Why learn Fortran?

Fortran is the dominant programming language used in engineering applications. It is therefore important for engineering graduates to be able to read and modify Fortran code. From time to time, so-called experts predict that Fortran will rapidly fade in popularity and soon become extinct. These predictions have always failed. Fortran is the most enduring computer programming language in history. One of the main reasons Fortran has survived and will survive is software inertia. Once a company has spent many man-years and perhaps millions of dollars on a software product, it is unlikely to try to translate the software to a different language. Reliable software translation is a very difficult task.

Portability

A major advantage Fortran has is that it is standardized by ANSI and ISO (see footnotes). Consequently, if your program is written in ANSI Fortran 77, using nothing outside the standard, then it will run on any computer that has a Fortran 77 compiler. Thus, Fortran programs are portable across machine platforms. (If you want to read some Fortran Standards Documents, click here.)

Footnotes

ANSI = American National Standards Institute
ISO = International Standards Organization

Program organization

A Fortran program generally consists of a main program (or driver) and possibly several subprograms (procedures or subroutines). For now we will place all the statements in the main program; subprograms will be treated later. The structure of a main program is:

      program name

      declarations

      statements

      stop
      end

In this tutorial, words that are in italics should not be taken as literal text, but rather as a description of what belongs in their place.

The stop statement is optional and may seem superfluous since the program will stop when it reaches the end anyway, but it is recommended to always terminate a program with the stop statement to emphasize that the execution flow stops there.

You should note that you cannot have a variable with the same name as the program.

Column position rules

Fortran 77 is not a free-format language, but has a very strict set of rules for how the source code should be formatted. The most important rules are the column position rules:

 Col. 1    : Blank, or a "c" or "*" for comments
 Col. 1-5  : Statement label (optional)
 Col. 6    : Continuation of previous line (optional)
 Col. 7-72 : Statements
 Col. 73-80: Sequence number (optional, rarely used today)

Most lines in a Fortran 77 program starts with 6 blanks and ends before column 72, i.e. only the statement field is used.

Note that many examples in this document have been slightly reformatted to improve their layout in the WIKI environment. A single space character has been added to the beginning of each code line to prevent the wiki framework from interpreting the first character as a markup command. In particular, '*' is interpreted as a bullet item and 'c' is interpreted as ordinary text. Empty lines terminate formatting of code examples but a single space followed by newline displays as an empty line of code and continues the example.

Annoyingly, another deficiency in the wiki framework is the requirement to tack a closing element tag on to the end of the element content for elements such as pre or code to prevent an additional blank line being added to the output box.

Comments

A line that begins with the letter "c" or an asterisk in the first column is a comment. Comments may appear anywhere in the program. Well-written comments are crucial to program readability. Commercial Fortran codes often contain about 50% comments. You may also encounter Fortran programs that use the exclamation mark (!) for comments. This is not a standard part of Fortran 77, but is supported by several Fortran 77 compilers and is explicitly allowed in Fortran 90. When understood, the exclamation mark may appear anywhere on a line (except in positions 2-6).

Continuation

Sometimes, a statement does not fit into the 66 available columns of a single line. One can then break the statement into two or more lines, and use the continuation mark in position 6. Example:

c23456789 (This demonstrates column position!)

c The next statement goes over two physical lines
      area = 3.14159265358979
     +       * r * r

Any character can be used instead of the plus sign as a continuation character. It is considered good programming style to use either the plus sign, an ampersand, or digits (using 2 for the second line, 3 for the third, and so on).

Blank spaces

Blank spaces are ignored in Fortran 77. So if you remove all blanks in a Fortran 77 program, the program is still acceptable to a compiler but almost unreadable to humans.

Variable names

Variable names in Fortran consist of 1-6 characters chosen from the letters a-z and the digits 0-9. The first character must be a letter. Fortran 77 does not distinguish between upper and lower case, in fact, it assumes all input is upper case. However, nearly all Fortran 77 compilers will accept lower case. If you should ever encounter a Fortran 77 compiler that insists on upper case it is usually easy to convert the source code to all upper case.

Unlike most other languages, Fortran does not have any reserved words. Although it is possible to use words that the language is built upon as variable names if the context shows that it can't be meant as a keyword, this can become very confusing and is therefore not recommended.^[1]

Types and declarations

Every variable should be defined in a declaration. This establishes the type of the variable. The most common declarations are:

      integer             list of variables
      real                list of variables
      double precision    list of variables
      complex             list of variables
      logical             list of variables
      character           list of variables

The list of variables should consist of variable names separated by commas. Each variable should be declared exactly once. If a variable is undeclared, Fortran 77 uses a set of implicit rules to establish the type. This means all variables starting with the letters i-n are integers and all others are real. Many old Fortran 77 programs uses these implicit rules, but you should not! The probability of errors in your program grows dramatically if you do not consistently declare your variables.

Integers and floating point variables

Fortran 77 has only one type for integer variables. Integers are usually stored as 32 bits (4 bytes) variables. Therefore, all integer variables should take on values in the range [-m,m] where m is approximately 2*10^9.

Fortran 77 has two different types for floating point variables, called real and double precision. While real is often adequate, some numerical calculations need very high precision and double precision should be used. Usually a real is a 4 byte variable and the double precision is 8 bytes, but this is machine dependent. Some non-standard Fortran versions use the syntax real*8 to denote 8 byte floating point variables.

The parameter statement

Some constants appear many times in a program. It is then often desirable to define them only once, in the beginning of the program. This is what the parameter statement is for. It also makes programs more readable. For example, the circle area program should rather have been written like this:

      program circle
      real r, area, pi
      parameter (pi = 3.14159)
 
c This program reads a real number r and prints
c the area of a circle with radius r.
 
      write (*,*) 'Give radius r:'
      read  (*,*) r
      area = pi*r*r
      write (*,*) 'Area = ', area
 
      stop
      end

The syntax of the parameter statement is

      parameter (name = constant, ... , name = constant)

The rules for the parameter statement are:

The name defined in the parameter statement is not a variable but a constant, which means you cannot change its value later in the program. A name can appear in at most one parameter statement. The parameter statement(s) must come before the first executable statement. Some good reasons to use the parameter statement are:

• It helps reduce the number of typos.
• It makes it easier to change a constant that appears many times in a program.
• It increases the readability of your program.

Constants

The simplest form of an expression is a constant. There are 6 types of constants, corresponding to the 6 data types. Here are some integer constants:

      1
      0
      -100
      32767
      +15

Then we have real constants:

      1.0
      -0.25
      2.0E6
      3.333E-1

The E-notation means that you should multiply the constant by 10 raised to the power following the "E". Hence, 2.0E6 is two million, while 3.333E-1 is approximately one third.

