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Arrays

Arrays

Topics

• Terminology.

• Specifications.

• Array sections.

• Vector subscripts.

• Array storage.

• Array assignment.

• Zero-sized arrays.

• Initialising arrays.

• WHERE.

• Array intrinsic functions.

Terminology

Arrays and elements

• The subscript (or index) of an array element is the position of that element within the array, for example:
  • the first object is 5 and has a subscript 1,

  • the second object is 7 and has a subscript 2.

Arrays properties

• Rank (dimension) - the number of subscripts needed to locate an element within an array.
  • A scalar has rank 0, a vector rank 1, a matrix rank 2...
    
    

• Section - a subset of elements.

• Bounds - the upper and lower subscript in each dimension.

• Extent - the number of elements in a dimension.

• Shape - a vector containing the extents.

• Size - total number of elements.
  • Product of extents,

  • may be zero.

• Conformance - Two objects are conformable if they have the same shape or if one is a scalar.

Example

• Rank 1

• Extent 8

• Shape (8)

• Size 8
  
  

• Rank 2

• Extent 2 and 4

• Shape (2,4)

• Size 8

Specifications

• Alternate (and equivalent) forms:

  type, DIMENSION(bound)[, attr] :: name

  type [, attr] :: name(bound)

  Where:

• TYPE - intrinsic or derived type of the elements,

• DIMENSION - (attribute used to specify the dimensions of the array (optional),

• bound - lower and upper bounds for each dimension,

• attr - other type attributes (as required),

• name - name of the array variable.

Examples

INTEGER, DIMENSION(6) :: a

REAL, DIMENSION(0:9) :: b

LOGICAL, DIMENSION(2,2) :: yes_no

INTEGER :: a(6)

REAL :: b(0:9)

LOGICAL :: yes_no(2,2)

Further examples:

INTEGER, DIMENSION(8) :: x, y

REAL :: alpha(1:3), beta(4:9)

REAL, DIMENSION(0:5,12:45,6) :: data

CHARACTER(len=10) :: names(25)

TYPE(point)

REAL :: position(3)

END TYPE(point)

TYPE(point) :: object(10)

Array sections

General

Each array subscript is either:

• A simple subscript - single elements.

• A subscript triplet - sections of an array.

  array( [lower]:[upper][:step] [...])

  lower and upper default to the declared dimensions and step defaults to 1.

• A vector subscript.

Elements

• Example

INTEGER, DIMENSION(5,4) :: b



Sections

• Example

INTEGER, DIMENSION(5,4) :: b



Vector subscripts

• Vector subscripts are integer expressions of rank 1.

• Form:

• Example:

INTEGER, DIMENSION(3) :: random

...

random = (/6,3,8/)

a( random ) = 0.0

a( (/7,8,9/) ) =

1.2

• Care must be taken not to duplicate values in a vector subscript when used in the LHS of an expression:

INTEGER, DIMENSION(3) :: list

...

list = (/2,3,2/)

a(list) = (/ 1.1,1.2,1.3 /) !illegal

Array storage

• Physical storage - the details of how an array is stored in memory varies with implementation.

• Element order - conceptually linear sequence of array elements, counting through early dimensions first.

• Example:

REAL, DIMENSION(3,5) :: a

Array assignment

Whole array assignment

• Operands in array expressions must be conformable.

• A scalar is broadcast to all elements of an array.

• Example:

REAL :: d(10,10)=0.0

...

b = 2*a+4 !array expression

a = 2.0 !scalar broadcast

c = b*a !element*element

...

c = d !illegal

Array section assignment

• Conforming array sections may appear in expressions.

• Example:

REAL :: gamma(20)

...

alpha(1:5) = 2.0

alpha(1:10:2) = beta(1:5)/6

alpha(2:10) = alpha(1:9)

...

gamma(11:20) = beta

Elemental intrinsic procedures

• Elemental procedures are specified for scalar arguments, but may take conforming array arguments:
  • As if the intrinsic were applied to each element.

• Example:

REAL, DIMENSION(3,3) :: a

INTEGER :: length(5)

CHARACTER(len=10) :: c(5)

...

x = SQRT( num ) !single variable

a = SQRT( a ) !all elements

...

!find string length for each element

!no trailing blanks

length = LEN( TRIM( c ) )

Zero-sized arrays

• An array has zero size when its lower bound is greater than its upper bound.

• A zero-sized array has no elements, holds no data, but is valid and always defined.

