Nonempty Intervals and Generalized Arrays

Bradley J. Lucier

This SRFI is currently in *final* status. Here is an explanation of each status that a SRFI can hold. To provide input on this SRFI, please send email to `srfi-122@nospamsrfi.schemers.org`

. To subscribe to the list, follow these instructions. You can access previous messages via the mailing list archive. There is a git repository of this document, a reference implementation, a test file, and other materials.

- Received: 2015/7/23
- Draft #1 published: 2015/7/27
- Draft #2 published: 2015/7/31
- Draft #3 published: 2015/7/31
- Draft #4 published: 2015/9/03
- Draft #5 published: 2015/9/18
- Draft #6 published: 2015/10/19
- Draft #7 published: 2016/8/15
- Draft #8 published: 2016/8/19
- Draft #9 published: 2016/8/25
- Draft #10 published: 2016/8/30
- Draft #11 published: 2016/9/7
- Draft #12 published: 2016/9/17
- Draft #13 published: 2016/11/18
- Draft #14 published: 2016/11/28
- Draft #15 published: 2016/12/15
- Finalized: 2016/12/24
- Note that the author has made a revised version available. See his message. The new changes are not fixes to errors, so they have not been incorporated here.

This SRFI specifies an array mechanism for Scheme. Arrays as defined here are quite general; at their most basic, an array is simply a mapping, or function, from multi-indices of exact integers $i_0,\ldots,i_{d-1}$ to Scheme values. The set of multi-indices $i_0,\ldots,i_{d-1}$ that are valid for a given array form the *domain* of the array. In this SRFI, each array's domain consists of a rectangular interval $[l_0,u_0)\times[l_1,u_1)\times\cdots\times[l_{d-1},u_{d-1})$, a subset of $\mathbb Z^d$, $d$-tuples of integers. Thus, we introduce a data type called *intervals*, which encapsulate the cross product of nonempty intervals of exact integers. Specialized variants of arrays are specified to provide portable programs with efficient representations for common use cases.

In a 1993 post to the news group comp.lang.scheme, Alan Bawden gave a simple implementation of multi-dimensional arrays in R4RS scheme. The only constructor of new arrays required specifying an initial value, and he provided the three low-level primitives `array-ref`

, `array-set!`

, and `array?`

. His arrays were defined on rectangular intervals in $\mathbb Z^d$ of the form $[0,u_0)\times\cdots\times [0,u_{d-1})$. I'll note that his function `array-set!`

put the value to be entered into the array at the front of the variable-length list of indices that indicate where to place the new value. He offered an intriguing way to "share" arrays in the form of a routine `make-shared-array`

that took a mapping from a new interval of indices into the domain of the array to be shared. His implementation incorporated what he called an *indexer*, which was a function from the interval $[0,u_0)\times\cdots\times [0,u_{d-1})$ to an interval $[0,N)$, where the *body* of the array consisted of a single Scheme vector of length $N$. Bawden called the mapping specified in `make-shared-array`

*linear*, but I prefer the term *affine*, as I explain later.

Mathematically, Bawden's arrays can be described as follows. We'll use the vector notation $\vec i$ for a multi-index $i_0,\ldots,i_{d-1}$. (Multi-indices correspond to Scheme `values`

.) Arrays will be denoted by capital letters $A,B,\ldots$, the domain of the array $A$ (in Bawden's case $[0,u_0)\times \cdots\times [0,u_{d-1})$) will be denoted by $D_A$, and the indexer of $A$, mapping $D_A$ to the interval $[0,N)$ will be denoted by $I_A$. Initially, Bawden constructs $I_A$ such that $I_A(\vec i)$ steps consecutively through the values $0,1,\ldots,N-1$ as $\vec i$ steps through the multi-indices $(0,\ldots,0,0)$, $(0,\ldots,0,1)$, $\ldots$, $(0,\ldots,1,0)$, etc., in lexicographical order, which means that if $\vec i$ and $\vec j$ are two multi-indices, then $\vec i<\vec j$ iff the first coordinate $k$ where $\vec i$ and $\vec j$ differ satisfies $\vec i_k<\vec j_k$. In fact, $I_A(\vec i)=\vec v\cdot\vec i$ for some specially-constructed vector $\vec v$ that depends only on $D_A$, the domain of $A$, where $\vec v\cdot\vec i$ is the dot product of $\vec v$ and $\vec i$.