For constants that are larger than the largest real allowed, or that requires high precision, double precision should be used. The notation is the same as for real constants except the "E" is replaced by a "D". Examples:

      2.0D-1
      1D99

Here 2.0D-1 is a double precision one-fifth, while 1D99 is a one followed by 99 zeros.

The next type is complex constants. This is designated by a pair of constants (integer or real), separated by a comma and enclosed in parentheses. Examples are:

      (2, -3)
      (1., 9.9E-1)

The first number denotes the real part and the second the imaginary part.

The fifth type is logical constants. These can only have one of two values:

      .TRUE.
      .FALSE.

Note that the dots enclosing the letters are required.

The last type is character constants. These are most often used as an array of characters, called a string. These consist of an arbitrary sequence of characters enclosed in apostrophes (single quotes):

      'ABC'
      'Anything goes!'
      'It is a nice day'

Strings and character constants are case sensitive. A problem arises if you want to have an apostrophe in the string itself. In this case, you should double the apostrophe:

      'It''s a nice day'

Expressions

The simplest non-constant expressions are of the form

      operand operator operand

and an example is

      x + y

The result of an expression is itself an operand, hence we can nest expressions together like

      x + 2 * y

This raises the question of precedence: Does the last expression mean x + (2*y) or (x+2)*y? The precedence of arithmetic operators in Fortran 77 are (from highest to lowest):

      **   {exponentiation}
      *,/  {multiplication, division}
      +,-  {addition, subtraction}

All these operators are calculated left-to-right, except the exponentiation operator **, which has right-to-left precedence. If you want to change the default evaluation order, you can use parentheses.

The above operators are all binary operators. There is also the unary operator - for negation, which takes precedence over the others. Hence an expression like -x+y means what you would expect.

Extreme caution must be taken when using the division operator, which has a quite different meaning for integers and reals. If the operands are both integers, an integer division is performed, otherwise a real arithmetic division is performed. For example, 3/2 equals 1, while 3./2. equals 1.5 (note the decimal points).

Assignment

The assignment has the form

     variable_name = expression

The interpretation is as follows: Evaluate the right hand side and assign the resulting value to the variable on the left. The expression on the right may contain other variables, but these never change value! For example,

     area = pi * r**2

does not change the value of pi or r, only area.

Type conversion

When different data types occur in the same expression, type conversion has to take place, either explicitly or implicitly. Fortran will do some type conversion implicitly. For example,

      real x
      x = x + 1

will convert the integer one to the real number one, and has the desired effect of incrementing x by one. However, in more complicated expressions, it is good programming practice to force the necessary type conversions explicitly. For numbers, the following functions are available:

      int
      real
      dble
      ichar
      char

The first three have the obvious meaning. ichar takes a character and converts it to an integer, while char does exactly the opposite.

      w = dble(x)*dble(y)

      w = dble(x*y)

Logical variables and assignment

Truth values can be stored in logical variables. The assignment is analogous to the arithmetic assignment. Example:

      logical a, b
      a = .TRUE.
      b = a .AND. 3 .LT. 5/2

The order of precedence is important, as the last example shows. The rule is that arithmetic expressions are evaluated first, then relational operators, and finally logical operators. Hence b will be assigned .FALSE. in the example above. Among the logical operators the precedence (in the absence of parenthesis) is that .NOT. is done first, then .AND., then .OR. is done last.

Logical variables are seldom used in Fortran. But logical expressions are frequently used in conditional statements like the if statement.

Nested if statements

if statements can be nested in several levels. To ensure readability, it is important to use proper indentation. Here is an example:

      if (x .GT. 0) then
         if (x .GE. y) then
            write(*,*) 'x is positive and x >= y'
         else
            write(*,*) 'x is positive but x < y'
         endif
      endif

You should avoid nesting many levels of if statements since things get hard to follow.

do-loops

The do-loop is used for simple counting. Here is a simple example that prints the cumulative sums of the integers from 1 through n (assume n has been assigned a value elsewhere):

      integer i, n, sum
 
      sum = 0
      do 10 i = 1, n
         sum = sum + i
         write(*,*) 'i =', i
         write(*,*) 'sum =', sum
  10  continue

The number 10 is a statement label. Typically, there will be many loops and other statements in a single program that require a statement label. The programmer is responsible for assigning a unique number to each label in each program (or subprogram). Recall that column positions 1-5 are reserved for statement labels. The numerical value of statement labels have no significance, so any integers can be used, in any order. Typically, most programmers use consecutive multiples of 10.

The variable defined in the do-statement is incremented by 1 by default. However, you can define the step to be any number but zero. This program segment prints the even numbers between 1 and 10 in decreasing order:

      integer i
 
      do 20 i = 10, 1, -2
         write(*,*) 'i =', i
  20  continue

The general form of the do loop is as follows:

      do label  var =  expr1, expr2, expr3
         statements
label continue

var is the loop variable (often called the loop index) which must be integer. expr1 specifies the initial value of var, expr2 is the terminating bound, and expr3 is the increment (step).

Note: The do-loop variable must never be changed by other statements within the loop! This will cause great confusion.

The loop index can be of type real, but due to round off errors may not take on exactly the expected sequence of values.

Many Fortran 77 compilers allow do-loops to be closed by the enddo statement. The advantage of this is that the statement label can then be omitted since it is assumed that an enddo closes the nearest previous do statement. The enddo construct is widely used, but it is not a part of ANSI Fortran 77.

It should be noted that unlike some programming languages, Fortran only evaluates the start, end, and step expressions once, before the first pass thought the body of the loop. This means that the following do-loop will multiply a non-negative j by two (the hard way), rather than running forever as the equivalent loop might in another language.

      integer i,j
 
      read (*,*) j
      do 20 i = 1, j
         j = j + 1
  20  continue
      write (*,*) j

while-loops

The most intuitive way to write a while-loop is

      while (logical expr) do
         statements
      enddo

or alternatively,

      do while (logical expr) 
         statements
      enddo

The program will alternate testing the condition and executing the statements in the body as long as the condition in the while statement is true. Even though this syntax is accepted by many compilers, it is not ANSI Fortran 77. The correct way is to use if and goto:

 label if (logical expr) then
         statements
         goto label
       endif

Here is an example that calculates and prints all the powers of two that are less than or equal to 100:

     integer n
     n = 1
  10 if (n .le. 100) then
        write (*,*) n
        n = 2*n
        goto 10
     endif

until-loops

If the termination criterion is at the end instead of the beginning, it is often called an until-loop (analogous to the do-while loop in some other programming languages). The pseudocode looks like this:

     do
        statements
     until (logical expr)

Again, this should be implemented in Fortran 77 by using if and goto:

  label continue
           statements
        if (logical expr) goto label

Note that the logical expression in the latter version should be the negation of the expression given in the pseudocode!

