# Sergey Budaev

## How to make an array and initialize it with a sequence of values in Fortran?

How do you make an array and initialize it with a sequence of values? For example, I want a list from 0.25 to 1.5 that is separated with 0.25. In other words I want something similar to seq(0.25,5,0.5) in R.

## Equally spaced real array with fixed increment in Fortran

Producing an equally spaced array from V1 to VN with increments ΔV

1. Each of the values in the above vector can be calculated as:

2. The total number of values N in the array ending with a fixed known VN is equal to

3. It is not possible to use a simple piece of code like this to produce real type array in Fortran:

Array = [V1:VN:Incr]


4. Such a construction cannot be used in modern Fortran, even though old versions could accept a similar construction based on implied loop with real type index counter:

real :: r ! Index must be integer in loops!
print *, (r, r=V1,VN,Incr)


5. In modern Fortran standard do loops can only have integer indexing variable. Real indexing in do loops is one of the very few features that had been deleted from the language because it can create lots of problems in float point computations due to finite precision in computer hardware.

The old code might work with modern compilers but it may require special legacy compiler options. The printing-only code as above may still work but would issue a compiler warning.

6. Initialising such equally spaced real type arrays in Fortran implied loops must use the formulas defined in 1. and 2.

# Produce exactly N_VALS values starting from INIT with increments INCR
Array = [( INIT + INCR * (i-1), i=1,N_VALS )]


Where the number of array elements N_VALS is calculated as:

N_VALS = floor( (END - INIT) / INCR + 1 )

N_VALS = ceiling( (END - INIT) / INCR + 1 )


The floor and ceiling functions convert real value to integer as the lower or upper nearest integer; they can give different values when division cannot be done without the remainder

# All values starting from INIT with increments INCR and up to the limit END
Array = [( INIT + INCR * (i-1), i=1,floor((END-INIT)/INCR+1) )]


7. This code does not seem to be a very simple and elegant solution. Ideally, the code should be packaged into a function returning the desired grid array. But such function could not be used in declarations of array parameters. In the later case the one-liner code should be used as above.

## Integer arrays

By the way, it is quite easy to produce an integer array, e.g. here is an initialisation for array from 1 to 100 (|1,2,3,...,100|). This can be useful for indexing arrays.

integer, parameter, dimension(*) :: IDX_ARRAY = (/(i,i=1,100)/)


## Examples:

A. Produce an array of 10 values starting from 1.0 with increments 0.1

Array = [( 1.0 + (i-1) * 0.1, i=1,10 )]


Result:

1.00000000 1.10000002 1.20000005 1.29999995 1.39999998
1.50000000 1.60000002 1.70000005 1.79999995 1.90000010


Declaration of a parameter array:

real, parameter, dimension(*) :: Array = [( 1.0 + (i-1) * 0.1, i=1,10 )]


However, note that not all compilers may support assumed array size dimension(*) in such array declaration statement, this requires newer Fortran standard (fortunately, recent versions of Intel and GNU Fortran do support assumed size arrays). In such a case declaration must explicitly set the number of array elements:

real, parameter, dimension(10) :: Array = [( 1.0 + (i-1) * 0.1, i=1,10 )]


B. Produce an array of starting from 1.0 to 2.0 with increments 0.145; note that lower value (floor) for the array size is used:

Array = [( 1.0 + 0.145 * (i-1), i=1, floor((2.0-1.0)/0.145 + 1) )]


Result:

1.00000000 1.14499998 1.28999996 1.43499994 1.57999992
1.72499990 1.87000000


C. The same as (B) but the upper value (ceiling) for the array size is used:

Array = [( 1.0 + 0.145 * (i-1), i=1, ceiling((2.0-1.0)/0.145 + 1) )]


Result:

1.00000000 1.14499998 1.28999996 1.43499994 1.57999992
1.72499990 1.87000000 2.01499987


D. In the case B., declarations of parameter arrays can be done like this:

real, parameter, dimension(*) :: Array =                                  &
[( 1.0 + 0.145 * (i-1), i=1, floor((2.0-1.0)/0.145 + 1) )]


or, if the compiler does not support assumed size arrays (*), with explicitly calculated array size:

real, parameter, dimension(floor((2.0-1.0)/0.145 + 1)) :: Array =         &
[( 1.0 + 0.145 * (i-1), i=1, floor((2.0-1.0)/0.145 + 1) )]


## Test program

! This program illustrates how to produce equally spaced real vectors with
! fixed increment in Fortran.
!
! 1. Produce exactly N_VALS values starting from INIT with increments INCR
! Array = [( INIT + INCR * (i-1), i=1,N_VALS )]
!
! 2. All values starting from INIT with increments INCR and up to the limit END
! Array = [( INIT + INCR * (i-1), i=1,floor((END-INIT)/INCR+1) )]
!-------------------------------------------------------------------------------
program spaced_array

