15. Arrays in subprograms

Fortran subprogram calls are based on call by reference. This means that the calling parameters are not copied to the called subprogram, but rather that the addresses of the parameters (variables) are passed. This saves a lot of memory space when dealing with arrays. No extra storage is needed as the subroutine operates on the same memory locations as the calling (sub-)program. However, you as a programmer has to know about this and take it into account.

It is possible to declare local arrays in Fortran subprograms, but this feature is rarely used. Typically, all arrays are declared (and dimensioned) in the main program and then passed on to the subprograms as needed.

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
c
do 10 i = 1, n
y(i) = alpha*x(i) + y(i)
10 continue
c
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
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).

Exercises

Exercise A
Write a main program that declares a matrix A by
```       integer nmax
parameter (nmax=40)
real A(nmax, nmax)
```
Declare appropriate vectors x and y and initialize m=10, n=20, A(i,j) = i+j-2 for 1<=i<=m and 1<=j<=n, x(j) = 1 for 1<=j<=n. Compute y = A*x by calling the matvec subroutine given in the text. Print the answer y.

Exercise B
Write a subroutine that computes y = A*x by scalar products, i.e. the j index should be in the innermost loop. Test your routine on the example in Exercise A and compare your answers.