In `make-shared-array`

, Bawden allows you to specify a new $r$-dimensional interval $D_B$ as the domain of a new array $B$, and a mapping $T_{BA}:D_B\to D_A$ of the form $T_{BA}(\vec i)=M\vec i+\vec b$; here $M$ is a $d\times r$ matrix of integer values and $\vec b$ is a $d$-vector. So this mapping $T_{BA}$ is *affine*, in that $T_{BA}(\vec i)-T_{BA}(\vec j)=M(\vec i-\vec j)$ is *linear* (in a linear algebra sense) in $\vec i-\vec j$. The new indexer of $B$ satisfies $I_B(\vec i)=I_A(T_{BA}(\vec i))$.

A fact Bawden exploits in the code, but doesn't point out in the short post, is that $I_B$ is again an affine map, and indeed, the composition of *any* two affine maps is again affine.

We incorporate Bawden-style arrays into this SRFI, but extend them in two relatively minor ways that we find quite useful.

First, we allow the intervals of multi-indices that form the domains of arrays to have nonzero lower bounds as well as upper bounds, so domains are rectangular, $d$-dimensional intervals $[l_0,u_0)\times\cdots\times[l_{d-1},u_{d-1})$.

Second, we introduce the notion of a *storage class*, an object that contains functions that manipulate, store, check, etc., different types of values. A `generic-storage-class`

can manipulate any Scheme value, whereas,e.g., a `u1-storage-class`

can store only the values 0 and 1 in each element of a body.

We also require that our affine maps be one-to-one, so that if $\vec i\neq\vec j$ then $T(\vec i)\neq T(\vec j)$. Without this property, modifying the $\vec i$th component of $A$ would cause the $\vec j$th component to change.

Requiring the transformations $T_{BA}:D_B\to D_A$ to be affine may seem esoteric and restricting, but in fact many common and useful array transformations can be expressed in this way. We give several examples below:

**Restricting the domain of an array:**If the domain of $B$, $D_B$, is a subset of the domain of $A$, then $T_{BA}(\vec i)=\vec i$ is a one-to-one affine mapping. We define`array-extract`

to define this common operation; it's like looking at a rectangular sub-part of a spreadsheet. We use it to extract the common part of overlapping domains of three arrays in an image processing example below.**Translating the domain of an array:**If $\vec d$ is a vector of integers, then $T_{BA}(\vec i)=\vec i-\vec d$ is a one-to-one affine map of $D_B=\{\vec i+\vec d\mid \vec i\in D_A\}$ onto $D_A$. We call $D_B$ the*translate*of $D_A$, and we define`array-translate`

to provide this operation.**Permuting the coordinates of an array:**If $\pi$ permutes the coordinates of a multi-index $\vec i$, and $\pi^{-1}$ is the inverse of $\pi$, then $T_{BA}(\vec i)=\pi (\vec i)$ is a one-to-one affine map from $D_B=\{\pi^{-1}(\vec i)\mid \vec i\in D_A\}$ onto $D_A$. We provide`array-permute`

for this operation. The only nonidentity permutation of a two-dimensional spreadsheet turns rows into columns and vice versa.**Currying an array:**Let's denote the cross product of two intervals $\text{Int}_1$ and $\text{Int}_2$ by $\text{Int}_1\times\text{Int}_2$; if $\vec j=(j_0,\ldots,j_{r-1})\in \text{Int}_1$ and $\vec i=(i_0,\ldots,i_{s-1})\in \text{Int}_2$, then $\vec j\times\vec i$, which we define to be $(j_0,\ldots,j_{r-1},i_0,\ldots,i_{s-1})$, is in $\text{Int}_1\times\text{Int}_2$. If $D_A=\text{Int}_1\times\text{Int}_2$ and $\vec j\in\text{Int}_1$, then $T_{BA}(\vec i)=\vec j\times\vec i$ is a one-to-one affine mapping from $D_B=\text{Int}_2$ into $D_A$. For each vector $\vec j$ we can compute a new array in this way; we provide`array-curry`