! Integer counter for implied loops defining vectors.
integer :: i

! Example A. Produce an array of 10 values
! starting from 1.0 with increments 0.1
real, parameter, dimension(*) :: Array1 = [( 1.0 + (i-1) * 0.1, i=1,10 )]

! Example B. Produce an array of starting from 1.0 to 2.0
! with increments 0.145.
! Note that lower value (floor) for the array size is used.
real, parameter, dimension(*) :: Array2 = &
[( 1.0 + 0.145 * (i-1), i=1, floor((2.0-1.0)/0.145 + 1) )]

! Example C. The same as (B) but the upper value (ceiling) for the
! array size is used.
real, parameter, dimension(*) :: Array3 = &
[( 1.0 + 0.145 * (i-1), i=1, ceiling((2.0-1.0)/0.145 + 1) )]

! Print the sizes of the arrays that were declared above.
print *, "Array sizes (Array1, Array2, Array3)", &
size(Array1), size(Array2), size(Array3)

! Print the parameter arrays that were declared above.
print *, "Array1", Array1
print *, "Array2", Array2
print *, "Array3", Array3

end program spaced_array


## How to produce a reverse of a vector in Fortran?

Let's we have a vector A, e.g.

A = [1,2,3,4,5]


How to produce a vector with reverse indices, e.g.

B = [5,4,3,2,1]


B = A( size(A):1:-1 )


To reverse A itself do

A = A( size(A):1:-1 )


## Gaussian random numbers in Fortran

The HEDTOOLS tools library has a module for working with random numbers BASE_RANDOM​. There is, in particular, a set of procedures for generating Gaussian random values: ​RNORM and RNORM_ARRAY. These are based on the Kinderman & Monahan, augmented with quadratic bounding curves method (Leva, 1992: algorithm 712, Trans. Math. Software, 18, 4, 434-435​).

I have made a quick comparison of the quality of the Gaussian random numbers generated by the simple Box-Muller method (Box & Muller, 1958​)

Classical (ancient) Fortran code:

normrand_number = dsqrt(-2.*dlog(drand(0)))*dcos(2.*pi*drand(0))


that has been used in TEG codes so far...

and the algorithm 712 as implemented in HEDTOOLS using this test program (see attachment).

Fortran code for the test program:

program test_bm
use csv_io
use base_random, rand_x => rand     ! Alias rand() as rand_x() for ifort.
!use IFPORT, only : rand_x => rand  ! This is the Intel Fortran tweak.

integer, parameter :: prec = 8, arrsize=100000
character(len=255), parameter :: filename1="file_01.csv", filename2="file_02.csv"
real(kind=prec), dimension(arrsize) :: norand1, norand2
real :: timer_start, timer_end
!-------------------------------------------------------------------------------
! Generating Box-Muller random numbers
call cpu_time(timer_start)  ! START
do i=1, arrsize
norand1(i) = sqrt(-2.*log(rand_x(0)))*cos(2.*pi*rand_x(0))
end do
call cpu_time(timer_end)    ! END
print *, "Box-Muller took: ", timer_end - timer_start
! Write random normal data to CSV
call CSV_MATRIX_WRITE(norand1, filename1)
!-------------------------------------------------------------------------------

!-------------------------------------------------------------------------------
! Generating based on algorithm 712
call cpu_time(timer_start)  ! START
call RNORM_ARRAY(norand2)
call cpu_time(timer_end)    ! END
print *, "Alg. 712 took: ", timer_end - timer_start
! Write random normal data to CSV
call CSV_MATRIX_WRITE(norand2, filename2)
!-------------------------------------------------------------------------------

end program test_bm


## Comparison of Box-Muller and A712

The alg. 712 looks slightly faster than the simple Box-Muller transform.

alg. 712 is much better, as the Box-Muller significantly deviates from the normal distribution, alg. 712 does not (using the Anderson-Darling test from the nortest R package).

  # Gaussian random numbers by Box-Muller deviate from the Normal distribution:
> ad.test(data_bm$X1) Anderson-Darling normality test data: data_bm$X1
A = 581.7, p-value < 2.2e-16
# Gaussian random numbers by Kinderman & Monahan's A712 do not deviate from the Normal distribution:
> ad.test(data_a712$X1) Anderson-Darling normality test data: data_a712$X1
A = 0.46975, p-value = 0.2474


So, the alg. 712 procedure implemented in HEDTOOLS should be used instead of the Box-Muller method.

## References

• Box, G. E. P., & Muller, M. E. (1958). A note on the generation of random normal deviates. The Annals of Mathematical Statistics, 29(2), 610–611. ​http://doi.org/10.1214/aoms/1177706645

• Leva, J. L. (1992). Algorithm 712; a normal random number generator. ACM Transactions on Mathematical Software, 18(4), 454–455. ​http://doi.org/10.1145/138351.138367