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Chapter 8
ARRAYS
Array subscript expressions

Each subscript in an array element designator is an expression of
integer type. It need not be a constant. The value of each subscript
expression must be within the declared subscript range for the
corresponding dimension:
 Each subscript value must be less than or equal to the upper
bound specified for the corresponding array dimension.
 Each subscript value must be greater than or equal to the specified
Lower bound; or, if no Lower bound is explicitly specified, it must
be greater than or equal to 1 (the default Lower bound).
Example for Array Element Names
integer, parameter :: EXTENT = 20
integer, parameter :: K1 = 1, L3 = 3, M1 = 1, I2 = 2, &
J1 = 1, L0 = 0, L16 = 16, I4 = 4, M2 = 2
real :: X0
real, dimension(EXTENT) :: X1
real, dimension(EXTENT, EXTENT) :: X2
2
Array element names
real, dimension(EXTENT, EXTENT, EXTENT) :: X3
read (unit = *, fmt = *) X1(6)
write (unit = *, fmt = *) X1(I4+2), X1(3*M2)
read (unit =*, fmt = *) X3(L3-1, 2*M1, 13*I2-7)
write (unit =*, fmt = *) X3(2, 2, 19)
read (unit =*, fmt = *) X2(3*J1-1, 3*L0+L16)
write (unit =*, fmt = *) X2(2, 16)
read (unit =*, fmt = *) X1(3)
read (unit =*, fmt = *) X2(K1, L0+2)
X1(1) = X1(3) + L16
X1(I4) = 17.42
X0 = X1(6) + X2(1,2)
write (unit =*, fmt = *) X1(K1), X1(4), X0
3
Vectors, matrices and cubes
One-dimensional and two-dimensional arrays may be interpreted in
special ways according to the rules of linear algebra. An array of rank 1
is a vector, an array of rank 2 is a matrix and an array of rank 3 may be
viewed as a stack of cubes forming a large block (parallelepiped).
Examples:
One-dimensional array; vector:

integer, dimension(3) :: A
Two-dimensional array;
integer, dimension(3,3) :: B
A(1)
A(2)
A(3)
B(1,1) B(1,2) B(1,3)
B(2,1) B(2,2) B(2,3)
B(3,1) B(3,2) B(3,3)
4
Vectors, matrices and cubes
Three-dimensional array;
Integer, dimension(3,3,3) :: C
C(1,1,1) C(1,1,2)
C(1,1,3)
C(1,2,1) C(1,2,2)
C(1,2,3)
C(1,3,1) C(1,3,2) C(1,3,3)
C(2,1,1) C(2,1,2)
C(2,1,3)
C(2,2,1) C(2,2,2)
C(2,2,3)
C(2,3,1) C(2,3,2)
C(2,3,3)
C(3,1,1) C(3,1,2)
C(3,1,3)
C(3,2,1) C(3,2,2)
C(3,2,3)
C(3,3,1) C(3,3,2)
C(3,3,3)
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Example

