An introduction to

arrays

WEEK 7

IntroductionIn scientific and engineering computing, it is very common to need to manipulate ordered sets of values, such as vectors and matrices.

 There is a common requirement in many applications to repeat the same sequence of operations on successive sets of data.

 In order to handle both of these requirements, F provides extensive facilities for grouping a set of items of the same type into an array which can be operated on

  either as an object in its own right

 or by reference to each of its individual elements.

The array concept

1 2 3 4 5 6

A

The array concept

A set with n (=6) object, named A In mathematical terms, we can call A, a

vector And we refer to its elements as

A1, A2, A3, …

The array concept

In F, we call such an ordered set of related variables an array

Individual items within an arrayarray elements

A(1), A(2), A(3), …, A(n)

Array ConceptIn all the programs we have used one name to refer to one location in the computer’s memory.

 It is possible to refer a complete set of data obtained for example from repeating statements, etc.

 To do this is to define a group called “array” with locations in the memory so that its all elements are identified by an index or subscript of integer type.

 An array element is designated by the name of the array along with a parenthesized list of subscript expressions.

The array concept

Subscripts can be:x(10)

y(i+4)

z(3*i+max(i, j, k))

x(int(y(i)*z(j)+x(k)))

x(1)=x(2)+x(4)+1

print*,x(5)

The array concept

A

1

2

3

4

-1 0 1 2 3

A20

A31

Array declarations

type, dimension(subscript bounds) :: list_of_array_names

type, dimension(n:m) :: variable_name_list

real, dimension(0:4) :: gravity, pressure

integer, dimension(1:100) :: scores

logical, dimension(-1, 3) :: table

Examples for Shape Arrays

real, dimension ( 0 : 49 ) : : z

! the array “z” has 50 elements

 

real, dimension ( 11 : 60 ) : : a, b, c

! a,b,c has 50 elements

 

real, dimension ( -20 : -1 ) : : x

! the array “x” has 20 elements

 

real, dimension ( -9 : 10 ) : : y

! the array “y” has 20 elements

Array declarations Up to 7 dimensions Number of permissible subscripts:

rank Number of elements in a particular dimension:

extent Total number of elements of an array:

size Shape of an array is determined by its rank

and the extent of each dimension

Array constants and initial values

Since an array consists of a number of array elements, it is necessary to provide values for each of these elements by means of an “ array constructor “.

 In its simplest form, an array constructor consists of a list of values enclosed between special delimiters :

                the first of which consists of the 2 characters : ( /

               the second of the 2 characters : / )

 

(/ value_1, value_2, ……………………., value_n /)

Examples

integer, dimension ( 10 ) : : arr

 

arr = (/ 3, 5, 7, 3, 27, 8, 12, 31, 4, 22 /)

 

 arr ( 1 ) = 3

 arr ( 5 ) = 27

! the value 27 is storen in the 5th location of the array “arr”

 arr ( 7 ) = 12

arr ( 10 ) = 22

Array constructors

(/ value_1, value_2, … /) arr = (/ 123, 234, 567, 987 /) Regular patterns are common:

implied do(/ value_list, implied_do_control /)

arr = (/ (i, i = 1, 10) /) arr = (/ -1, (0, i = 1, 48), 1 /)

Initialization

You can declare and initialize an array at the same time:

integer,dimension(50)::my_array=(/(0,i=1,50)/)

Input and output

Array elements treated as scalar variables

Array name may appear: whole array Subarrays can be referred too

EXAMPLE :

integer, dimension ( 5 ) : : value

read *, value

read *, value(3)

real, dimension(5) :: p, qinteger, dimension(4) :: rprint *, p, q(3), rread *, p(3), r(1), qprint *, p, q (3), q (4), rprint *, q (2)

! displays the value in the 2nd location of the array “q”

print *, p! displays all the values storen in the array “p”

Examples

Using arrays and array elements...  The use of array variables within a loop , therefore, greatly increases the

power and flexibility of a program. F enables an array to be treated as a single object in its own right, in much the same way as a scalar object. Assignment of an array constant to an array variable will be performed as is seen below :

  array_name = (/ list_of_values /) array_name = (/ value_1, value_2, ….., value_n /) An array element can be used anywhere that a scalar variable can be

useda(2) = t(5) - t(3)*q(2) a(i) = t(j) - f(k)

Using arrays and array elements...