for this operation, which returns an array whose domain is $\text{Int}_1$ and whose elements are themselves arrays, each of which is defined on $\text{Int}_2$. Currying a two-dimensional array would be like organizing a spreadsheet into a one-dimensional array of rows of the spreadsheet.**Traversing some indices in a multi-index in reverse order:**Consider an array $A$ with domain $D_A=[l_0,u_0)\times\cdots\times[l_{d-1},u_{d-1})$. Fix $D_B=D_A$ and assume we're given a vector of booleans $F$ ($F$ for "flip?"). Then define $T_{BA}:D_B\to D_A$ by $i_j\to i_j$ if $F_j$ is`#f`

and $i_j\to u_j+l_j-1-i_j$ if $F_j$ is`#t`

.In other words, we reverse the ordering of the $j$th coordinate of $\vec i$ if and only if $F_j$ is true. $T_{BA}$ is an affine mapping from $D_B\to D_A$, which defines a new array $B$, and we can provide`array-reverse`

for this operation. Applying`array-reverse`

to a two-dimensional spreadsheet might reverse the order of the rows or columns (or both).**Uniformly sampling an array:**Assume that $A$ is an array with domain $[0,u_1)\times\cdots\times[0,u_{d-1})$ (i.e., an interval all of whose lower bounds are zero). We'll also assume the existence of vector $S$ of scale factors, which are positive exact integers. Let $D_B$ be a new interval with $j$th lower bound equal to zero and $j$th upper bound equal to $\operatorname{ceiling}(u_j/S_j)$ and let $T_{BA}(\vec i)_j=i_j\times S_j$, i.e., the $j$th coordinate is scaled by $S_j$. ($D_B$ contains precisely those multi-indices that $T_{BA}$ maps into $D_A$.) Then $T_{BA}$ is an affine one-to-one mapping, and we provide`interval-scale`

and`array-sample`

for these operations.

We make several remarks. First, all these operations could have been computed by specifying the particular mapping $T_{BA}$ explicitly, so that these routines are simply "convenience" procedures. Second, because the composition of any number of affine mappings are again affine, accessing or changing the elements of a restricted, translated, curried, permuted array is no slower than accessing or changing the elements of the original array itself. Finally, we note that by combining array currying and permuting, say, one can come up with simple expressions of powerful algorithms, such as extending one-dimensional transforms to multi-dimensional separable transforms, or quickly generating two-dimensional slices of three-dimensional image data. Examples are given below.

Bawden-style arrays are clearly useful as a programming construct, but they do not fulfill all our needs in this area. An array, as commonly understood, provides a mapping from multi-indices $(i_0,\ldots,i_{d-1})$ of exact integers
in a nonempty, rectangular, $d$-dimensional interval $[l_0,u_0)\times[l_1,u_1)\times\cdots\times[l_{d-1},u_{d-1})$ (the *domain* of the array) to Scheme objects.
Thus, two things are necessary to specify an array: an interval and a mapping that has that interval as its domain.

Since these two things are often sufficient for certain algorithms, we introduce in this SRFI a minimal set of interfaces for dealing with such arrays.

Specifically, an array specifies a nonempty, multi-dimensional interval, called its *domain*, and a mapping from this domain to Scheme objects. This mapping is called the *getter* of the array, accessed with the procedure `array-getter`

; the domain of the array (more precisely, the domain of the array's getter) is accessed with the procedure `array-domain`

.

If this mapping can be changed, the array is said to be *mutable* and the mutation is effected
by the array's *setter*, accessed by the procedure `array-setter`

. We call an object of this type a mutable array. Note: If an array does not have a setter, then we call it immutable even though the array's getter might not be a "pure" function, i.e., the value it returns may not depend solely on the arguments passed to the getter.

In general, we leave the implementation of generalized arrays completely open. They may be defined simply by closures, or they may have hash tables or databases behind an implementation, one may read the values from a file, etc.