Declare an array of rank 3 which might me suitable for
representing a hotel with 8 floors and 16 rooms on each floor
and two beds in each room. How would the second bed in the
5th room on floor 7 be referenced?
SOLUTION
Declaration:
INTEGER, DIMENSION(8,16,2) :: Hotel
(or any permutation of 8, 16 and 2)
Reference:
Hotel(7,5,2)
6
Whole-array operations
An array is a variable; thus, the array name (without a subscript)
represents all elements of the array. For instance, an array name in an
input or output list causes input or output of all the array elements.
There is a rich set of operations on whole arrays; most scalar
operations are extended to whole-array operands. A whole-array
operation is applied to all elements of the array. A whole-array, denoted
by the array name without a subscript, is an array variable. F language
permits an array variable in most contexts where a scalar variable is
permitted. Assignment is permitted between whole arrays of the same
shape:
1) The shape of an array is determined by its rank (number of dimension)
and by its extent (number of elements) along each dimension.
2) Two arrays have the same shape if they have the same rank and if their
extents agree along each dimension. The subscript ranges (upper and
lower subscript bounds) are not required to agree.
7
Whole-array operations
Whole-array assignment
real, dimension(5, 7) :: A, B
real, dimension(0 : 4, 0 : 6) :: C
integer, dimension(5, 7) :: I
read(unit = *, fmt = *) B
C=B
I=C
! Elementwise type coercion, with truncation, occurs.
A=I
write (unit = *, fmt = *) A
Exercise: Study exercises 5.1 (Meissner`s
book)
8
Arithmetic and relational
operations on whole arrays
real, dimension(5, 7) :: A, B, C
logical, dimension(5, 7) :: T
real, dimension(20) :: V, V_Squared
read (unit = *, fmt = *) B, C, V
A=B+C
T=B >C
C=A*B
V_Squared = V * V
write (unit = *, fmt = *) T, C, V, V_Squared
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Elemental intrinsic functions
Many intrinsic functions (see Appendix A, Meissner`s book), notably
including the mathematical functions, are classified as elemental. An
elemental intrinsic function accepts either a scalar or an array as its
argument. When the argument is an array, the function performs its
operation element wise, applying the scalar operation to every array
element and producing a result array of the same shape.
Example:
real, dimension(5, 7) :: A, B, C
logical, dimension(5, 7) :: T

..
.
A = max(B,C)
T=sin(A) > B

Check elemental intrinsic functions
between pages 423-436 (Meissner`s book)
10
Array sections with increment
(stride) designator
An array section designator may include a third expression, analogous
to the third control parameter of an indexed do construct. This
increment (in this context, often called the stride) gives the spacing
between those elements of the underlying parent array that are to be
selected for the array section:
real, dimension(8) :: A
real, dimension(18) :: V
A = V(1: 15: 2)
By analogy to an indexed do with explicit increment, we see that the values
assigned to A are those of V(1), V(3), …, V(15). A negative increment
value reverses the normal array element sequence:
A = V(9: 2: -1)
the eight elements are V(9), V(8), V(7),…, V(2)

11
Array sections with increment
(stride) designator
real, dimension (18) :: V
V(2:9)
V(1: 15: 2)
V(2: 17: 5)
12
Array sections with increment
(stride) designator
real, dimension(5, 7) :: M
M(1: 4, 1: 4)
real, dimension(3, 5) :: E
real, dimension(2, 3) :: F
F = E (1 : 3 : 2, 1 : 5 : 2)
13
Array sections of reduced rank

Array section designators may be combined with single subscripts to
designate an array object of lower rank
V( : 5) = M(2, 1: 5)
Here, the elements M(2,1) through M(2,5) form a one-dimensional array
section that is assigned to the first five elements of V


Do exercise 5.2.1 on page 231 (Meissener`s book)
Read pages between 360 - 368 on Ellis’s book.
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Intrinsic and array inquiry
functions

Intrinsic functions perform many operations that would otherwise have
to be programmed with loops. A selection of these functions is listed on
pages 233-234 on Meissner’s book.

For example:
transpose(matrix) forms the transpose of a matrix (rank-2 array) of any
type.
ubound(array [, dim]) returns the upper bound [in the specified
dimension], or a vector of upper bounds.
mask argument in sum, product, any, all, count, maxloc, and minloc is
a logical array with the same shape as array, and the function is applied
to elements where mask is true.
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Sum and product (sum & dim)