An array can be treated as a single object– Two arrays are conformable if they have

the same shape– A scalar, including a constant, is

conformable with any array– All intrinsic operations are defined between

two conformable objects

Using arrays and array elements...

.real, dimension(20) :: a, b, c..a = b*c

do i = 1, 20 a(i) = b(i)*c(i)end do

Arrays having the same number of elements may be applied to arrays and simple expressions. In this case, operation applied toan array are carried out elementwise.

Exampleinteger, dimension ( 4 ) : : a, b

integer, dimension ( 0 : 3 ) : : c

integer, dimension ( 6 : 9 ) : : d

 

a = (/ 1, 2, 3, 4 /)

b = (/ 5, 6, 7, 8 /)

c = (/ -1, 3, -5, 7 /)

c(0) c(1) c(2) c(3)

a = a + b ! will result a = (/ 6, 8, 10, 12 /)

d = 2 * abs ( c ) + 1 ! will result d = (/ 3, 7, 11, 15 /)

d(6) d(7) d(8) d(9)

Intrinsic procedures with arrays

Elemental intrinsic procedures with arraysarray_1 = sin(array_2)

arr_max = max(100.0, a, b, c, d, e)

Some special intrinsic functions:maxval(arr)

maxloc(arr)

minval(arr)

minloc(arr)

size(arr)

sum(arr)

Sub-arrays

Array sections can be extracted from a parent array in a rectangular grid usinf subscript triplet notation or using vector subscript notation

Subscript triplet:subscript_1 : subscript_2 : stride

Similar to the do – loops, a subscript triplet defines an ordered set of subscripts

beginning with subscript_1,     ending with subscript_2 and

 considering a seperation of stride between the consecutive subscripts. The value of stride must not be “zero”.

Sub-arrays Subscript triplet:

subscript_1 : subscript_2 : stride Simpler forms:

subscript_1 : subscript_2

subscript_1 :

subscript_1 : : stride

: subscript_2

: subscript_2 : stride

: : stride

:

Examplereal, dimension ( 10 ) : : arr

 

arr ( 1 : 10 )

! rank-one array containing all the elements of arr.

arr ( : )

! rank-one array containing all the elements of arr.

arr ( 3 : 5 )

! rank-one array containing the elements arr (3), arr(4), arr(5).

arr ( : 9 )

! rank-one array containing the elements arr (1), arr(2),…., arr(9).

arr ( : : 4 )

! rank-one array containing the elements arr (1), arr(5),arr(9).

Exampleinteger, dimension ( 10 ) : : a

integer, dimension ( 5 ) : : b, i

integer : : j

 

a = (/ 11, 22, 33, 44, 55, 66, 77, 88, 99, 110 /)

i = (/ 6, 5, 3, 9, 1 /)

  b = a ( i ) ! will result b = ( 66, 55, 33, 99, 11 /)

a ( 2 : 10 : 2 ) ! will result a = ( 22, 44, 66, 88, 110 /)

  a ( 1 : 10 : 2 ) = (/ j ** 2, ( j = 1, 5 ) /)

! will result a = ( 1, 4, 9, 16, 25 /)

a(1) a(3) a(5) a(7) a(9)

Type of printing sub-arrayswork = ( / 3, 7, 2 /)

print *, work (1), work (2), work (3)

! or

print *, work ( 1 : 3 )

! or

print *, work ( : : 1 )