• Useful in certain situations - no need for extra code:

...

a( 1:COUNT(a==1) ) = 0

!new a(0,0,0,1,3)

...

a( 1:COUNT(a==4) ) = 0

!unchanged a(0,0,0,1,3)

Initialising arrays

Constructors

• Constructors are used to assign values to rank 1 arrays, form:

  array = (/ list /)

  Where list may be:

• a list of values of the appropriate type,

• variable expression(s),

• array expression(s),

• implied DO loop(s).

• Example:

  INTEGER :: a(6)=(/1,2,3,6,7,8/)

  REAL :: b(2)=(/SIN(x),COS(x)/)

  INTEGER :: c(5)=(/0,a(1:3),4/)

  REAL :: d(100)=(/REAL(i),i=1,100/)

Reshape

• Used to assign values to arrays with rank > 1.

• Form:

  RESHAPE( list, shape [,PAD] [,ORDER])

  Where:

• list is a rank 1 array or constructor containing the data,

• shape is a rank 1 array or vector subscript containing the new shape of the data.

• Example:

  INTEGER, DIMENSION(2,3) :: a

  a = RESHAPE( (/i,i=0,5/), (/3,2/) )

DATA statement

• Form:

• Example:

...

DATA a/4,3,2,1/ !by value

...

DATA a/4*0/ !whole array

...

DATA b(1,:)/0,0/ !by section

DATA b(2,:)/1,1/

...

DATA (c(i),i=1,10,2)/5*1/

DATA (c(i),i=2,10,2)/5*2/

• DATA statement may be used for other variables as well as arrays.

WHERE

Statement

• Form:

• Example:

...

WHERE( a<0 ) a=0

WHERE( 3*a>10 ) a=99

Construct

• Form:

block1

[ELSEWHERE

block2]

ENDWHERE

• Example:

WHERE( b<=0 )

b = 0

ELSEWHERE

b = 1/b

ENDWHERE

Array intrinsic functions

List

• Intrinsic functions save time and effort when programming.

• Seven basic categories:
  • vector and matrix multiplication,

  • array reduction,

  • array inquiry,

  • array construction,

  • array reshape,

  • array manipulation,

  • array location.

Example of reduction

• Determines whether all elements in a given dimension satisfy a condition.

• Example:

REAL, DIMENSION(3,2) :: a

DATA a/5,9,6,10,8,12/

...

test = ALL( a>5 ) !false

test2 = ALL( a>5, DIM=1 ) !f,t,t

test3 = ALL( a>5, DIM=2 ) !f,t



Example of inquiry

• Returns the size of an array or the extent in a specified dimension.

• Example:

...

num = SIZE( a ) !num=6

num = SIZE( a, DIM=1 ) !num=2

num = SIZE( a, DIM=2 ) !num=3

Example of construction

• Returns an array with a rank one greater than array.

• Example:

REAL, DIMENSION(3,3) :: b, c

...

b = SPREAD( a, DIM=1, NCOPIES=3 )

c = SPREAD( a, DIM=2, NCOPIES=3 )



Example of location

• Returns the subscript(s) of the element with the maximum value.

• Example:

a = (/2.0,8.0,5.0,3.0,4.0/)

...

num = MAXLOC( a ) !num=2

num = MAXLOC( a, MASK=a<5 ) !num=5

num = MAXLOC( a(2:4) ) !num=1

List

• Reduction:

  ALL - True if all elements are true.

  ANY - True if any element is true.

  COUNT - Number of true elements.

  MAXVAL - Maximum element value.

  MINVAL - Minimum element value.

  PRODUCT - Product of array elements.

  SUM - Sum of array elements.

• Inquiry:

  ALLOCATED - True if array has storage allocated.

  LBOUND - Lower bounds of array.

  SHAPE - Shape of array (or scalar).

  SIZE - Size of array.

  UBOUND - Upper bounds of array.

• Construction:

  MERGE - Merge arrays.

  PACK - Pack elements into a vector.

  SPREAD - Duplicate an array section.

  UNPACK - Unpack elements of a vector.

• Reshape:

  RESHAPE - Reshapes array.

• Location:

  MAXLOC - Location of maximum element.

  MINLOC - Location of minimum element.

• Manipulation:

  CSHIFT - Circular shift.

  EOSHIFT - End-off shift.

  TRANSPOSE - Transpose matrix.

• Vector and matrix arithmetic:

  DOT_PRODUCT - Calculate dot product.

  MATMUL - Matrix multiplication.