Nested Loops in Fortran

Loops are a fundamental concept in programming, allowing repeated execution of a block of code. In many applications, particularly in scientific and engineering computations, nested loops—loops inside loops—are essential. Nested loops enable iteration over multiple dimensions, such as arrays, matrices, grids, and more complex structures.

This post explores nested loops in Fortran in detail, including syntax, examples, multidimensional arrays, practical applications, and advanced techniques.

1. Introduction to Nested Loops

A nested loop is a loop placed inside another loop. The inner loop executes completely for every single iteration of the outer loop. Nested loops are commonly used for:

• Iterating over multidimensional arrays
• Performing matrix operations
• Generating tables or grids
• Running complex computations repeatedly in multiple dimensions

Understanding nested loops is essential for efficient programming in Fortran.

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2. Syntax of Nested Loops in Fortran

The general structure of a nested loop is:

do outer_variable = start, end, step
! Outer loop block
do inner_variable = start, end, step
    ! Inner loop block
end do
end do

• outer_variable: Loop control variable for the outer loop
• inner_variable: Loop control variable for the inner loop
• start, end, step: Define the loop range and increment (step is optional; default is 1)

The inner loop completes all iterations for each iteration of the outer loop.

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3. Basic Example of Nested Loops

program nested_loop_example
  integer :: i, j
  do i = 1, 3
  do j = 1, 2
      print *, "i =", i, ", j =", j
  end do
  end do
end program nested_loop_example

Output:

i = 1 , j = 1
i = 1 , j = 2
i = 2 , j = 1
i = 2 , j = 2
i = 3 , j = 1
i = 3 , j = 2

Explanation: For each value of i in the outer loop, the inner loop executes fully for all values of j.

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4. Using Nested Loops for Arrays

Nested loops are particularly useful for iterating through multidimensional arrays.

4.1 Example: 2D Array Initialization

program array_2d
  integer, dimension(3,2) :: matrix
  integer :: i, j

  do i = 1, 3
  do j = 1, 2
      matrix(i,j) = i * j
      print *, "matrix(", i, ",", j, ") =", matrix(i,j)
  end do
  end do
end program array_2d

Output:

matrix( 1 , 1 ) = 1
matrix( 1 , 2 ) = 2
matrix( 2 , 1 ) = 2
matrix( 2 , 2 ) = 4
matrix( 3 , 1 ) = 3
matrix( 3 , 2 ) = 6

Explanation: Nested loops traverse all elements of the 2D array systematically.

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5. Nested Loops for Matrices

5.1 Example: Matrix Addition

program matrix_addition
  integer, dimension(2,2) :: A = reshape((/1,2,3,4/), (/2,2/))
  integer, dimension(2,2) :: B = reshape((/5,6,7,8/), (/2,2/))
  integer, dimension(2,2) :: C
  integer :: i, j

  do i = 1, 2
  do j = 1, 2
      C(i,j) = A(i,j) + B(i,j)
      print *, "C(", i, ",", j, ") =", C(i,j)
  end do
  end do
end program matrix_addition

Output:

C( 1 , 1 ) = 6
C( 1 , 2 ) = 8
C( 2 , 1 ) = 10
C( 2 , 2 ) = 12

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6. Nested Loops for Multiplication Tables

Nested loops are excellent for generating tabular outputs.

program multiplication_table
  integer :: i, j

  do i = 1, 5
  do j = 1, 5
      print *, i, "x", j, "=", i*j
  end do
  print *, "----------------"
  end do
end program multiplication_table

Output:

1 x 1 = 1
1 x 2 = 2
1 x 3 = 3
1 x 4 = 4
1 x 5 = 5
----------------
2 x 1 = 2
...

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7. Nested Loops in Scientific Computations

Nested loops are widely used in numerical simulations:

• Solving systems of equations
• Computing distances in 2D or 3D grids
• Performing finite difference or finite element computations

7.1 Example: Distance Matrix

program distance_matrix
  real, dimension(3,2) :: points = reshape((/0.0,0.0, 1.0,0.0, 0.0,1.0/), (/3,2/))
  real :: distance
  integer :: i, j

  do i = 1, 3
  do j = i+1, 3
      distance = sqrt((points(i,1)-points(j,1))**2 + (points(i,2)-points(j,2))**2)
      print *, "Distance between point", i, "and", j, "=", distance
  end do
  end do
end program distance_matrix

Output:

Distance between point 1 and 2 = 1.0
Distance between point 1 and 3 = 1.0
Distance between point 2 and 3 = 1.4142135

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8. Nested Loops for Pattern Printing

Nested loops are often used to print patterns in programming exercises.

8.1 Example: Star Triangle

program star_triangle
  integer :: i, j

  do i = 1, 5
  do j = 1, i
      write(*,'(A)', advance="no") "*"
  end do
  print *, ""
  end do
end program star_triangle

Output:

*
**
***
****
*****

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9. Nested Loops with Arrays and Conditional Statements

Nested loops can be combined with IF statements to process array elements conditionally.

program conditional_nested
  integer, dimension(3,3) :: matrix
  integer :: i, j

  do i = 1, 3
  do j = 1, 3
      matrix(i,j) = i*j
      if (matrix(i,j) > 4) then
          print *, "matrix(", i, ",", j, ") is greater than 4:", matrix(i,j)
      end if
  end do
  end do
end program conditional_nested

Output:

matrix( 2 , 3 ) is greater than 4: 6
matrix( 3 , 2 ) is greater than 4: 6
matrix( 3 , 3 ) is greater than 4: 9

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10. Advanced Applications of Nested Loops

10.1 Image Processing

Nested loops are used to traverse pixels in 2D images for operations like filtering, thresholding, and convolution.

10.2 Physics Simulations

Loops inside loops are essential for iterating over space and time grids in simulations like:

• Heat distribution
• Wave propagation
• Particle simulations

10.3 Data Analysis

Nested loops can iterate through multiple datasets, compute pairwise comparisons, or calculate statistical measures.

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11. Best Practices for Nested Loops

1. Keep loops readable: Use descriptive loop variables (i, j, row, col).
2. Avoid excessive nesting: Deeply nested loops can be inefficient and hard to read.
3. Use exit and cycle statements when appropriate to optimize loops.
4. Combine with arrays: Nested loops are ideal for multidimensional array operations.
5. Document purpose: Add comments explaining what each loop does.
6. Consider vectorization: Fortran compilers can optimize some nested loops when vectorized.

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12. Nested Loops with More Than Two Levels

Nested loops can have three or more levels for higher-dimensional data:

program three_level_nested
  integer :: i, j, k

  do i = 1, 2
  do j = 1, 2
      do k = 1, 2
          print *, "i =", i, ", j =", j, ", k =", k
      end do
  end do
  end do
end program three_level_nested

Output:

i = 1 , j = 1 , k = 1
i = 1 , j = 1 , k = 2
i = 1 , j = 2 , k = 1
i = 1 , j = 2 , k = 2
i = 2 , j = 1 , k = 1
i = 2 , j = 1 , k = 2
i = 2 , j = 2 , k = 1
i = 2 , j = 2 , k = 2