! or

integer : : i

do i = 1, 3, 1

print *, work (i)

end do

! or another example

print *, sum ( work (1: 3) )

! or another example

print *, sum ( work (1: 3 : 2) )

Type of INPUT of sub-arrays

integer, dimension ( 10 ) : : a

integer, dimension ( 3 ) : : b

.

b = a ( 4 : 6 ) ! b (1 ) = a (4)

! b (2 ) = a (5)

! b (3 ) = a (6)

  .

a ( 1 : 3 ) = 0 ! a(1) = a(2) = a(3) = 0

a ( 1 : : 2 ) = 1 ! a(1) = a(3) =…….= a(9) = 1

a ( 1 : : 2 ) = a ( 2 : : 2 ) + 1 ! a(1) = a(2) +1

! a(3) = a(4) +1

! a(5) = a(6) +1

! etc.

PROBLEM : Write a function that takes 2 real arrays as its arguments returns an array in which each element is the maximum of the 2 corresponding elements in the input array

 function max_array ( array_1, array_2 ) result (maximum_array)

 ! this function returns the maximum of 2 arrays on an element-by- element basis

 ! dummy arguments

real, dimension ( : ) , intent (in) : : array_1, array_2

! result variable

real, dimension ( size (array_1) ) : : maximum_array

! use the intrinsic function “max” to compare elements

maximum_array = max ( array_1, array_2)

! or use a do-loop instead of the intrinsic function max to compare elements

! do i = 1, size (array_2)

! maximum_array ( i ) = max ( array_1 ( i ), array_2 ( i ) )

! end do

end function max_array

program normal_probabilityinteger, parameter :: ndim = 100real, parameter :: pi = 3.1415927, two_pi = 2.0*pireal, dimension(ndim) :: phi_x, xreal :: del_x integer :: i, n , stepstep = 5! use do constructi = 1x(i) = -3.0del_x = 0.2do if (x(i) > 3.0 + del_x) thenexitend ifphi_x(i) = exp(-x(i)**2/2.0)/sqrt(two_pi)i = i + 1x(i) = x(i-1) + del_xend do n = i-1do i = 1 , nwrite(unit=*,fmt="(1x,i3,2x,f8.5,2x,f8.5)") i,x(i),phi_x(i)end dodo i = 1 , n , stepprint "(5f8.5)", phi_x(i:i+step-1)end doend program normal_probability

program vector_manipulationreal, dimension(3) :: a, b, creal :: dot_proa = (/1., -1., 0./)b = (/2., 3., 5./)dot_pro = dots_product(a,b)call vector_product(a,b,c)print *, "the dot product of two vectors is:",dot_proprint *, "the vector product of two vectors is:",cend program vector_manipulation

! *****************************************************real function dots_product(a,b) result(result)real, dimension(3),intent(in):: a, b

result = a(1)*b(1) + a(2)*b(2) + a(3)*b(3)end function dots_product

! ******************************************************subroutine vector_product(a,b,c)real, dimension(3),intent(in):: a, breal, dimension(3),intent(out) :: c

c(1) = a(2)*b(3) - a(3)*b(2) c(2) = a(3)*b(1) - a(1)*b(3) c(3) = a(1)*b(2) - a(2)*b(1)

end subroutine vector_product

If we wish to use an array as an argument to a procedure, it is needed to know the size or shape of the actual arguments used. 

On the other hand, as mentioned before 

• a procedure does not need to know the details of the calling program unit

• the program unit called, in turn, also does not need to know anything about the procedure except the information about its arguments which form part of the procedure’s interface.

In fact, if an array is a procedure dummy argument, it must be declared in the procedure as an assumed-shape array. 

An assumed-shape array is a dummy argument array wose shape, as its mame implies, is not known but which assumes the shape as that of any actual argument that becomes associated with it. 

