# Mathematics of arrays **URL:** \ **Category:** Tutorials\ **Created:** [August 12, 2021, 8:39am UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685 "2021-08-12T08:39:30Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![wyphan](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/wyphan/32/632_2.png) [@wyphan](https://fortran-lang.discourse.group/u/wyphan)\ **Post date:** [September 2, 2021, 2:13pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/21 "2021-09-02T14:13:36Z")
The second lecture (and the rest) in the series will be held on Zoom: Lenore Mullin is inviting you to a scheduled Zoom meeting. Topic: MoA Lecture 1 Time: Sep 3, 2021 02:30 PM Eastern Time (US and Canada) Every 2 weeks on Fri, until Nov 12, 2021, 6 occurrence(s) Sep 3, 2021 02:30 PM Sep 17, 2021 02:30 PM Oct 1, 2021 02:30 PM Oct 15, 2021 02:30 PM Oct 29, 2021 02:30 PM Nov 12, 2021 02:30 PM Please download and import the following iCalendar (.ics) files to your calendar system. Weekly: [https://albany.zoom.us/meeting/tJwkfumuqT8sGdVCPn9GavN8j88zWjS399rF/ics?icsToken=98tyKuCgpzIqHNORthqGRow-HYr4c-7wtildjadrvy\_rWgdSdC2uPLoaKIV1I4uJ](https://albany.zoom.us/meeting/tJwkfumuqT8sGdVCPn9GavN8j88zWjS399rF/ics?icsToken=98tyKuCgpzIqHNORthqGRow-HYr4c-7wtildjadrvy_rWgdSdC2uPLoaKIV1I4uJ) Join Zoom Meeting [Launch Meeting - Zoom](https://albany.zoom.us/j/98990674631?pwd=TzkwNUhDR0ZkQzZIMzNQWTkzaGJPdz09) Meeting ID: 989 9067 4631 Passcode: 122039 ---
**Author:** ![certik](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/certik/32/4_2.png) [@certik](https://fortran-lang.discourse.group/u/certik)\ **Post date:** [September 3, 2021, 8:40pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/22 "2021-09-03T20:40:25Z")
Sorry I couldn’t make the meeting this time, too busy at work. If you could post the slides, that would be very helpful for me. ---
**Author:** ![wyphan](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/wyphan/32/632_2.png) [@wyphan](https://fortran-lang.discourse.group/u/wyphan)\ **Post date:** [September 3, 2021, 8:46pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/23 "2021-09-03T20:46:35Z")
Hi @certik, yes I will post the slides sometime this weekend after some corrections (mainly typos). Also, since this time it is hosted on Zoom, we recorded it, which I will also post once it’s ready! ---
**Author:** ![certik](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/certik/32/4_2.png) [@certik](https://fortran-lang.discourse.group/u/certik)\ **Post date:** [September 3, 2021, 10:44pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/24 "2021-09-03T22:44:34Z")
@wyphan perfect, thank you! Much appreciated. ---
**Author:** ![wyphan](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/wyphan/32/632_2.png) [@wyphan](https://fortran-lang.discourse.group/u/wyphan)\ **Post date:** [September 5, 2021, 12:49pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/25 "2021-09-05T12:49:50Z")
Hi everyone, The slides for the second lecture can be accessed here: > **[MoA-Lecture 1.pdf](https://drive.google.com/file/d/10m-AUnpx5Dl5cjFK33apLxCpnYzG5lS4/view?usp=sharing)** > > Google Drive file. And here’s the link for the recording: > **[Video Conferencing, Web Conferencing, Webinars, Screen Sharing](https://albany.zoom.us/rec/share/4raZmAe4Q_4gCkIMTY7dbqlclhvPfXJyGMlJN_znclBT4HEIqRYG2MXLvcH9UlGW.KQaOFDeJBFKJHLb5)** > > Zoom is the leader in modern enterprise video communications, with an easy, reliable cloud platform for video and audio conferencing, chat, and webinars across mobile, desktop, and room systems. Zoom Rooms is the original software-based conference... Passcode: `P2=fQfK2` ---
**Author:** ![FortranFan](https://avatars.discourse-cdn.com/v4/letter/f/96bed5/32.png) [@FortranFan](https://fortran-lang.discourse.group/u/FortranFan)\ **Post date:** [September 5, 2021, 2:57pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/26 "2021-09-05T14:57:22Z")
> [@Lenore](#): > > Hoping to generate interest and things that came to mind. > > There are so many array languages vying to be “the” array language. Fortran has been around > for a while, has grown, and is used by many. I know there is an enhanced subset that is based > on mathematics, and with mathematics there are no debates. Furthermore, much, if not most, can be automated to assist compilation