In this SRFI, Bawden-style arrays are called *specialized*. A specialized array is an example of a mutable array.

Even if an array $A$ is not a specialized array, then it could be "shared" by specifying a new interval $D_B$ as the domain of a new array $B$ and an affine map $T_{BA}:D_B\to D_A$. Each call to $B$ would then be computed as $B(\vec i)=A(T_{BA}(\vec i))$.

One could again "share" $B$, given a new interval $D_C$ as the domain of a new array $C$ and an affine transform $T_{CB}:D_C\to D_B$, and then each access $C(\vec i)=A(T_{BA}(T_{CB}(\vec i)))$. The composition $T_{BA}\circ T_{CB}:D_C\to D_A$, being itself affine, could be precomputed and stored as $T_{CA}:D_C\to D_A$, and $C(\vec i)=A(T_{CA}(\vec i))$ can be computed with the overhead of computing a single affine transformation.

So, if we wanted, we could share generalized arrays with constant overhead by adding a single layer of (multi-valued) affine transformations on top of evaluating generalized arrays. Even though this could be done transparently to the user, we do not do that here; it would be a compatible extension of this SRFI to do so. We provide only the routine `specialized-array-share`

, not a more general `array-share`

.

Certain ways of sharing generalized arrays, however, are relatively easy to code and not that expensive. If we denote `(array-getter A)`

by `A-getter`

, then if B is the result of `array-extract`

applied to A, then `(array-getter B)`

is simply `A-getter`

. Similarly, if A is a two-dimensional array, and B is derived from A by applying the permutation $\pi((i,j))=(j,i)$, then `(array-getter B)`

is `(lambda (i j) (A-getter j i))`

. Translation and currying also lead to transformed arrays whose getters are relatively efficiently derived from `A-getter`

, at least for arrays of small dimension.

Thus, while we do not provide for sharing of generalized arrays for general one-to-one affine maps $T$, we do allow it for the specific functions `array-extract`

, `array-translate`

, `array-permute`

, `array-curry`

, `array-reverse`

, and `array-sample`

, and we provide relatively efficient implementations of these functions for arrays of dimension no greater than four.

Daniel Friedman and David Wise wrote a famous paper CONS should not Evaluate its Arguments. In the spirit of that paper, our procedure `array-map`

does not immediately produce a specialized array, but a simple immutable array, whose elements are recomputed from the arguments of `array-map`

each time they are accessed. This immutable array can be passed on to further applications of `array-map`

for further processing, without generating the storage bodies for intermediate arrays.

We provide the procedure `array->specialized-array`

to transform a generalized array (like that returned by `array-map`

) to a specialized, Bawden-style array, for which accessing each element again takes $O(1)$ operations.

**Relationship to nonstrict arrays in Racket.**It appears that what we call simply arrays in this SRFI are called nonstrict arrays in the math/array library of Racket, which in turn was influenced by an array proposal for Haskell. Our "specialized" arrays are related to Racket's "strict" arrays.**Indexers.**The argument new-domain->old-domain to`specialized-array-share`

is, conceptually, a multi-valued array.**Source of function names.**The function`array-curry`

gets its name from the curry operator in programming---we are currying the getter of the array and keeping careful track of the domains.`interval-projections`

can be thought of as currying the characteristic function of the interval, encapsulated here as`interval-contains-multi-index?`

.**Choice of functions on intervals.**The choice of functions for both arrays and intervals was motivated almost solely by what I needed for arrays. There are natural operations on intervals, like

(the inverse of`(interval-cross-product interval1 interval2 ...)`

`interval-projections`

), which don't seem terribly natural for arrays.**No empty intervals.**This SRFI considers arrays over only nonempty intervals of positive dimension. The author of this proposal acknowledges that other languages and array systems allow either zero-dimensional intervals or empty intervals of positive dimension, but prefers to leave such empty intervals as possibly compatible extensions to the current proposal.**Multi-valued arrays.**While this SRFI restricts attention to single-valued arrays, wherein the getter of each array returns a single value, allowing multi-valued immutable arrays would a compatible extension of this SRFI.**No low-level specialized-array constructor.**While the author of the SRFI uses mainly`(make-array ...)`

,`array-map`

, and`array->specialized-array`

to construct arrays, and while there are several other ways to construct arrays, there is no really low-level interface given for constructing specialized arrays (where one specifies a body, an indexer, etc.). It was felt that certain difficulties, some surmountable (such as checking that a given body is compatible with a given storage class) and some not (such as checking that an indexer is indeed affine), made a low-level interface less useful. At the same time, the simple`(make-array ...)`

mechanism is so general, allowing one to specify getters and setters as general functions, as to cover nearly all needs.