The array specification for an “assumed-shape array” can take 2 forms as can be seen below , where the 2nd form is equivalent to the 1st with a “lower bound = 1” and on the other hand, the upper bound in both cases will only be established on entry to the procedure and will be whatever value is necessary to ensure that the extent of dummy array is the same as that of the actual array argument :

* ( lower bound : )

Arrays and procedures

Explicit-shape array: no freedom! Assumed-shape array

–If an array is a procedure dummy argument, it must be declared in the procedure as an assumed-shape array!

Arrays and procedures

Calling program

Procedure

Assumed-shape

Explicit-shape

Arrays and procedures

Example

main program unit: real, dimension ( b : 40 ) : : a

subprogram: real, dimension ( d : ) : : x

x ( d ) ---------------------- a ( b )

x ( d + 1 ) ---------------------- a ( b + 1 )

x ( d + 2 ) ---------------------- a ( b + 2 )

x ( d + 3 ) ---------------------- a ( b + 3 )

.

.

Examplemain program unit:

real, dimension ( 10 : 40 ) : : a, b

.

call array_example_l (a,b)

subroutine array_example_1 ( dummy_array_1, dummy_array_2)

real, intent (inout), dimension : : dummy_array_1, dummy_array_2

.

.

subprogram:

Examplemain program unit

real : : p (-5 : 5 ), q ( 100 ) . call array_example_2 ( p, q )

. .

 subprogram subroutine array_example_2 ( dummy_array_1, dummy_array_2)real, intent (inout), dimension : : dummy_array_1, dummy_array_2 .

Example

integer , dimension ( 4 ) : : section = (/ 5, 1, 2, 3 /)

real , dimension ( 9 ) : : a

call array_example_3 ( a ( 4 : 8 : 2 ), a ( section ) )

dummy_array_1 = (/ a (4), a (6), a (8) /) 

dummy_array_2 = (/ a (5), a (1), a (2), a (3) /) 

dummy_argument_2 (4) dummy_argument_2 (3)

dummy_argument_2 (2)

dummy_argument_2 (1)

ExamplePROBLEM : Write a subroutine that will sort a set of names into alphabetic order.

İnitial order 7 1 8 4 6 3 5 2

After 1st exc. 1 7 8 4 6 3 5 2

After 2nd exc. 1 2 8 4 6 3 5 7

After 3rd exc. 1 2 3 4 6 8 5 7

After 4th exc. 1 2 3 4 6 8 5 7

After 5th exc. 1 2 3 4 5 8 6 7

After 6th exc. 1 2 3 4 5 6 8 7

After 7th exc. 1 2 3 4 5 6 7 8

 

Array-valued functionsAs is well known, arrays can be passed to procedures as arguments and a subroutine can return information by means of an array.

 

It is convenient for a function to return its result in the form of an array of values rather than as a single scalar value ------------ array-valued function, as can be seen below :

 

  function name ( ……………………..) result (arr)

real, dimension ( dim ) : : arr

.

.

end function name

Array-valued functions

A function can return an array Such a function is called an

array-valued function

function name(…..) result(arr)real, dimension(dim) :: arr...

end function nameExplicit-shape

ExampleConsider a trivial subroutine that simply adds 2 arrays together, returning the result through a third dummy argument array.

subroutine trivial_fun ( x, y ) result (sumxy)

 

real, dimension ( : ) , intent (in) : : x, y

real, dimension ( size (x) ) : : sumxy

 

end subroutine trivial_fun

ExampleWrite a function that takes 2 real arrays as its arguments, returns an array in which each element is the maximum of the 2 corresponding elements in the input array

function max_array ( array_1, array_2 ) result (maximum_array)

! this function returns the maximum of 2 arrays on an element-by-element basis dummy arguments

real, dimension ( : ) , intent (in) : : array_1, array_2

! result variable

real, dimension ( size (array_1) ) : : maximum_array

! use the intrinsic function “max” to compare elements

maximum_array = max ( array_1, array_2)

! or use a do-loop instead of the intrinsic function max to compare elements

do i = 1, size (array_2)

maximum_array ( i ) = max ( array_1 ( i ), array_2 ( i ) )

end do

end function max_array

Matrices and 2-d arrays

In mathematics, a matrix is a two-dimensional rectangular array of elements (that obeys to certain rules!)