to hardware or software. Hoping Fortran folks attend an intro lecture to discuss these topics. @Lenore , The math with MoA is most interesting, there is a lot there to be studied by compiler writers, those working on compiler optimizations, and certain library authors. Some questions re: Fortran and its leaning toward an “array language” since its inception and all the work you have done with MoA: - if you can do a quick review (or perhaps you’ve done it already) of **current Fortran standard published in 2018** , what comes across to you that the Fortran base language itself can do better to be a reasonably full-fledged “array language”? - Are there any improvements in terms of semantics or syntax or intrinsic facilities (e.g., functions, etc.) that can be introduced in the Fortran standard to enable better practice of MoA using Fortran? A summary or a paper (a short one or a very detailed) on **recommended Fortran language standard enhancements** , should there be any, by you and those working with you on MoA will be highly valuable for the language. ---
**Author:** ![Lenore](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/lenore/32/762_2.png) [@Lenore](https://fortran-lang.discourse.group/u/Lenore)\ **Post date:** [September 5, 2021, 3:16pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/27 "2021-09-05T15:16:06Z")
I would love to work with you and others on this. ---
**Author:** ![Lenore](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/lenore/32/762_2.png) [@Lenore](https://fortran-lang.discourse.group/u/Lenore)\ **Post date:** [September 5, 2021, 3:17pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/28 "2021-09-05T15:17:23Z")
When I get home I’ll respond more on Fortran discourse. ---
**Author:** ![Beliavsky](https://avatars.discourse-cdn.com/v4/letter/b/ba8739/32.png) [@Beliavsky](https://fortran-lang.discourse.group/u/Beliavsky)\ **Post date:** [September 5, 2021, 3:27pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/29 "2021-09-05T15:27:27Z")
> [@FortranFan](#): > > A summary or a paper (a short one or a very detailed) on **recommended Fortran language standard enhancements** , should there be any, by you and those working with you on MoA will be highly valuable for the language. Looking at the slides, it’s not clear on p23 how to express “catenation” in Fortran. Another issue, which should be solvable, is that MoA assumes array indexing starting at 0, while Fortran arrays start at 1 by default. One can declare a lower bound other than 1, but when passing such an array to a procedure the lower bound is typically lost (example below). Thanks to Lenore for her lectures. ```auto module m contains function take(ivec,itake) result(jvec) integer, intent(in) :: ivec(:),itake(:) integer :: jvec(size(itake)) jvec = ivec(itake) end function take end module m program main use m integer :: ivec(0:2) ivec = [10,20,30] print*,take(ivec,[1,2]) ! gives [10,20] end program main ``` ---
**Author:** ![wyphan](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/wyphan/32/632_2.png) [@wyphan](https://fortran-lang.discourse.group/u/wyphan)\ **Post date:** [September 5, 2021, 7:34pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/30 "2021-09-05T19:34:22Z")
@Beliavsky Yes, I encountered the 1-based indexing in Fortran to be an issue when testing the Fortran analog for the “take” operation (p.19). Thankfully the fix is pretty simple, i.e. manually add `+1`. For instance, here’s the “take” operation implemented as a pointer function for the default `real` data type. ```auto PROGRAM taketest IMPLICIT NONE INTEGER, PARAMETER :: n = 10 INTEGER :: i, s REAL, TARGET :: v(0:n-1) REAL, POINTER :: p(:) ! Init v v = [( 0.1 * REAL(i), i = 0, n-1 )] PRINT *, 'vvec = ' PRINT *, ( v(i), ' ' , i = 0, n-1 ) ! Positive take s = 2 p => take( s, v ) PRINT *, s, ' take vvec = ' PRINT *, ( p(i), ' ' , i = 1, s ) NULLIFY(p) ! Negative take s = -2 p => take( s, v ) PRINT *, s, ' take vvec = ' PRINT *, ( p(i), ' ' , i = 1, ABS(s) ) NULLIFY(p) STOP CONTAINS FUNCTION take( sigma, vec ) IMPLICIT NONE ! Arguments INTEGER, INTENT(IN) :: sigma REAL, INTENT(IN), TARGET :: vec(:) REAL, POINTER :: take(:) ! Internal variables INTEGER :: n n = SIZE( vec, 1 ) IF( ABS(sigma) <= n ) THEN IF( sigma > 0 ) THEN ! Positive take take => vec( :sigma ) ELSE ! Negative take ! Note how +1 is needed here to account for 1-based Fortran indexing take => vec( (n+sigma+1): ) END IF ELSE PRINT *, 'Error[take]: out of bounds ', sigma, ' of ', n END IF END FUNCTION take END PROGRAM taketest ``` ---