The intrinsic functions sum and product compute the sum or product of
all elements of an array or array section of any numerical type.
Sum of selected elements (optional argument mask). mask is an array
of the same shape as the first argument array or it is a scalar, and its
type is logical. When mask is present, sum and product operate only on
the array elements selected by the mask.
The conditional sum SXi>0.0 xi is denoted by the expression
sum (X, mask = (X > 0.0))
The mask locates the positive elements of the vector X. The value of
this expression is 0.7 if X is (0.0, -0.1, 0.2, 0.3, 0.2, -0.1, 0.0).
The optional argument dim(dimension) simplifies computation of the
sums or products of all rows or columns of a matrix, or along any one
dimension of an array of higher rank. The result array has one less
dimension than Matrix; the dimension designated by dim is deleted, or
rather it is reduced to a scalar. If the rank of Matrix is 2, sum(Matrix,
dim =1) produces a vector containing the column sums, and
sum(Matrix, dim=2) produces a vector containing the row sums.
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Sum and product (sum & dim)
Example: Let A be the following 3 by 3 matrix:
1 2 3
4 5 6
2 1 1
The function sum with two different dim argument values returns the vector
of column sums (7, 8, 10) and the vector of row sums (6, 15, 4).
 The sum of products is denoted by the expression
sum( product (A, dim = 1) )
The argument dim = 1 means that the product is to be computed down
each column of A. The result of product is the vector (8, 10, 18), whose
sum is 36.
 The product of sums is denoted by the expression
product( sum (A, dim =2 ) )
The argument dim = 2 means that the sum is to be computed along each
row of A. The result of sum is the vector (6, 15, 4), whose product is
360.
17
STUDY & EXERCISES
Study intrinsic functions any, all, count,
maxloc, minloc, maxval and minval on
pages 237-241 (Meissner`s book).
 Study
matmul, transpose, reshape
functions on Ellis’s book.

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The where construct

A where construct may be used to assign values to arrays depending
on the value of a logical array expression. where construct has the
following form:
where (logical-array-expr) array-var1 = array-expr1
or
where (logical-array-expr)
array-var1 = array-expr1
.
.
.
array-varm = array-exprm
elsewhere
array-varm+1 = array-exprm+1
.
.
.
array-varn = array-exprn
end where
where each array-varm has the same size as the value of logical-arrayexpr, in the second form, the elsewhere part is optional.
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The where construct

The logical-array-expr is evaluated element by element, and whenever
the expression is true (false), the value of the corresponding element of
each array-expri (in the elsewhere part) is assigned to the
corresponding element of array-vari. In the first form, all other elements
are left unchanged.
For example: Assume that A and B are declared as
integer, dimension(5) :: A = (/ 0, 2, 5, 0, 10 /)
real, dimension(5) :: B
the where construct is
where (A > 0)
B = 1.0 / real (A)
elsewhere
B = -1.0
end where
construct assigns to B the sequence -1.0, 0.5, 0.2, -1.0, 0.1.

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Array intrinsics for linear
algebra applications
The intrinsic functions transpose, dot_product, and matmul
perform essential operations of linear algebra.
real, dimension(3, 5) :: A
real, dimension(5, 3) :: B

..
.
B = transpose(A)
dot_product forms the sum of products of corresponding elements
of two vector, which must have the same length.
matmul forms the product of a p by r matrix (or a vector of length r)
and an r by s matrix (or a vector of length r); the result is a p by
s matrix. The arguments must not both be vectors. For example;
real, dimension(5) :: A, C
real, dimension(5, 3) :: X
C = matmul(A, X)
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Array constructors and
array-valued constants
An array constructor consists of a List of expressions and
implied do loops, enclosed by the symbols (/ and /):
(/ List /)
The expressions in the List may be scalar expressions or array
expressions, and all must have the same type and kind. Let`s
see the array-valued constants in the following. The first two of
the following array constructors form constant vectors of real
type, while the third is a constant vector of integer type.
( / 3.2, 4.01, 6.4 /)
( / 4.5, 4.5 /)
( / 3, 2 /)

The implied do loop form that is permitted in an array constructor is
so called because of its resemblance to the control statement of
an indexed do construct:
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Array constructors and
array-valued constants
(Sublist, Index = Initial value, Limit [ , Increment])
The Sublist consists of a sequence of items of any form that is permitted in
the List
The Index, the Initial value, and the optional Increment are subject to the
same rules that apply to indexed do constructs.
Example:
(/ (I, I = 1, N) /)
(/ ((0.0, J=1, I-1), 1.0, (0.0, J = I+1, N), I = 1, N) /)
It must be kept in mind that the rank of an array constructor is always 1.
An array constructor always creates an array of rank 1, but the intrinsic
function reshape, which changes the shape of an array, can be applied
to an array constructor to create an array of higher rank. The result
from reshape is an array whose rank is the number of elements in the
shape vector and whose shape is given by the shape vector.