1,23,42,41,4

4,33,32,31,3

4,23,22,21,2

4,13,12,11,1

AAAA

AAAA

AAAA

AAAA

A=

Row number Column number

F extends the concept of a one-dimensional array in a natural manner, by means of the “dimension” attribute.

 

Thus, to define a two-dimensional array that could hold the elements of the matrix A, we would write :

 

real, dimension ( 3, 4 ) : : A

 

numbers of rows numbers of columns

 

 

 

Matrices and 2-d arrays

Similarly, a vector is (given a basis) a one-dimensional array of elements (that obeys to certain rules!)

4321 AAAA

4

3

2

1

A

A

A

A

2-d arrays

dimension attribute:real, dimension(3, 4) :: alogical, dimension(10,4) :: b, c, d

You can refer to a particular element of a 2-d array by specifing the row and column numbers:

a(2,3)b(9,3)

Intrinsic functions

matmul(array, array)dot_product(array, array)transpose(array)maxval(array[,dim,mask])maxloc(array[,dim]) minval(array[,dim,mask]) minloc(array[,dim]) product(array[,dim,mask])sum(array[,dim,mask])

program vectors_and_matrices

integer, dimension(2,3) :: matrix_a

integer, dimension(3,2) :: matrix_b

integer, dimension(2,2) :: matrix_ab

integer, dimension(2) :: vector_c

integer, dimension(3) :: vector_bc

! Set initial value for vector_c

vector_c = (/1, 2/)! Set initial value for matrix_a

matrix_a(1,1) = 1 ! matrix_a is the matrix:

matrix_a(1,2) = 2

matrix_a(1,3) = 3 ! [1 2 3]

matrix_a(2,1) = 2 ! [2 3 4]

matrix_a(2,2) = 3

matrix_a(2,3) = 4

! Set matrix_b as the transpose of matrix_a

matrix_b = transpose(matrix_a)

! Calculate matrix products

matrix_ab = matmul(matrix_a, matrix_b) vector_bc = matmul(matrix_b, vector_c)

end program vectors_and_matrices

Some basic array concepts...

Ranknumber of its dimensions

Examples:real, dimension(8) :: ainteger, dimension(3, 100, 5) :: blogical, dimension(2,5,3,100,5) :: c

Some basic array concepts...

Extentnumber of elements in each dimension

Examples:real, dimension(8) :: areal, dimension(-4:3) :: b

integer, dimension(3,-50:50,40:44) :: cinteger, dimension(0:2, 101, -1:3) :: d

Some basic array concepts...

Sizetotal number of elements

Examples:real, dimension(8) :: a

integer, dimension(3, 100, 5) :: b

logical, dimension(2,5,4,10,5) :: c

8

1500

2000

Some basic array concepts...

Shape– Rank– Extent in each dimension

Shape of an array is representable as a rank-one integer array whose elements are the extents

Examplelogical, dimension(2,-3:3,4,0:9,5) :: cshape_c = (/ 2,7,4,10,5 /)

Some basic array concepts...

Array element orderfirst index of the element specification is varying most rapidly, …

Exampleinteger, dimension(3,4) :: a

a(1,1), a(2,1), a(3,1) a(1,2), a(2,2), a(3,2) ...

Array constructors for rank-n arrays An array constructor always creates a

rank-one array of values Solution? reshape functionreshape((/ (i, i=1,6) /), (/ 2, 3 /))

1 2 3

4 5 6

Example : The implied-do loop given below provides the following matrix

integer : : i, j

 

real, save, dimension ( 2,2 ) : : a = &

reshape ( ( / (10*i + j, i = 1,2 ), j = 1, 2) / ), (/ 2, 2 /) )

11 12

21 22

Array I/O

As a list of individual array elements:a(1,1), a(4,3), …

As the complete array:real, dimension(2,4) :: aprint *, a

a(1,1), a(2,1), a(1,2), a(2,2), ...