**Author:** ![FortranFan](https://avatars.discourse-cdn.com/v4/letter/f/96bed5/32.png) [@FortranFan](https://fortran-lang.discourse.group/u/FortranFan)\ **Post date:** [September 5, 2021, 8:34pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/32 "2021-09-05T20:34:56Z")
> [@wyphan](#): > > ```auto > CONTAINS > > FUNCTION take( sigma, vec ) > IMPLICIT NONE > > ! Arguments > INTEGER, INTENT(IN) :: sigma > REAL, INTENT(IN), TARGET :: vec(:) > REAL, POINTER :: take(:) > > ``` @wyphan , @Beliavsky , and especially anyone interested in **library** development: I would “debate” instead the flexibility provided in the Fortran standard with `assumed-shape` arrays is “good enough” for the needs here. The “take” procedure truly belongs in a library and under the circumstances, that library can have named constant for the desired lower bound (zero-based or whatever) and work with assumed-shaped arrays accordingly. ```Fortran integer, parameter :: LB = 0 ! Desired lower bound .. function take (..) .. real, .., intent(in) :: vec(LB:) ! Set the lower bound .. ``` While the above may fail to tickle some tummies, it is hardly the painful barriers in 0-based indexing languages (Djikstra-inspired ones) when it comes to a library author wanting to use a lower bound other than that hardwired into the language. With Fortran language standard and MoA, the question instead is **whether something deeper is lacking.** ---
**Author:** ![Lenore](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/lenore/32/762_2.png) [@Lenore](https://fortran-lang.discourse.group/u/Lenore)\ **Post date:** [September 6, 2021, 11:13am UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/33 "2021-09-06T11:13:27Z")
@certik @gak and other Fortran experts who know the standard : we could do this together and publish to help others get on board. BTW index origin is arbitrary in a language. The theory is behind the scenes. ---
**Author:** ![wyphan](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/wyphan/32/632_2.png) [@wyphan](https://fortran-lang.discourse.group/u/wyphan)\ **Post date:** [September 9, 2021, 2:13pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/34 "2021-09-09T14:13:08Z")
@Beliavsky Sorry I missed the question on catenation. Yes, I don’t have any idea yet on how to make it work properly for cases such as catenating a vector with a scalar. ---
**Author:** ![Arjen](https://avatars.discourse-cdn.com/v4/letter/a/b9bd4f/32.png) [@Arjen](https://fortran-lang.discourse.group/u/Arjen)\ **Post date:** [September 9, 2021, 2:24pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/35 "2021-09-09T14:24:44Z")
I am picking up this challenge 🙂 ---
**Author:** ![wyphan](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/wyphan/32/632_2.png) [@wyphan](https://fortran-lang.discourse.group/u/wyphan)\ **Post date:** [September 9, 2021, 5:56pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/36 "2021-09-09T17:56:39Z")
@Arjen Please DM me your GitHub username, so I can add you to a common repo that I made for the MoA Fortran Library ---
**Author:** ![Arjen](https://avatars.discourse-cdn.com/v4/letter/a/b9bd4f/32.png) [@Arjen](https://fortran-lang.discourse.group/u/Arjen)\ **Post date:** [September 17, 2021, 7:34am UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/37 "2021-09-17T07:34:54Z")
Okay, it took me some time to acutally sit down for it, but I have a (slightly disfunctional) example of how you could do the catenate operation (something is still wrong, need to figure that out). Here is the code: ```auto ! moa_cat.f90 -- ! First version of a cat operation ! module moa_operations implicit none ! ! Type for defining "views" on arrays - to allow for a#b#c: pointers to other views ! type view_type integer, dimension(:), pointer :: left_array => null() integer, dimension(:), pointer :: right_array => null() type(view_type), pointer :: left_view => null() type(view_type), pointer :: right_view => null() contains ! procedure, nopass :: cataa => catenate_array_array ! procedure, nopass :: catav => catenate_array_view ! procedure, nopass :: catva => catenate_view_array ! procedure, nopass :: catvv => catenate_view_view procedure :: elem => get_elem ! generic :: cat => cataa, catav, catva, catvv end type view_type interface operator(//) module procedure catenate_array_array, catenate_array_view, catenate_view_array, catenate_view_view end interface contains function catenate_array_array( array1, array2 ) result(new_view) integer, dimension(:), target, intent(in) :: array1 integer, dimension(:), target, intent(in) :: array2 class(view_type), allocatable :: new_view allocate( new_view ) new_view%left_array => array1 new_view%right_array => array2 end function catenate_array_array function catenate_view_array( view1, array2 ) result(new_view) class(view_type), target, intent(in) :: view1 integer, dimension(:), target, intent(in) :: array2 class(view_type), allocatable :: new_view allocate( new_view ) new_view%left_view => view1 new_view%right_array => array2 end function catenate_view_array function catenate_array_view( array1, view2 ) result(new_view) integer, dimension(:), target, intent(in) :: array1 class(view_type), target, intent(in) :: view2 class(view_type), allocatable :: new_view allocate( new_view ) new_view%left_array => array1 new_view%right_view => view2 end function catenate_array_view function catenate_view_view( view1, view2 ) result(new_view) class(view_type), target, intent(in) :: view1 class(view_type), target, intent(in) :: view2 class(view_type), allocatable :: new_view allocate( new_view ) new_view%left_view => view1 new_view%right_view => view2 end function catenate_view_view recursive function get_elem(view, i) result(elem) class(view_type), intent(inout) :: view integer, intent(in) :: i integer, pointer :: elem integer :: sz if ( associated(view%left_array) ) then sz = size(view%left_array) if ( i <= sz ) then elem => view%left_array(i) else if ( associated(view%right_array) ) then elem => view%right_array(i-sz) else elem => view%right_view%elem(i-sz) endif endif else elem => view%left_view%elem(i) endif end function end module moa_operations program test_cat use moa_operations implicit none type(view_type) :: x, y integer, dimension(10) :: a, b, c integer :: i, j ! 1 2 3 4 5 6 7 8 9 10 a = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] b = [-2, -3, -5, -7, -11, -13, -17, -19, -23, -29] c = [31, 37, 41, 43, 47, 53, 59, 61, 67, 71] x = a // b y = x // c do i = 1,30 write(*,*) i, y%elem(i) enddo end program test_cat ``` ---
**Author:** ![everythingfunctional](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/everythingfunctional/32/176_2.png) [@everythingfunctional](https://fortran-lang.discourse.group/u/everythingfunctional)\ **Post date:** [September 17, 2021, 12:00pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/38 "2021-09-17T12:00:02Z")
> [@Arjen](#): > > ```auto > else > elem => view%left_view%elem(i) > endif > > ``` In this branch you still need to determine if `i <=` size of the left view. I think a recursive procedure to calculate the size of a view is needed. ```auto if (associated(view%left_array) then if (associated(view%right_array) then view_size = size(view%left_array) + size(view%right_array) else view_size = size(view%left_array) + view%right_view%size() end if else if (associated(view%right_array) then view_size = view%left_view%size() + size(view%right_array) else view_size = view%left_view%size() + view%right_view%size() end if end if ``` An additional thought, should you check for index out of bounds and `error stop`? Or should this behave like normal arrays and only crash if you’ve got bounds checking on? ---
**Author:** ![Arjen](https://avatars.discourse-cdn.com/v4/letter/a/b9bd4f/32.png) [@Arjen](https://fortran-lang.discourse.group/u/Arjen)\ **Post date:** [September 17, 2021, 12:01pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/39 "2021-09-17T12:01:17Z")
@everythingfunctional , yes, I found that out myself. Trying to wrap my poor brains around this, while als o doing my regular job 😉 ---