- Miscellaneous Functions
- translation?, permutation?.
- Intervals
- make-interval, interval?, interval-dimension, interval-lower-bound, interval-upper-bound, interval-lower-bounds->list, interval-upper-bounds->list, interval-lower-bounds->vector, interval-upper-bounds->vector, interval=, interval-volume, interval-subset?, interval-contains-multi-index?, interval-projections, interval-for-each, interval-dilate, interval-intersect, interval-translate, interval-permute, interval-scale.
- Storage Classes
- make-storage-class, storage-class?, storage-class-getter, storage-class-setter, storage-class-checker, storage-class-maker, storage-class-length, storage-class-default, generic-storage-class, s8-storage-class, s16-storage-class, s32-storage-class, s64-storage-class, u1-storage-class, u8-storage-class, u16-storage-class, u32-storage-class, u64-storage-class, f32-storage-class, f64-storage-class, c64-storage-class, c128-storage-class.
- Arrays
- make-array, array?, array-domain, array-getter, array-dimension, mutable-array?, array-setter, specialized-array-default-safe?, make-specialized-array, specialized-array?, array-storage-class, array-indexer, array-body, array-safe?, specialized-array-share, array->specialized-array, array-curry, array-extract, array-translate, array-permute, array-reverse, array-sample, array-map, array-for-each, array-fold, array-fold-right, array-any, array-every, array->list, list->specialized-array.

This document refers to *translations* and *permutations*.
A translation is a vector of exact integers. A permutation of dimension $n$
is a vector whose entries are the exact integers $0,1,\ldots,n-1$, each occurring once, in any order.

**Procedure: **`translation? `

`object`

Returns `#t`

if

is a translation, and `object``#f`

otherwise.

**Procedure: **`permutation? `

`object`

Returns `#t`

if

is a permutation, and `object``#f`

otherwise.

An interval represents the set of all multi-indices of exact integers
$i_0,\ldots,i_{d-1}$
satisfying
$l_0\leq i_0<u_0,\ldots,l_{d-1}\leq i_{d-1}<u_{d-1}$,
where the *lower bounds*
$l_0,\ldots,l_{d-1}$
and the *upper bounds*
$u_0,\ldots,u_{d-1}$
are specified multi-indices of exact integers. The positive integer $d$ is the *dimension*
of the interval. It is required that
$l_0<u_0,\ldots,l_{d-1}<u_{d-1}$.

Intervals are a data type distinct from other Scheme data types.

**Procedure: **`make-interval `

`lower-bounds` `upper-bounds`

Create a new interval;

and `lower-bounds`

are nonempty vectors (of the same length) of exact integers that satisfy`upper-bounds`

` (< (vector-ref ``lower-bounds` i) (vector-ref `upper-bounds` i))

for
$0\leq i<{}$`(vector-length `

. It is an error if
`lower-bounds`)

and `lower-bounds`

do not satisfy these conditions.`upper-bounds`

**Procedure: **`interval? `

`obj`

Returns `#t`

if

is an interval, and `obj``#f`

otherwise.

