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This note attempts to describe the motivation for and design of an implementation of Fortran 90 (and later) array expression evaluation that minimizes the use of dynamically allocated temporary storage for the results of calls to transformational intrinsic functions, and making them more amenable to acceleration.

The transformational intrinsic functions of Fortran of interest to us here include:

Reductions to scalars (SUM(X), also ALL, ANY, COUNT, DOT_PRODUCT, IALL, IANY, IPARITY, MAXVAL, MINVAL, PARITY, PRODUCT)
Axial reductions (SUM(X,DIM=), &c.)
Location reductions to indices (MAXLOC, MINLOC, FINDLOC)
Axial location reductions (MAXLOC(DIM=, &c.)
TRANSPOSE(M) matrix transposition
RESHAPE without ORDER=
RESHAPE with ORDER=
CSHIFT and EOSHIFT with scalar SHIFT=
CSHIFT and EOSHIFT with array-valued SHIFT=
PACK and UNPACK
MATMUL
SPREAD

Other Fortran intrinsic functions are technically transformational (e.g., COMMAND_ARGUMENT_COUNT) but not of interest for this note. The generic REDUCE is also not considered here.

Arrays as functions

A whole array can be viewed as a function that maps its indices to the values of its elements. Specifically, it is a map from a tuple of integers to its element type. The rank of the array is the number of elements in that tuple, and the shape of the array delimits the domain of the map.

REAL :: A(N,M) can be seen as a function mapping ordered pairs of integers (J,K) with 1<=J<=N and 1<=J<=M to real values.

Array expressions as functions

The same perspective can be taken of an array expression comprising intrinsic operators and elemental functions. Fortran doesn't allow one to apply subscripts directly to an expression, but expressions have rank and shape, and one can view array expressions as functions over index tuples by applying those indices to the arrays and subexpressions in the expression.

Consider B = A + 1.0 (assuming REAL :: A(N,M), B(N,M)). The right-hand side of that assignment could be evaluated into a temporary array T and then subscripted as it is copied into B.

REAL, ALLOCATABLE :: T(:,:)
ALLOCATE(T(N,M))
DO CONCURRENT(J=1:N,K=1:M)
  T(J,K)=A(J,K) + 1.0
END DO
DO CONCURRENT(J=1:N,K=1:M)
  B(J,K)=T(J,K)
END DO
DEALLOCATE(T)

But we can avoid the allocation, population, and deallocation of the temporary by treating the right-hand side expression as if it were a statement function F(J,K)=A(J,K)+1.0 and evaluating

DO CONCURRENT(J=1:N,K=1:M)
  A(J,K)=F(J,K)
END DO

In general, when a Fortran array assignment to a non-allocatable array does not include the left-hand side variable as an operand of the right-hand side expression, and any function calls on the right-hand side are elemental or scalar-valued, we can avoid the use of a temporary.

Transformational intrinsic functions as function composition

Many of the transformational intrinsic functions listed above can, when their array arguments are viewed as functions over their index tuples, be seen as compositions of those functions with functions of the "incoming" indices -- yielding a function for an entire right-hand side of an array assignment statement.

For example, the application of TRANSPOSE(A + 1.0) to the index tuple (J,K) becomes A(K,J) + 1.0.

Partial (axial) reductions can be similarly composed. The application of SUM(A,DIM=2) to the index J is the complete reduction SUM(A(J,:)).

More completely:

Reductions to scalars (SUM(X) without DIM=) become runtime calls; the result needs no dynamic allocation, being a scalar.
Axial reductions (SUM(X,DIM=d)) applied to indices (J,K) become scalar values like SUM(X(J,K,:)) if d=3.
Location reductions to indices (MAXLOC(X) without DIM=) do not require dynamic allocation, since their results are either scalar or small vectors of length RANK(X).
Axial location reductions (MAXLOC(X,DIM=), &c.) are handled like other axial reductions like SUM(DIM=).
TRANSPOSE(M) exchanges the two components of the index tuple.
RESHAPE(A,SHAPE=s) without ORDER= must precompute the shape vector S, and then use it to linearize indices into offsets in the storage order of A (whose shape must also be captured). These conversions can involve division and/or modulus, which can be optimized into a fixed-point multiplication using the usual technique.
RESHAPE with ORDER= is similar, but must permute the components of the index tuple; it generalizes TRANSPOSE.
CSHIFT applies addition and modulus.
EOSHIFT applies addition and a conditional move (MERGE).
PACK and UNPACK are likely to require a runtime call.
MATMUL(A,B) can become DOT_PRODUCT(A(J,:),B(:,K)), but might benefit from calling a highly optimized runtime routine.
SPREAD(A,DIM=d,NCOPIES=n) for compile-time d simply applies A to a reduced index tuple.