One-dimensional arrays

The simplest array is the one-dimensional array, which is just a sequence of elements stored consecutively in memory. For example, the declaration

      real a(20)

declares a as a real array of length 20. That is, a consists of 20 real numbers stored contiguously in memory. By convention, Fortran arrays are indexed from 1 and up. Thus the first number in the array is denoted by a(1) and the last by a(20). However, you may define an arbitrary index range for your arrays using the following syntax:

      real b(0:19), weird(-162:237)

Here, b is exactly similar to a from the previous example, except the index runs from 0 through 19. weird is an array of length 237-(-162)+1 = 400.

The type of an array element can be any of the basic data types. Examples:

      integer i(10)
      logical aa(0:1)
      double precision x(100)

Each element of an array can be thought of as a separate variable. You reference the i'th element of array a by a(i). Here is a code segment that stores the 10 first square numbers in the array sq:

      integer i, sq(10)
      do 100 i = 1, 10
         sq(i) = i**2
  100 continue

A common bug in Fortran is that the program tries to access array elements that are out of bounds or undefined. This is the responsibility of the programmer, and the Fortran compiler will not detect any such bugs!

Two-dimensional arrays

Matrices are very important in linear algebra. Matrices are usually represented by two-dimensional arrays. For example, the declaration

      real A(3,5)

defines a two-dimensional array of 3*5=15 real numbers. It is useful to think of the first index as the row index, and the second as the column index. Hence we get the graphical picture:

  (1,1)  (1,2)  (1,3)  (1,4)  (1,5)
  (2,1)  (2,2)  (2,3)  (2,4)  (2,5)
  (3,1)  (3,2)  (3,3)  (3,4)  (3,5)

Two-dimensional arrays may also have indices in an arbitrary defined range. The general syntax for declarations is:

     name (low_index1 : hi_index1, low_index2 : hi_index2)

The total size of the array is then

     size = (hi_index1-low_index1+1)*(hi_index2-low_index2+1)

It is quite common in Fortran to declare arrays that are larger than the matrix we want to store. (This is because Fortran does not have dynamic storage allocation.) This is perfectly legal. Example:

      real A(3,5)
      integer i,j
c
c     We will only use the upper 3 by 3 part of this array.
c
      do 20 j = 1, 3
         do 10 i = 1, 3
            a(i,j) = real(i)/real(j)
   10    continue
   20 continue

The elements in the submatrix A(1:3,4:5) are undefined. Do not assume these elements are initialized to zero by the compiler (some compilers will do this, but not all).

Storage format for 2-dimensional arrays

Fortran stores higher dimensional arrays as a contiguous sequence of elements. It is important to know that 2-dimensional arrays are stored by column. So in the above example, array element (1,2) will follow element (3,1). Then follows the rest of the second column, thereafter the third column, and so on.

Consider again the example where we only use the upper 3 by 3 submatrix of the 3 by 5 array A(3,5). The 9 interesting elements will then be stored in the first nine memory locations, while the last six are not used. This works out neatly because the leading dimension is the same for both the array and the matrix we store in the array. However, frequently the leading dimension of the array will be larger than the first dimension of the matrix. Then the matrix will not be stored contiguously in memory, even if the array is contiguous. For example, suppose the declaration was A(5,3) instead. Then there would be two "unused" memory cells between the end of one column and the beginning of the next column (again we are assuming the matrix is 3 by 3).

This may seem complicated, but actually it is quite simple when you get used to it. If you are in doubt, it can be useful to look at how the address of an array element is computed. Each array will have some memory address assigned to the beginning of the array, that is element (1,1). The address of element (i,j) is then given by

      addr[A(i,j)] = addr[A(1,1)] + (j-1)*lda + (i-1)

where lda is the leading (i.e. row) dimension of A. Note that lda is in general different from the actual matrix dimension. Many Fortran errors are caused by this, so it is very important you understand the distinction!

Multi-dimensional arrays

Fortran 77 allows arrays of up to seven dimensions. The syntax and storage format are analogous to the two-dimensional case, so we will not spend time on this.

The dimension statement

There is an alternate way to declare arrays in Fortran 77. The statements

      real A, x
      dimension x(50)
      dimension A(10,20)

are equivalent to

      real A(10,20), x(50)

This dimension statement is considered old-fashioned style today.

Functions

Fortran functions are quite similar to mathematical functions: They both take a set of input arguments (parameters) and return a value of some type. In the preceding discussion we talked about user defined subprograms. Fortran 77 also has some intrinsic (built-in) functions.

A simple example illustrates how to use a function:

      x = cos(pi/3.0)

Here cos is the cosine function, so x will be assigned the value 0.5 (if pi has been correctly defined; Fortran 77 has no built-in constants). There are many intrinsic functions in Fortran 77. Some of the most common are:

      abs     absolute value
      min     minimum value
      max     maximum value
      sqrt    square root
      sin     sine
      cos     cosine
      tan     tangent
      atan    arctangent
      exp     exponential (natural)
      log     logarithm (natural)

In general, a function always has a type. Most of the built-in functions mentioned above, however, are generic. So in the example above, pi and x could be either of type real or double precision. The compiler would check the types and use the correct version of cos (real or double precision). Unfortunately, Fortran is not really a polymorphic language so in general you have to be careful to match the types of your variables and your functions!

Now we turn to the user-written functions. Consider the following problem: A meteorologist has studied the precipitation levels in the Bay Area and has come up with a model r(m,t) where r is the amount of rain, m is the month, and t is a scalar parameter that depends on the location. Given the formula for r and the value of t, compute the annual rainfall.

The obvious way to solve the problem is to write a loop that runs over all the months and sums up the values of r. Since computing the value of r is an independent subproblem, it is convenient to implement it as a function. The following main program can be used:

      program rain
      real r, t, sum
      integer m
 
      read (*,*) t
      sum = 0.0
      do 10 m = 1, 12
         sum = sum + r(m, t)
  10  continue
      write (*,*) 'Annual rainfall is ', sum, 'inches'
 
      stop
      end

Note that we have declared 'r' to be 'real' just as we would a variable. In addition, the function r has to be defined as a Fortran function. The formula the meteorologist came up with was

      r(m,t) = t/10 * (m**2 + 14*m + 46) if this is positive
      r(m,t) = 0                         otherwise

The corresponding Fortran function is

      real function r(m,t)
      integer m
      real t
 
      r = 0.1*t * (m**2 + 14*m + 46)
      if (r .LT. 0) r = 0.0
 
      return
      end

We see that the structure of a function closely resembles that of the main program. The main differences are:

Functions have a type. This type must also be declared in the calling program. The return value should be stored in a variable with the same name as the function. Functions are terminated by the return statement instead of stop. To sum up, the general syntax of a Fortran 77 function is:

      type function name (list-of-variables)
      declarations
      statements
      return
      end

The function has to be declared with the correct type in the calling program unit. If you use a function which has not been declared, Fortran will try to use the same implicit typing used for variables, probably getting it wrong. The function is called by simply using the function name and listing the parameters in parenthesis.