**Procedure: **`interval-dimension `

`interval`

If

is an interval built with `interval`

`(make-interval ``lower-bounds` `upper-bounds`)

then `interval-dimension`

returns `(vector-length `

. It is an error to call `lower-bounds`)`interval-dimension`

if

is not an interval.`interval`

**Procedure: **`interval-lower-bound `

`interval` `i`

**Procedure: **`interval-upper-bound `

`interval` `i`

If

is an interval built with `interval`

`(make-interval ``lower-bounds` `upper-bounds`)

and

is an exact integer that satisfies`i`

`$0 \leq i<$ ``(vector-length ``lower-bounds`)

,

then `interval-lower-bound`

returns
`(vector-ref `

and `lower-bounds` `i`)`interval-upper-bound`

returns
`(vector-ref `

. It is an error to call `upper-bounds` `i`)`interval-lower-bound`

or `interval-upper-bound`

if

and `interval`

do not satisfy these conditions.`i`

**Procedure: **`interval-lower-bounds->list `

`interval`

**Procedure: **`interval-upper-bounds->list `

`interval`

If

is an interval built with `interval`

`(make-interval ``lower-bounds` `upper-bounds`)

then `interval-lower-bounds->list`

returns `(vector->list `

and `lower-bounds`)`interval-upper-bounds->list`

returns `(vector->list `

. It is an error to call
`upper-bounds`)`interval-lower-bounds->list`

or `interval-upper-bounds->list`

if

does not satisfy these conditions.`interval`

**Procedure: **`interval-lower-bounds->vector `

`interval`

**Procedure: **`interval-upper-bounds->vector `

`interval`

If

is an interval built with `interval`

`(make-interval ``lower-bounds` `upper-bounds`)

then `interval-lower-bounds->vector`

returns a copy of

and `lower-bounds``interval-upper-bounds->vector`

returns a copy of

. It is an error to call
`upper-bounds``interval-lower-bounds->vector`

or `interval-upper-bounds->vector`

if

does not satisfy these conditions.`interval`

**Procedure: **`interval-volume `

`interval`

If

is an interval built with `interval`

`(make-interval ``lower-bounds` `upper-bounds`)

then, assuming the existence of `vector-map`

, `interval-volume`

returns

`(apply * (vector->list (vector-map - ``upper-bounds` `lower-bounds`)))

It is an error to call `interval-volume`

if

does not satisfy this condition.`interval`

**Procedure: **`interval= `

`interval1` `interval2`

If

and `interval1`

are intervals built with `interval2`

`(make-interval ``lower-bounds1` `upper-bounds1`)

and

`(make-interval ``lower-bounds2` `upper-bounds2`)

respectively, then `interval=`

returns

`(and (equal? ``lower-bounds1` `lower-bounds2`) (equal? `upper-bounds1` `upper-bounds2`))

It is an error to call `interval=`

if

or `interval1`

do not satisfy this condition.`interval2`

**Procedure: **`interval-subset? `

`interval1` `interval2`

If

and `interval1`

are intervals of the same dimension $d$, then `interval2``interval-subset?`

returns `#t`

if

`(<= (interval-lower-bound ``interval1` j) (interval-lower-bound `interval2` j))

and

`(<= (interval-upper-bound ``interval1` j) (interval-upper-bound `interval2` j))

for all $0\leq j<d$, otherwise it returns `#f`

. It is an error if the arguments do not satisfy these conditions.

**Procedure: **`interval-contains-multi-index? `

`interval` `index-0` `index-1` `...`

If

is an interval with dimension $d$ and `interval`

, `index-0`

, ..., is a multi-index of length $d$,
then `index-1``interval-contains-multi-index?`

returns `#t`

if

`(interval-lower-bound`

$\leq$intervalj)`$<$`

index-j`(interval-upper-bound`

intervalj)

for $0\leq j < d$, and `#f`

otherwise.

It is an error to call `interval-contains-multi-index?`

if

and `interval`

,..., do not satisfy this condition.`index-0`

**Procedure: **`interval-projections `

`interval` `right-dimension`

Conceptually, `interval-projections`

takes a $d$-dimensional interval
$[l_0,u_0)\times [l_1,u_1)\times\cdots\times[l_{d-1},u_{d-1})$
and splits it into two parts

$[l_0,u_0)\times\cdots\times[l_{d-\text{right-dimension}-1},u_{d-\text{right-dimension}-1})$

and

$[l_{d-\text{right-dimension}},u_{d-\text{right-dimension}})\times\cdots\times[l_{d-1},u_{d-1})$

This function, the inverse of Cartesian products or cross products of intervals, is used to keep track of the domains of curried arrays.

