1. Multi-Dimensional Arrays

2. Matrices Arrays seen so far are single dimensional
A linear list of items
One subscript for accessing individual elements
Multidimensional arrays are useful organizations in many applications
values of temperatures at different locations in different times
number of runs in different sessions on different days in a test match
trajectory of a planet in space - time
Matrices
simplest and useful multidimensional arrays
Two dimensions

3. Two-Dimensional Data Structures
Fortran allows variables that can assume two dimensional array values
Typical values are:
true. .true.
false. .false.
true. .true.
An integer 3 by 3 matrix: 3 rows and 3 columns of integer values
4 by 3 real matrix
3 by 2 logical matrix

4. Matrix Declaration Matrix entries can be of different types
No. of rows and columns can be arbitrary
Fortran Declaration defines these two attributes for a matrix variable
Real, Dimension(3,6):: Sum
Logical, Dimension(23:100,-2:5):: flag
integer, Dimension(15:19,5):: runs
Accessing an element
Sum(2,5), flag(56,0), runs(18,3)
Two subscripts are needed

5. Matrix Representation
Memory is Single-Dimensional
Linear representation of matrix elements
Row major order: One row after another
Column major order: one column after another (assumed representation)
Example: 15 20 3 56 54 45 67 77 86 76 11 22

6. Initialization/Assignment
Using array constructor (column major order)
A = (\ 2,3,4,3,4,5,4,5,6 \)
Explicit nested do loops 
do I = 1, 5
do J = 1,8
A(I,J) = I + J
end do
Whole row/column assignment 
do I =1,5
A(I,:) = I
Initialization can be done at the time of declaration

7. Assignment Whole, part or individual array elements can be assigned
A(:,i) = ...
A(i,j) = …
The rhs should be appropriate type
Lifting of scalar to suitable types may be required
with READ statements
read *, ((A(i,j),j = 1, 3), I = 1, 4)

8. Using matrix variables
Matrix variables can be accessed, individually, in sections or as a whole
Used in expressions on the right hand side of any assignment
Intrinsic functions on base types lifted to matrix types
Eg. Abs, Sin, Cos, Tan, Log, Sqrt can be applied to a matrix variable
Sin(A) is a similar matrix in which each entry is Sin of the corresponding entries in A

9. Other intrinsic functions
LBOUND(A,d) – lower bound of index in the dimension d
LBOUND(A) - list of lower bounds
UBOUND - upper bound, similar
SIZE(A,d) - number of entries in dimension d
size(A) - total no. of elements
SHAPE(A) - returns the `shape' of A
Extent in a dimension is the no. of entries in that dimension
Shape is an one dimensional array of extents(one entry per dimension)

10. Transformational Intrinsic Functions
Dot_Product(A,B) - dot product of one dimensional arrays of A and B
MATMUL(A,B) - Matrix multiplication on conformable matrices A and B
RESHAPE(A,B) - B is a one-dimensional shape array
Entries in A are converted into an array of shape B
Column major order used
Example: Suppose A = and B = [6,2]
Reshape(A,B) =
Usually used for initializing a matrix from a linear list of elements

11. Masked Assignment
A more generalized selection is possible using WHERE construct
Suppose, you want to multiply all entries > 0 by 2,
and all other entries set to 0 then
where (A > 0)
A = 2 * A
elsewhere
A = 0
end where
Note: Reference to an array has many connotations – context dependent

12. Accessing Matrix variables
When a matrix is accessed, the index values should be valid
illegal values lead to run-time error - Array index exceeding bounds
Compiler optionally inserts code for the check

13. Generalized Arrays Multi-dimensional arrays are possible
All array operations, like declaration, initialization and use are similar
Rank of an array is the number of dimensions
One-dimensional array is of rank 1, matrix of rank 2 etc.
Arrays of up to rank 7 are allowed in Fortran

14. Memory Allocation Arrays are allocated memory by the compiler
The allocation is static - done at compile time
More and larger arrays, larger the memory requirement
Array use and their size requirements should be done with care
indiscriminate use would result in wastage of memory
Use arrays only when all the data in the arrays are required at the same time in memory

15. Allocatable Arrays
Allocatable arrays is a solution to memory wastage problem
When the size of an input data set is unknown, better to use the following:
real, dimension(), allocatable :: A
integer, dimension(:), allocatable :: B
This is a declaration of an Array A and B of rank 1 and 2 respectively
Only ranks specified, the extent decided dynamically
When A ( and B) initialized, the extents are decided and allocated
This enables avoiding static allocation of conservative estimate of required space

16. Allocate Construct
Allocatable arrays are allocated memory using explicit instruction:
allocate(A(n), i)
This indicates the extent of A is the value of n
This is an external routine that may succeed or fail
Succeeds when the memory allocation is successful
The external procedure sends a value 0 via the parameter i
Typically the size is input by the user which is used in the allocation
Allocated memory can be deallocated using
deallocate(A,i)
This succeeds provided the value returned via i is 0

17. Illustration Linear Systems of Equations
Most commonly occurring problem, the original motivation for matrices
solve a system of n equations in n variables
a11x1 + a12x a1nxn = b1
a12x2 + a22x a2nxn = b2
...
an1x1 + an2x2 + … + annxn = bn
aij,, bj are all real numbers

18. Gaussian Elimination
choose any equation and any variable with non zero coefficient ( called pivot) in the equation
eliminate this variable from all other equations by adding a multiple of chosen equation
repeat with remaining equations
solve one equation in one variable
back substitute value of variable to find others
two steps, elimination and back substitution

19. Program
real, dimension(:,:), allocatable :: a
integer :: n, i, j
real, dimension(:), allocatable :: x
read *, n ! Number of variables
allocate(x(n), stat=i)
if ( i /= 0) then
print *, "allocation of x failed"
stop
endif
allocate(a(n,n+1), stat=i)
print *, "allocation of array a failed"
do i = 1,n
print *, "enter coefficients of equation", i
read *, a(i, 1:n+1)
end do

20. do i = 1,n
a(i, i:n+1) = a(i,i:n+1)/a(i,i) ! make coefficient of x_i 1.0
do j = i+1,n
a(j,i:n+1) = a(j,i:n+1) - a(i,i:n+1)*a(j,i)
end do ! eliminate x_i
end do
do i = n, 1, -1
x(i) = a(i,n+1) - sum(a(i,i+1:n)*x(i+1:n))
end do ! sum of a null array is 0.0
print *, "solution is"
print *, x(1:n)
deallocate(a, x, stat=i)
if ( i /= 0) then
print *, "deallocation failed"
stop
endif

21. Elimination
a(1:n+1,1:n+1) contains the coefficients, x(1:n) the computed values of variables
assumes a(i,i) /= 0.0 at every step
true for many special types of matrices
problem if abs(a(i,i)) is too small
elimination needs to be done only once for different right hand sides
only back substitution needs to be done
reduces operations from n3 to n2

22. Pivoting swap rows and columns to make a(i,i) large
partial pivoting - swap row i with row containing maxval(abs(a(i:n,i)))
what if it is 0.0?
complete pivoting- swap row and column to make a(i,i) = maxval(abs(a(i:n,i:n)))
what if it is still 0.0?
this changes order of the variables
gives more accurate results but more time