It should be noted that strictly speaking Fortran 77 doesn't permit recursion (functions which call themselves). However, it is not uncommon for a compiler to allow recursion.

Subroutines

A Fortran function can essentially only return one value. Often we want to return two or more values (or sometimes none!). For this purpose we use the subroutine construct. The syntax is as follows:

      subroutine name (list-of-arguments)
      declarations
      statements
      return
      end

Note that subroutines have no type and consequently should not (cannot) be declared in the calling program unit. They are also invoked differently than functions, using the word call before their names and parameters.

We give an example of a very simple subroutine. The purpose of the subroutine is to swap two integers.

      subroutine iswap (a, b)
      integer a, b
 
C Local variables
 
      integer tmp
 
      tmp = a
      a = b
      b = tmp
 
      return
      end

Note that there are two blocks of variable declarations here. First, we declare the input/output parameters, i.e. the variables that are common to both the caller and the callee. Afterwards, we declare the local variables, i.e. the variables that can only be used within this subprogram. We can use the same variable names in different subprograms and the compiler will know that they are different variables that just happen to have the same names.

Call-by-reference

Fortran 77 uses the so-called call-by-reference paradigm. This means that instead of just passing the values of the function/subroutine arguments (call-by-value) which is the default in the C and C++ languages, the memory address of the arguments (pointers) are passed instead. A small example should show the difference:

      program callex
      integer m, n
C
      m = 1
      n = 2 
 
      call iswap(m, n)
      write(*,*) m, n
 
      stop
      end

The output from this program is "2 1", just as one would expect. However, if Fortran 77 had been using call-by-value then the output would have been "1 2", i.e. the variables m and n were unchanged! The reason for this is that only the values of m and n had been copied to the subroutine iswap, and even if a and b were swapped inside the subroutine the new values would not have been passed back to the main program.

In the above example, call-by-reference was exactly what we wanted. But you have to be careful about this when writing Fortran code, because it is easy to introduce undesired side effects. For example, sometimes it is tempting to use an input parameter in a subprogram as a local variable and change its value. Since the new value will then propagate back to the calling program with an unexpected value, you should never do this unless (like our iswap subroutine) the change is part of the purpose of the subroutine.

We will come back to this issue in a later section on passing arrays as arguments (parameters).

Variable length arrays

A basic vector operation is the saxpy operation. This calculates the expression

      y := alpha*x + y

where alpha is a scalar but x and y are vectors. Here is a simple subroutine for this:

       subroutine saxpy (n, alpha, x, y)
       integer n
       real alpha, x(*), y(*)
c
c Saxpy: Compute y := alpha*x + y,
c where x and y are vectors of length n (at least).
c
c Local variables
       integer i
 
       do 10 i = 1, n
          y(i) = alpha*x(i) + y(i)
    10 continue
 
       return
       end

The only new feature here is the use of the asterisk in the declarations x(*) and y(*). This notation says that x and y are arrays of arbitrary length. The advantage of this is that we can use the same subroutine for all vector lengths. Recall that since Fortran is based on call-by-reference, no additional space is allocated but rather the subroutine works directly on the array elements from the calling routine/program. It is the responsibility of the programmer to make sure that the vectors x and y really have been declared to have length n or more in some other program unit. A common error in Fortran 77 occurs when you try to access out-of-bounds array elements.

We could also have declared the arrays like this:

      real x(n), y(n)

Most programmers prefer to use the asterisk notation to emphasize that the "real array length" is unknown. Some old Fortran 77 programs may declare variable length arrays like this:

      real x(1), y(1)

This is legal syntax even if the array lengths are greater than one! But this is poor programming style and is strongly discouraged.

Passing subsections of arrays

Next we want to write a subroutine for matrix-vector multiplication. There are two basic ways to do this, either by using inner products or saxpy operations. Let us be modular and re-use the saxpy code from the previous section. A simple code is given below.

       subroutine matvec (m, n, A, lda, x, y)
       integer m, n, lda
       real x(*), y(*), A(lda,*)
c
c Compute y = A*x, where A is m by n and stored in an array
c with leading dimension lda.
c
c Local variables
       integer i, j
      
c Initialize y
       do 10 i = 1, m
          y(i) = 0.0
    10 continue
 
c Matrix-vector product by saxpy on columns in A.
c Notice that the length of each column of A is m, not n!
       do 20 j = 1, n
          call saxpy (m, x(j), A(1,j), y)
    20 continue
 
       return
       end

There are several important things to note here. First, note that even if we have written the code as general as possible to allow for arbitrary dimensions m and n, we still need to specify the leading dimension of the matrix A. The variable length declaration (*) can only be used for the last dimension of an array! The reason for this is the way Fortran 77 stores multidimensional arrays (see the section on arrays).

When we compute y = A*x by saxpy operations, we need to access columns of A. The j'th column of A is A(1:m,j). However, in Fortran 77 we cannot use such subarray syntax (but it is encouraged in Fortran 90!). So instead we provide a pointer to the first element in the column, which is A(1,j) (it is not really a pointer, but it may be helpful to think of it as if it were). We know that the next memory locations will contain the succeeding array elements in this column. The saxpy subroutine will treat A(1,j) as the first element of a vector, and does not know that this vector happens to be a column of a matrix.

Finally, note that we have stuck to the convention that matrices have m rows and n columns. The index i is used as a row index (1 to m) while the index j is used as a column index (1 to n). Most Fortran programs handling linear algebra use this notation and it makes it a lot easier to read the code!

Different dimensions

Sometimes it can be beneficial to treat a 1-dimensional array as a 2-dimensional array and vice versa. This is fairly simple to do in Fortran 77, some will say it is too easy!