**Author:** ![Arjen](https://avatars.discourse-cdn.com/v4/letter/a/b9bd4f/32.png) [@Arjen](https://fortran-lang.discourse.group/u/Arjen)\ **Post date:** [September 17, 2021, 12:30pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/40 "2021-09-17T12:30:20Z")
Okay, it took a bit serious thinking and debugging, but here is an improved version (not extensively tested, mind you): ```auto ! moa_cat.f90 -- ! First version of a cat operation ! module moa_operations implicit none ! ! Type for defining "views" on arrays - to allow for a#b#c: pointers to other views ! type view_type integer, dimension(:), pointer :: left_array => null() integer, dimension(:), pointer :: right_array => null() type(view_type), pointer :: left_view => null() type(view_type), pointer :: right_view => null() contains ! procedure, nopass :: cataa => catenate_array_array ! procedure, nopass :: catav => catenate_array_view ! procedure, nopass :: catva => catenate_view_array ! procedure, nopass :: catvv => catenate_view_view procedure :: elem => get_elem ! generic :: cat => cataa, catav, catva, catvv end type view_type interface operator(//) module procedure catenate_array_array, catenate_array_view, catenate_view_array, catenate_view_view end interface contains function catenate_array_array( array1, array2 ) result(new_view) integer, dimension(:), target, intent(in) :: array1 integer, dimension(:), target, intent(in) :: array2 class(view_type), allocatable :: new_view allocate( new_view ) new_view%left_array => array1 new_view%right_array => array2 end function catenate_array_array function catenate_view_array( view1, array2 ) result(new_view) class(view_type), target, intent(in) :: view1 integer, dimension(:), target, intent(in) :: array2 class(view_type), allocatable :: new_view allocate( new_view ) new_view%left_view => view1 new_view%right_array => array2 end function catenate_view_array function catenate_array_view( array1, view2 ) result(new_view) integer, dimension(:), target, intent(in) :: array1 class(view_type), target, intent(in) :: view2 class(view_type), allocatable :: new_view allocate( new_view ) new_view%left_array => array1 new_view%right_view => view2 end function catenate_array_view function catenate_view_view( view1, view2 ) result(new_view) class(view_type), target, intent(in) :: view1 class(view_type), target, intent(in) :: view2 class(view_type), allocatable :: new_view allocate( new_view ) new_view%left_view => view1 new_view%right_view => view2 end function catenate_view_view recursive subroutine get_pointer(view, i, elem, found) class(view_type), intent(inout) :: view integer, intent(inout) :: i integer, pointer :: elem logical, intent(out) :: found integer :: sz found = .false. if ( associated(view%left_array) ) then sz = size(view%left_array) if ( i <= sz ) then found = .true. elem => view%left_array(i) else i = i - sz if ( associated(view%right_array) ) then sz = size(view%right_array) if ( i <= sz ) then found = .true. elem => view%right_array(i) else i = i - sz return endif else call get_pointer( view%right_view, i, elem, found ) endif endif else call get_pointer(view%left_view, i, elem, found ) if ( .not. found ) then if ( associated(view%right_array) ) then sz = size(view%right_array) if ( i <= sz ) then found = .true. elem => view%right_array(i) else error stop endif else call get_pointer( view%right_view, i, elem, found ) endif endif endif end subroutine get_pointer function get_elem(view, i) result(elem) class(view_type), intent(inout) :: view integer, intent(in) :: i integer, pointer :: elem integer :: inew logical :: found inew = i call get_pointer( view, inew, elem, found ) end function end module moa_operations program test_cat use moa_operations implicit none type(view_type) :: x, y integer, dimension(10) :: a, b, c integer :: i, j ! 1 2 3 4 5 6 7 8 9 10 a = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] b = [-2, -3, -5, -7, -11, -13, -17, -19, -23, -29] c = [31, 37, 41, 43, 47, 53, 59, 61, 67, 71] x = a // b y = x // c do i = 1,30 write(*,*) i, y%elem(i) enddo end program test_cat ``` ---
**Author:** ![Xingzhi](https://avatars.discourse-cdn.com/v4/letter/x/dec6dc/32.png) [@Xingzhi](https://fortran-lang.discourse.group/u/Xingzhi)\ **Post date:** [September 17, 2021, 1:26pm UTC](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685/41 "2021-09-17T13:26:37Z")
Google’s [Dex language](https://github.com/google-research/dex-lang) looks interesting. Is it somehow related to MoA? [Previous page](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685.md?page=1) [Next page](https://fortran-lang.discourse.group/t/mathematics-of-arrays/1685.md?page=3)