As an array section

Could be handled in three different ways

The four classes of arrays

Explicit-shape arrays Assumed-shape arrays Allocatable (deferred-shape) arrays Automatic arrays

Explicit-shape arrays

Arrays whose index bounds for each dimension are specified when the array is declared in a type declaration statement

real, dimension(35, 0:26, 3) :: a

Assumed-shape arrays

Only assumed shape arrays can be procedure dummy arguments

Extents of such arrays is defined implicitly when an actual array is associated with an assumed-shape dummy argument

subroutine example(a, b) integer, dimension(:,:), intent(in) :: a integer, dimension(:), intent(out) :: b

Automatic arrays

A special type of explicit-shape array It can only be declared in a procedure It is not a dummy argument At least one index bound is not constant The space for the elements of an automatic

array is created dynamically when the procedure is entered and is removed upon exit from the procedure

Automatic arrays

subroutine abc(x)

! Dummy arguments

real, dimension(:), intent(inout):: x

! Assumed shape

! Local variables

real, dimension(size(x)) :: e !Automatic

real, dimension(size(x), size(x)) :: f !Automatic

real, dimension(10) :: !Explicit shape

.

.

end subroutine abc

Allocatable arrays

Allocation and deallocation of space for their elements us completely under user control

Flexible: can be defined in main programs, procedures and modules

Allocatable arrays

Steps:– Array is specified in a type declaration

statement– Space is diynamically allocated for its

elements in a separate allocation statement

– The arrays is used (like any other array!)– Space for the elements is deallocated by a

deallocation statement

Allocatable arrays

Declaration

type, allocatable, dimension(:,:,:) :: allocatable array

real, allocatable, dimension(:,:,:) :: my_array

type(person), allocatable, dimension(:) :: personel_list

integer, allocatable, dimension(:,:) :: big_table

Allocatable arrays

Allocation

allocate(list_of_array_specs, [stat=status_variable])

allocate(my_array(1:10,-2:3,5:15), personel_list(1:10))

allocate(big_table(0:50000,1:100000), stat=check)

if (check /= 0) then

print *, "No space for big_table"

stop

end if

Allocatable arrays

Deallocation

deallocate(list_of_currently_allocated_arrays,

[stat=status variable])

deallocate(my_array, personel_list, big_table)

Whole-array operations

Whole array operations can be used with conformable arrays of any rank– Two arrays are conformable if they have

the same shape– A scalar, including a constant, is

conformable with any array– All intrinsic operations are defined between

conformable arrays

Whole-array operations

Relational operators follow the same rules:

Example:

26

51

43

21BA

....

....

truefalse

truetrue

BA

Masked array assignment

where (mask_expression)array_assignment_statements

end where where (my_array > 0)

my_array = exp(my_array)end where

Masked array assignment

where (mask_expression)array_assignment_statements

elsewherearray_assignment_statements

end where where (my_array > 0)

my_array = exp(my_array)elsewhere

my_array = 0.0end where

Sub-arrays

Array sections can be extracted from a parent array of any rank

Using subscript triplet notation:first_index, last_index, stride

Resulting array section is itself an array

Sub-arrays

integer, dimension(-2:2, 0:4) :: tablo

2 3 4 5 64 5 6 7 81 2 3 4 5 tablo(-1:2, 1:3)6 7 8 9 64 4 6 6 8

Rank-2

integer, dimension(-2:2, 0:4) :: tablo

2 3 4 5 64 5 6 7 81 2 3 4 5 tablo(2, 1:3)6 7 8 9 64 4 6 6 8

Sub-arrays

Rank-1