Let us look at a very simple example. Another basic vector operation is scaling, i.e. multiplying each element in a vector by the same constant. Here is a subroutine for this:

       subroutine scale(n, alpha, x)
       integer n
       real alpha, x(*)
c
c Local variables
       integer i
 
       do 10 i = 1, n
          x(i) = alpha * x(i)
    10 continue
 
       return
       end

Now suppose we have a m by n matrix we want to scale. Instead of writing a new subroutine for this, we can simply treat the matrix as a vector and use the subroutine scale. A simple version is given first:

       integer m, n
       parameter (m=10, n=20)
       real alpha, A(m,n)
 
c Some statements here define A...
 
c Now we want to scale A
       call scale(m*n, alpha, A)

Note that this example works because we assume the declared dimension of A equals the actual dimension of the matrix stored in A. This does not hold in general. Often the leading dimension lda is different from the actual dimension m, and great care must be taken to handle this correctly. Here is a more robust subroutine for scaling a matrix that uses the subroutine scale:

       subroutine mscale(m, n, alpha, A, lda)
       integer m, n, lda
       real alpha, A(lda,*)
c
c Local variables
       integer j
 
       do 10 j = 1, n
          call scale(m, alpha, A(1,j) )
    10 continue
 
       return
       end

This version works even when m is not equal to lda since we scale one column at a time and only process the m first elements of each column (leaving the rest untouched).

Example

Suppose you have two parameters alpha and beta that many of your subroutines need. The following example shows how it can be done using common blocks.

      program main
      some declarations
      real alpha, beta
      common /coeff/ alpha, beta
 
      statements
      stop
      end
 
      subroutine sub1 (some arguments)
      declarations of arguments
      real alpha, beta
      common /coeff/ alpha, beta
 
      statements
      return
      end
 
      subroutine sub2 (some arguments)
      declarations of arguments
      real alpha, beta
      common /coeff/ alpha, beta
 
      statements
      return
      end

Here we define a common block with the name coeff. The contents of the common block are the two variables alpha and beta. A common block can contain as many variables as you like. They do not need to all have the same type. Every subroutine that wants to use any of the variables in the common block has to declare the whole block.

Note that in this example we could easily have avoided common blocks by passing alpha and beta as parameters (arguments). A good rule is to try to avoid common blocks if possible. However, there are a few cases where there is no other solution.

Syntax

      common / name / list-of-variables

You should know that the common statement should appear together with the variable declarations, before the executable statements. Different common blocks must have different names (just like variables). A variable cannot belong to more than one common block. The variables in a common block do not need to have the same names each place they occur (although it is a good idea to do so), but they must be listed in the same order and have the same type and size. To illustrate this, look at the following continuation of our example:

      subroutine sub3 (some arguments)
      declarations of arguments
      real a, b
      common /coeff/ a, b
 
      statements
      return
      end

This declaration is equivalent to the previous version that used alpha and beta. It is recommended that you always use the same variable names for the same common block to avoid confusion. Here is a dreadful example:

      subroutine sub4 (some arguments)
      declarations of arguments
      real alpha, beta
      common /coeff/ beta, alpha
 
      statements
      return
      end

Now alpha is the beta from the main program and vice versa. If you see something like this, it is probably a mistake. Such bugs are very hard to find.

Arrays in common blocks

Common blocks can include arrays, too. But again, this is not recommended. The major reason is flexibility. An example shows why this is such a bad idea. Suppose we have the following declarations in the main program:

      program main
      integer nmax
      parameter (nmax=20)
      integer n
      real A(nmax, nmax)
      common /matrix/ A, n

This common block contains first all the elements of A, then the integer n. Now assume you want to use the matrix A in some subroutines. Then you have to include the same declarations in all these subroutines, e.g.

      subroutine sub1 (...)
      integer nmax
      parameter (nmax=20)
      integer n
      real A(nmax, nmax)
      common /matrix/ A, n

Arrays with variable dimensions cannot appear in common blocks, thus the value of nmax has to be exactly the same as in the main program. Recall that the size of a matrix has to be known at compile time, hence nmax has to be defined in a parameter statement.

This example shows there is usually nothing to gain by putting arrays in common blocks. Hence the preferred method in Fortran 77 is to pass arrays as arguments to subroutines (along with the leading dimensions).

The data statement

The data statement is another way to input data that are known at the time when the program is written. It is similar to the assignment statement. The syntax is:

      data list-of-variables/ list-of-values/, ...

where the three dots means that this pattern can be repeated. Here is an example:

      data m/10/, n/20/, x/2.5/, y/2.5/

We could also have written this

      data m,n/10,20/, x,y/2*2.5/

We could have accomplished the same thing by the assignments

      m = 10
      n = 20
      x = 2.5
      y = 2.5

The data statement is more compact and therefore often more convenient. Notice especially the shorthand notation for assigning identical values repeatedly.

The data statement is performed only once, right before the execution of the program starts. For this reason, the data statement is mainly used in the main program and not in subroutines.

The data statement can also be used to initialize arrays (vectors, matrices). This example shows how to make sure a matrix is all zeros when the program starts:

      real A(10,20)
      data A/ 200 * 0.0/

Some compilers will automatically initialize arrays like this but not all, so if you rely on array elements to be zero it is a good idea to follow this example. Of course you can initialize arrays to other values than zero. You may even initialize individual elements:

      data A(1,1)/ 12.5/, A(2,1)/ -33.3/, A(2,2)/ 1.0/

Or you can list all the elements for small arrays like this:

      integer v(5)
      real B(2,2)
      data v/10,20,30,40,50/, B/1.0,-3.7,4.3,0.0/

The values for two-dimensional arrays will be assigned in column-first order as usual.

The block data statement

The data statement cannot be used for variables contained in a common block. There is a special "subroutine" for this purpose, called block data. It is not really a subroutine, but it looks a bit similar because it is given as a separate program unit. Here is an example:

      block data
      integer nmax
      parameter (nmax=20)
      real v(nmax), alpha, beta
      common /vector/v,alpha,beta
      data v/20*100.0/, alpha/3.14/, beta/2.71/
      end

Just as the data statement, block data is executed once before the execution of the main program starts. The position of the block data "subroutine" in the source code is irrelevant (as long as it is not nested inside the main program or a subprogram).

Opening and closing a file

Before you can use a file you have to open it. The command is

    open (list-of-specifiers)

where the most common specifiers are:

    [UNIT=]  u
    IOSTAT=  ios
    ERR=     err
    FILE=    fname
    STATUS=  sta
    ACCESS=  acc
    FORM=    frm
    RECL=    rl

The unit number u is a number in the range 1-99 that denotes this file (the programmer may choose any number but he/she has to make sure it is unique).

• iostat is the I/O status identifier and should be an integer variable. Upon return, ios is zero if the statement was successful and returns a non-zero value otherwise.
• err is a label which the program will jump to if there is an error.
• fname is a character string denoting the file name.
• sta is a character string that has to be either NEW, OLD or SCRATCH. It shows the prior status of the file. A scratch file is a file that is created when opened and deleted when closed (or the program ends).
• acc must be either SEQUENTIAL or DIRECT. The default is SEQUENTIAL.
• frm must be either FORMATTED or UNFORMATTED. The default is UNFORMATTED.
• rl specifies the length of each record in a direct-access file.