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Array constructors and
array-valued constants
For example, the following expression constructs an N by N identity matrix:
reshape( ( / ((0.0, J=1, I-1), 1.0, (0.0, J = I+1, N), I = 1, N) / ),( / N, N / )
An array constructor on the right side of an assignment statement (or in the
definition of a named constant) must agree in shape with the array on
the left.
A(1: 7 : 2) = (/ 11, 6, 14, 5 /)
result is: A(1)=11, A(3)=6, A(5)=14, A(7)=5

Study program F27 on page 254 (Meissner`s book)
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Array allocation
Allocatable arrays and allocated pointer target arrays are deferredshape arrays. Allocatable arrays are declared in the usual way, except
that the keyword allocatable appears in the list of attributes and that the
dimension attribute is only specified. Colons in the array shape
specification determine the rank (number of dimension) of the
allocatable array, but the Subscript bounds are omitted from the
declaration. The Subscript bounds remain to be established at the time
the array is actually allocated:
real, dimension( : , : ), allocatable :: Alimony
real, dimension( : ), allocatable :: Alpenglow
real, dimension( : , : , : ), allocatable :: Alabaster
At various points in the program, allocate statement can then be executed
to create the arrays:
allocate(Alimony(5, 1000))
allocate(Alpenglow(0 : Range), Alabaster(5, 1000, -17 : -6) )

25
Array allocation

Storage space that has been provided for the array can be released by
execution of a deallocate statement:
deallocate(Alimony, Alabaster)

An array that has been deallocated may be reallocated with different
Subscript bounds, but the rank cannot be changed.

It is an error to allocate an array that is currently allocated or to
deallocate an array that is not currently deallocated. The intrinsic
function allocated may be used to determine whether or not an array is
currently allocated. At the end of execution of the allocate statement,
the designated Integer variable will be set to zero if the allocation is
successful and to a positive value otherwise.
26
Exercises

do I = 1, 3
do J=1, 3
B(I,J) = I + J
end do
end do
RESULT:

2
3
4
do I =1, 3
do J = 1, 3
if ( I < J) then
A(I,J) = -1
else if (I == J) then
A(I,J)= 0
else
A(I,J) = 1
3
4
5
4
5
6
27
Exercises
end if
end do
end do


Read pages between 379 - 401 (Ellis).
Do self-test exercises 13.2 on page 402 (Ellis).
28
Examples


EXAMPLE 1: Write an array construct for the 5 element rank 1 array
BOXES containing the values 1, 4, 6, 12, 23.
BOXES = (/1, 4, 6, 12, 23/), or BOXES( : )=(/1, 4, 6, 12, 23/)
EXAMPLE 2: Using the intrinsic subroutine RANDOM_NUMBER, write
a program to simulate the throw of a dice.
PROGRAM throw-dice
IMPLICIT NONE
INTEGER :: throw, r
CALL RANDOM_NUMBER(r)
! R is returned 0 <= r <= 1
! NINT(r*5) gives an integer between 0 and 5
throw = 1 + NINT(r*5)
PRINT*, “die throw = “, throw
END PROGRAM
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Examples
EXAMPLE 3: MATMUL intrinsic
For the declarations
REAL, DIMENSION(100,100) :: A, B, C

WHAT is the difference between C=MATMUL(A,B) and C=A*B ?
C=MATMUL(A,B) performs “proper” mathematical matrix multiplication.
Element C(i,j) equals DOT_PRODUCT(A(i, : )
C=A*B, however, performs element-by-element multiplication. Element
C(i,j) equals A(i,j)*B(i,j)
30