Determination of rank and shape

An important part of evaluating array expressions without the use of temporary storage is determining the shape of the result prior to, or without, evaluating the elements of the result.

The shapes of array objects, results of elemental intrinsic functions, and results of intrinsic operations are obvious. But it is possible to determine the shapes of the results of many transformational intrinsic function calls as well.

SHAPE(SUM(X,DIM=d)) is SHAPE(X) with one element removed: PACK(SHAPE(X),[(j,j=1,RANK(X))]/=d) in general. (The DIM= argument is commonly a compile-time constant.)
SHAPE(MAXLOC(X)) is [RANK(X)].
SHAPE(MAXLOC(X,DIM=d)) is SHAPE(X) with one element removed.
SHAPE(TRANSPOSE(M)) is a reversal of SHAPE(M).
SHAPE(RESHAPE(..., SHAPE=S)) is S.
SHAPE(CSHIFT(X)) is SHAPE(X); same with EOSHIFT.
SHAPE(PACK(A,VECTOR=V)) is SHAPE(V)
SHAPE(PACK(A,MASK=m)) with non-scalar m and without VECTOR= is [COUNT(m)].
RANK(PACK(...)) is always 1.
SHAPE(UNPACK(MASK=M)) is SHAPE(M).
SHAPE(MATMUL(A,B)) drops one value from SHAPE(A) and another from SHAPE(B).
SHAPE(SHAPE(X)) is [RANK(X)].
SHAPE(SPREAD(A,DIM=d,NCOPIES=n)) is SHAPE(A) with n inserted at dimension d.

This is useful because expression evaluations that do require temporaries to hold their results (due to the context in which the evaluation occurs) can be implemented with a separation of the allocation of the temporary array and the population of the array. The code that evaluates the expression, or that implements a transformational intrinsic in the runtime library, can be designed with an API that includes a pointer to the destination array as an argument.

Statements like ALLOCATE(A,SOURCE=expression) should thus be capable of evaluating their array expressions directly into the newly-allocated storage for the allocatable array. The implementation would generate code to calculate the shape, use it to allocate the memory and populate the descriptor, and then drive a loop nest around the expression to populate the array. In cases where the analyzed shape is known at compile time, we should be able to have the opportunity to avoid heap allocation in favor of stack storage, if the scope of the variable is local.

Automatic reallocation of allocatables

Fortran 2003 introduced the ability to assign non-conforming array expressions to ALLOCATABLE arrays with the implied semantics of reallocation to the new shape. The implementation of this feature also becomes more straightforward if our implementation of array expressions has decoupled calculation of shapes from the evaluation of the elements of the result.

Rewriting rules

Let {...} denote an ordered tuple of 1-based indices, e.g. {j,k}, into the result of an array expression or subexpression.

Array constructors always yield vectors; higher-rank arrays that appear as constituents are flattened; so [X] => RESHAPE(X,SHAPE=[SIZE(X)}).
Array constructors with multiple constituents are concatenations of their constituents; so [X,Y]{j} => MERGE(Y{j-SIZE(X)},X{j},J>SIZE(X)).
Array constructors with implied DO loops are difficult when nested triangularly.
Whole array references can have lower bounds other than 1, so A => A(LBOUND(A,1):UBOUND(A,1),...).
Array sections simply apply indices: A(i:...:n){j} => A(i1+n*(j-1)).
Vector-valued subscripts apply indices to the subscript: A(N(:)){j} => A(N(:){j}).
Scalar operands ignore indices: X{j,k} => X. Further, they are evaluated at most once.
Elemental operators and functions apply indices to their arguments: (A(:,:) + B(:,:)){j,k} => A(:,:){j,k} + B(:,:){j,k}`.
TRANSPOSE(X){j,k} => X{k,j}.
SPREAD(X,DIM=2,...){j,k} => X{j}; i.e., the contents are replicated. If X is sufficiently expensive to compute elementally, it might be evaluated into a temporary.

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