For more details on these specifiers, see a good Fortran 77 book.

After a file has been opened, you can access it by read and write statements. When you are done with the file, it should be closed by the statement

      close ([UNIT=]u[,IOSTAT=ios,ERR=err,STATUS=sta])

where, as usual, the parameters in brackets are optional.

In this case sta is a character string which can be KEEP (the default) or DELETE.

Read and write revisited

The only necessary change from our previous simplified read/write statements, is that the unit number must be specified. But frequently one wants to add more specifiers. Here is how:

      read ([UNIT=]u, [FMT=]fmt, IOSTAT=ios, ERR=err, END=s)
      write([UNIT=]u, [FMT=]fmt, IOSTAT=ios, ERR=err, END=s)

where most of the specifiers have been described above. The END=s specifier defines which statement label the program jumps to if it reaches end-of-file.

Example

You are given a data file with xyz coordinates for a bunch of points. The number of points is given on the first line. The file name of the data file is points.dat. The format for each coordinate is known to be F10.4 (We'll learn about FORMAT statements in a later lesson). Here is a short program that reads the data into 3 arrays x,y,z:

       program inpdat
c
c  This program reads n points from a data file and stores them in 
c  3 arrays x, y, z.
c
       integer nmax, u
       parameter (nmax=1000, u=20)
       real x(nmax), y(nmax), z(nmax)
 
c  Open the data file
       open (u, FILE='points.dat', STATUS='OLD')
 
c  Read the number of points
       read(u,*) n
       if (n.GT.nmax) then
          write(*,*) 'Error: n = ', n, 'is larger than nmax =', nmax
          goto 9999
       endif
 
c  Loop over the data points
       do 10 i= 1, n
          read(u,100) x(i), y(i), z(i)
    10 continue
   100 format (3(F10.4))
 
c  Close the file
       close (u)
 
c  Now we should process the data somehow...
c  (missing part)
 
  9999 stop
       end

Read and write

Read is used for input, while write is used for output. A simple form is

      read (unit no, format no) list-of-variables
      write(unit no, format no) list-of-variables

The unit number can refer to either standard input, standard output, or a file. The format number refers to a label for a format statement, which will be described in the next lesson.

It is possible to simplify these statements further by using asterisks (*) for some arguments, like we have done in most of our examples so far. This is sometimes called list directed read/write.

      read (*,*) list-of-variables
      write(*,*) list-of-variables

The first statement will read values from the standard input and assign the values to the variables in the variable list, while the second one writes to the standard output.

Examples

Here is a code segment from a Fortran program:

      integer m, n
      real x, y, z(10)
 
      read(*,*) m, n 
      read(*,*) x, y
      read(*,*) z

We give the input through standard input (possibly through a data file directed to standard input). A data file consists of records according to traditional Fortran terminology. In our example, each record contains a number (either integer or real). Records are separated by either blanks or commas. Hence a legal input to the program above would be:

   -1  100
  -1.0 1e+2
  1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Or, we could add commas as separators:

   -1, 100
 -1.0, 1e+2
  1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0

Note that Fortran 77 input is line sensitive, so it is important not to have extra input elements (fields) on a line (record). For example, if we gave the first four inputs all on one line as

   -1, 100, -1.0, 1e+2
  1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0

then m and n would be assigned the values -1 and 100 respectively, but the last two values would be discarded, x and y would be assigned the values 1.0 and 2.0, ignoring the rest of the second line. This would leave the elements of z all undefined.

If there are too few inputs on a line then the next line will be read. For example

   -1
  100
 -1.0
 1e+2
  1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0

would produce the same results as the first two examples.

Other versions

For simple list-directed I/O it is possible to use the alternate syntax

      read  *, list-of-variables
      print *, list-of-variables

which has the same meaning as the list-directed read and write statements described earlier. This version always reads/writes to standard input/output so the * corresponds to the format.

Syntax

      write(*, label) list-of-variables
 label format format-code

A simple example demonstrates how this works. Say you have an integer variable you want to print in a field 4 characters wide and a real number you want to print in fixed point notation with 3 decimal places.

      write(*, 900) i, x
  900 format (I4,F8.3)

The format label 900 is chosen somewhat arbitrarily, but it is common practice to number format statements with higher numbers than the control flow labels. After the keyword format follows the format codes enclosed in parenthesis. The code I4 stands for an integer with width four, while F8.3 means that the number should be printed using fixed point notation with field width 8 and 3 decimal places.

The format statement may be located anywhere within the program unit. There are two programming styles: Either the format statement follows directly after the read/write statement, or all the format statements are grouped together at the end of the (sub-)program.

Common format codes

The most common format code letters are:

   A - text string
   D - double precision numbers, exponent notation
   E - real numbers, exponent notation
   F - real numbers, fixed point format
   I - integer
   X - horizontal skip (space)
   / - vertical skip (newline)

The format code F (and similarly D, E) has the general form Fw.d where w is an integer constant denoting the field width and d is an integer constant denoting the number of significant digits.

For integers only the field width is specified, so the syntax is Iw. Similarly, character strings can be specified as Aw but the field width is often dropped.

If a number or string does not fill up the entire field width, spaces will be added. Usually the text will be adjusted to the right, but the exact rules vary among the different format codes.

For horizontal spacing, the nX code is often used. This means n horizontal spaces. If n is omitted, n=1 is assumed. For vertical spacing (newlines), use the code /. Each slash corresponds to one newline. Note that each read or write statement by default ends with a newline (here Fortran differs from C).

Some examples

This piece of Fortran code

      x = 0.025
      write(*,100) 'x=', x
  100 format (A,F)
      write(*,110) 'x=', x
  110 format (A,F5.3)
      write(*,120) 'x=', x
  120 format (A,E)
      write(*,130) 'x=', x
  130 format (A,E8.1)

produces the following output when we run it:

x=      0.0250000
x=0.025
x=  0.2500000E-01
x= 0.3E-01

Note how blanks are automatically padded on the left and that the default field width for real numbers is usually 14. We see that Fortran 77 follows the rounding rule that digits 0-4 are rounded downwards while 5-9 are rounded upwards.

In this example each write statement used a different format statement. But it is perfectly fine to use the same format statement for many different write statements. In fact, this is one of the main advantages of using format statements. This feature is handy when you print tables for instance, and want each row to have the same format.

