and imaginary parts of a complex vector. The real part x should contain x(i) = sin(i/a) where a is a parameter to be read in. The imaginary part, y, should be initialised to zero. The arrays should be initialised in a subroutine called from the main program. Now extend the program to use a NAG routine to take the Fourier transform of the values and write the result out to disc so that the data can be plotted in gnuplot. For the purposes of this exercise you can simply adopt an arbitrary scale for the frequency. You can find a solution to this problem in the examples directory $PHYTEACH/part_2/examples/exercise2.f90 49