Format strings in read/write statements

Instead of specifying the format code in a separate format statement, one can give the format code in the read/write statement directly. For example, the statement

      write (*,'(A, F8.3)') 'The answer is x = ', x

is equivalent to

      write (*,990) 'The answer is x = ', x
  990 format (A, F8.3)

Sometimes text strings are given in the format statements, e.g. the following version is also equivalent:

      write (*,999) x
  999 format ('The answer is x = ', F8.3)

Implicit loops and repeat counts

Now let us do a more complicated example. Say you have a two-dimensional array of integers and want to print the upper left 5 by 10 submatrix with 10 values each on 5 rows. Here is how:

      do 10 i = 1, 5
         write(*,1000) (a(i,j), j=1,10)
   10 continue
 1000 format (I6)

We have an explicit do loop over the rows and an implicit loop over the column index j.

Often a format statement involves repetition, for example

  950 format (2X, I3, 2X, I3, 2X, I3, 2X, I3)

There is a shorthand notation for this:

  950 format (4(2X, I3))

It is also possible to allow repetition without explicitly stating how many times the format should be repeated. Suppose you have a vector where you want to print the first 50 elements, with ten elements on each line. Here is one way:

      write(*,1010) (x(i), i=1,50)
 1010 format (10I6)

The format statement says ten numbers should be printed. But in the write statement we try to print 50 numbers. So after the ten first numbers have been printed, the same format statement is automatically used for the next ten numbers and so on.

Implicit do-loops can be multi-dimensional and can be used to make an read or write statement difficult to understand. You should avoid using implicit loops which are much more complicated than the ones shown here.

Netlib

Netlib (the NET LIBrary) is a large collection of freely available software, documents, and databases of interest to the numerical, scientific computing, and other communities. The repository is maintained by AT&T Bell Laboratories, the University of Tennessee and Oak Ridge National Laboratory, and replicated at several sites around the world.

Netlib contains high-quality software that has been extensively tested, but (as all free software) it comes with no warranty and little (if any) support. In order to use the software, you first have to download it to your computer and then compile it yourself.

There are many ways to access Netlib. The most common methods are the World Wide Web, e-mail, and ftp:

World Wide Web (WWW) http://www.netlib.org/ e-mail Send the message: send index to

netlib@netlib.org to receive a contents summary and instructions.

ftp Anonymous ftp to: ftp.netlib.org Two of the most popular packages at Netlib are the BLAS and LAPACK libraries which we will describe in later sections.

Some commercial Fortran packages

In this section we briefly mention a few of the largest (commercial) software packages for general numerical computations.

BLAS

BLAS is an acronym for Basic Linear Algebra Subroutines. As the name indicates, it contains subprograms for basic operations on vectors and matrices. BLAS was designed to be used as a building block in other codes, for example LAPACK. The source code for BLAS is available through Netlib. However, many computer vendors will have a special version of BLAS tuned for maximal speed and efficiency on their computer. This is one of the main advantages of BLAS: the calling sequences are standardized so that programs that call BLAS will work on any computer that has BLAS installed. If you have a fast version of BLAS, you will also get high performance on all programs that call BLAS. Hence BLAS provides a simple and portable way to achieve high performance for calculations involving linear algebra. LAPACK is a higher-level package built on the same ideas.

Level 1: Vector-vector operations. O(n) data and O(n) work.
Level 2: Matrix-vector operations. O(n^2) data and O(n^2) work.
Level 3: Matrix-matrix operations. O(n^2) data and O(n^3) work.

     S      Real single precision.
     D      Real double precision.
     C      Complex single precision.
     Z      Complex double precision.

     double complex list-of-variables
     complex*16     list-of-variables

xCOPY - copy one vector to another
xSWAP - swap two vectors
xSCAL - scale a vector by a constant
xAXPY - add a multiple of one vector to another
xDOT - inner product
xASUM - 1-norm of a vector
xNRM2 - 2-norm of a vector
IxAMAX - find maximal entry in a vector

xGEMV - general matrix-vector multiplication
xGER - general rank-1 update
xSYR2 - symmetric rank-2 update
xTRSV - solve a triangular system of equations

xGEMM - general matrix-matrix multiplication
xSYMM - symmetric matrix-matrix multiplication
xSYRK - symmetric rank-k update
xSYR2K - symmetric rank-2k update

      call SSCAL ( n, a, x, incx )

      call SSCAL(100, a, x, 1)
      call SSCAL( 50, a, x(50), 1)
      call SSCAL( 50, a, x(2), 2)

      SDOT ( n, x, incx, y, incy )

      real A(lda,lda)
      real B(ldb,ldb)

     SDOT ( p, A(i,1), lda, B(1,j), 1 )

LAPACK

LAPACK is a collection of Fortran subprograms for advanced linear algebra problems like solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems. LAPACK replaces the older packages LINPACK and EISPACK. LAPACK subroutines were written to exploit BLAS as much as possible.

xGESV - Solve AX=B for a general matrix A
xPOSV - Solve AX=B for a symmetric positive definite matrix A
xGBSV - Solve AX=B for a general banded matrix A

       SUBROUTINE SGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )
 *
 *  -- LAPACK driver routine (version 2.0) --
 *     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
 *     Courant Institute, Argonne National Lab, and Rice University
 *     March 31, 1993 
 *
 *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, LDB, N, NRHS
 *     ..
 *     .. Array Arguments ..
       INTEGER            IPIV( * )
       REAL               A( LDA, * ), B( LDB, * )
 *     ..
 *
 *  Purpose
 *  =======
 *
 *  SGESV computes the solution to a real system of linear equations
 *     A * X = B,
 *  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
 *
 *  The LU decomposition with partial pivoting and row interchanges is
 *  used to factor A as
 *     A = P * L * U,
 *  where P is a permutation matrix, L is unit lower triangular, and U is
 *  upper triangular.  The factored form of A is then used to solve the
 *  system of equations A * X = B.
 *
 *  Arguments
 *  =========
 *
 *  N       (input) INTEGER
 *          The number of linear equations, i.e., the order of the
 *          matrix A.  N >= 0.
 *
 *  NRHS    (input) INTEGER
 *          The number of right hand sides, i.e., the number of columns
 *          of the matrix B.  NRHS >= 0.
 *
 *  A       (input/output) REAL array, dimension (LDA,N)
 *          On entry, the N-by-N coefficient matrix A.
 *          On exit, the factors L and U from the factorization
 *          A = P*L*U; the unit diagonal elements of L are not stored.
 *
 *  LDA     (input) INTEGER
 *          The leading dimension of the array A.  LDA >= max(1,N).
 *
 *  IPIV    (output) INTEGER array, dimension (N)
 *          The pivot indices that define the permutation matrix P;
 *          row i of the matrix was interchanged with row IPIV(i).
 *
 *  B       (input/output) REAL array, dimension (LDB,NRHS)
 *          On entry, the N-by-NRHS matrix of right hand side matrix B.
 *          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
 *
 *  LDB     (input) INTEGER
 *          The leading dimension of the array B.  LDB >= max(1,N).
 *
 *  INFO    (output) INTEGER
 *          = 0:  successful exit
 *          <0: if INFO="-i," the i-th argument had an illegal value *> 0:  if INFO = i, U(i,i)  is exactly zero.  The factorization
 *                has been completed, but the factor U is exactly
 *                singular, so the solution could not be computed.
 *
 *  =====================================================================

Using Libraries Under Unix

A Fortran package of subprograms may contain hundreds of files. It is very slow and inconvenient to recompile these files every time you want to use any of the subroutines. Under the Unix operating system you can avoid this by making a library file. The library file is an object file, so you only have to compile your additional main (driver) program and then link it with library. (Linking is much faster than compiling.)

Libraries have file names starting with lib and ending in .a. Some libraries have already been installed by your system administrator, usually in the directories /usr/lib and /usr/local/lib. For example, the BLAS library MAY be stored in the file /usr/local/lib/libblas.a. You use the -l option to link it together with your main program, e.g.

     f77 main.f -lblas

You can link several files with several libraries at the same time if you wish:

     f77 main.f mysub.f -llapack -lblas 

The order you list the libraries is significant. In the example above -llapack should be listed before -lblas since LAPACK calls BLAS routines.

If you want to create your own library, you can do so by compiling the source code to object code and then collecting all the object files into one library file. This example generates a library called my_lib:

      f77 -c *.f
      ar rcv libmy_lib.a *.o
      ranlib libmy_lib.a 
      rm *.o

Check the manual pages or a Unix book for more information on the commands ar and ranlib. If you have the library file in the current directory, you can link with it as follows:

      f77 main.f -L. -lmy_lib

One advantage of libraries is that you only compile them once but you can use them many times.

Some of the BLAS and LAPACK subroutines have been downloaded to the class account. The files are located in /usr/class/me390/src/. The routines have been compiled and collected into a library file called libmy_lib.a stored in /usr/class/me390/lib/libmy_lib.a.

Searching for Mathematical Software

A common situation is that you need a piece of code to solve a particular problem. You believe somebody must have written software to solve this type of problem already, but you do not know how to find it. Here are some suggestions:

Ask your co-workers and colleagues. Always do this first, otherwise you can waste lots of time trying to find something the person next-door may have already. Check Netlib. They have a search feature. Look it up in GAMS, the Guide to Available Mathematical Software from NIST.

Portability

To ensure portability, use only standard Fortran 77. The only exception we have allowed in this tutorial is to use lower case letters.

Program structure

The overall program structure should be modular. Each subprogram should solve a well-defined task. Many people prefer to write each subprogram in a separate file.

Comments

Write legible code, but also add comments in the source explaining what is happening! It is especially important to have a good header for each subprogram that explains each input/output argument and what the subprogram does.

Indentation

Always use proper indentation for loops and if blocks as demonstrated in this tutorial.

Variables

Always declare all variables. Implicit type declaration is bad! Try to stick to maximum 6 characters for variable names, or at the very least make sure the 6 first characters are unique.

Subprograms

Never let functions have "side effects", i.e. do not change the value of the input parameters. Use subroutines in such cases.

In the declarations, separate parameters, common blocks, and local variables.

Minimize the use of common blocks.

Goto

Minimize the use of goto. Unfortunately it is necessary to use goto in some loops since while is not standard Fortran.

Arrays

In many cases it is best to declare all large arrays in the main program and then pass them as arguments to the various subroutines. This way all the space allocation is done in one place. Remember to pass the leading dimensions. Avoid unnecessary "reshaping" of matrices.

Efficiency concerns

When you have a double (nested) loop accessing a two-dimensional array, it is often best to have the first (row) index in the innermost loop. This is because of the way in which Fortran maps multi-dimensional arrays into one-dimensional computer memory. Modern microprocessors implement optimizations such as memory caches used to speed up accesses to locations near recent accesses. Matrix processing for problems in linear algebra is one of the most common applications for Fortran programs and efficient techniques have been a major area of research over the past forty years. Naive implementations can generate worst-case memory access patterns which result in programs that are too slow to be worthwhile.

When you have if-then-elseif statements with multiple conditions, try to place the most likely conditions first. This exploits the branch-prediction logic in modern microprocessors. Branch prediction is used to keep the microprocessor pipeline as full as possible by starting to execute instructions following a predicted branch path. If the prediction turns out to be wrong, the pipeline stalls while intermediate results are discarded and instructions from the other path are scheduled.

If possible, floating point and integer statements can be mixed together so that the processor can have a mixture of floating point and integer instructions to keep the different execution units in each pipeline busy. Some of these optimizations can be done by the Fortran compiler but others must be done by hand. Compilers must consider the possibility that the arrays used for matrix operations may overlap, so many optimizations must be done by the programmer.

Useful compiler options

Most Fortran compilers will have a set of options you can turn on if you like. The following compiler options are specific to the Sun Fortran 77 compiler, but most compilers will have similar options (although the letters may be different).

• -ansi

This will warn you about all non-standard Fortran 77 extensions in your program.

• -u

This will override the implicit type rules and make all variables undeclared initially. If your compiler does not have such an option, you can achieve almost the same thing by adding the declaration

     implicit none (a-z)

in the beginning of each (sub-)program.

• -C

Check for array bounds. The compiler will try to detect if you access array elements that are out of bounds. It cannot catch all such errors, however.

Some common errors

Here are some common errors to watch out for:

• Make sure your lines end at column 72. The rest will be ignored!
• Do your parameter lists in the calling and the called program match?
• Do your common blocks match?
• Have you done integer division when you wanted real division (or vice versa)?
• Have you typed an o when you meant 0, an l when you meant 1?

Debugging tools

If you have a bug, you have to try to locate it. Syntax errors are easy to find. The problem is when you have run-time errors. The old-fashioned way to find errors is to add write statements in your code and try to track the values of your variables. This is a bit tedious since you have to recompile your source code every time you change the program slightly. Today one can use special debuggers which is a convenient tool. You can step through a program line by line or define your own break points, you can display the values of the variables you want to monitor, and much more. Most UNIX machines will have the debuggers dbx and gdb. Unfortunately, these are difficult to learn since they have an old-fashioned line-oriented user interface. Check if there is a graphical user interface available, like, e.g., xdbx or dbxtool.

References