More precisely, if

is an interval and `interval`

is an exact integer that satisfies `right-dimension``0 < `

then `right-dimension` < `d``interval-projections`

returns two intervals:

```
(values
(make-interval
(vector (interval-lower-bound
````interval` 0)
...
(interval-lower-bound `interval`
(- `d` `right-dimension` 1)))
(vector (interval-upper-bound `interval` 0)
...
(interval-upper-bound `interval`
(- `d` `right-dimension` 1))))
(make-interval
(vector (interval-lower-bound `interval`
(- `d` `right-dimension`))
...
(interval-lower-bound `interval`
(- `d` 1)))
(vector (interval-upper-bound `interval`
(- `d` `right-dimension`))
...
(interval-upper-bound `interval`
(- `d` 1)))))

It is an error to call `interval-projections`

if its arguments do not satisfy these conditions.

**Procedure: **`interval-for-each `

`f` `interval`

This routine assumes that

is an interval and `interval`

is a routine whose domain includes elements of `f`

. It is an error to call
`interval``interval-for-each`

if

and `interval`

do not satisfy these conditions.`f`

`interval-for-each`

calls

with each multi-index of `f`

as arguments, all in lexicographical order.`interval`

**Procedure: **`interval-dilate `

`interval` `lower-diffs` `upper-diffs`

If

is an interval with
lower bounds l`interval`_{0}, ..., l_{d-1} and
upper bounds u_{0}, ..., u_{d-1}, and

is a vector of exact integers L`lower-diffs`_{0}, ..., L_{d-1} and

is a vector of exact integers U`upper-diffs`_{0}, ..., U_{d-1}, then `interval-dilate`

returns a new interval with
lower bounds l_{0}+L_{0}, ..., l_{d-1}+L_{d-1} and
upper bounds u_{0}+U_{0}, ..., u_{d-1}+U_{d-1}, as long as this is a
nonempty interval. It is an error if the arguments do not satisfy these conditions.

Examples:

```
(interval=
(interval-dilate (make-interval '#(0 0) '#(100 100))
'#(1 1) '#(1 1))
(make-interval '#(1 1) '#(101 101))) => #t
(interval=
(interval-dilate (make-interval '#(0 0) '#(100 100))
'#(-1 -1) '#(1 1))
(make-interval '#(-1 -1) '#(101 101))) => #t
(interval=
(interval-dilate (make-interval '#(0 0) '#(100 100))
'#(0 0) '#(-50 -50))
(make-interval '#(0 0) '#(50 50))) => #t
(interval-dilate
(make-interval '#(0 0) '#(100 100))
'#(0 0) '#(-500 -50)) => error
```

**Procedure: **`interval-intersect `

`interval-1` `interval-2` `...`

If all the arguments are intervals of the same dimension and they have a nonempty intersection,
the `interval-intersect`

returns that intersection; otherwise it returns `#f`

.

It is an error if the arguments are not all intervals with the same dimension.

**Procedure: **`interval-translate `

`interval` `translation`

If

is an interval with
lower bounds l`interval`_{0}, ..., l_{d-1} and
upper bounds u_{0}, ..., u_{d-1}, and

is a translation with entries T`translation`_{0}, ..., T_{d-1}, then `interval-translate`

returns a new interval with
lower bounds l_{0}+T_{0}, ..., l_{d-1}+T_{d-1} and
upper bounds u_{0}+T_{0}, ..., u_{d-1}+T_{d-1}.
It is an error if the arguments do not satisfy these conditions.

One could define `(interval-translate interval translation)`

by `(interval-dilate interval translation translation)`

.

**Procedure: **`interval-permute `

`interval` `permutation`

The argument

must be an interval, and the argument `interval`

must be a valid permutation with the same dimension as `permutation`

. It is an error if the arguments do not satisfy these conditions.`interval`

Heuristically, this function returns a new interval whose axes have been permuted in a way consistent with

.
But we have to say how the entries of `permutation`

are associated with the new interval.`permutation`

We have chosen the following convention: If the permutation is $(\pi_0,\ldots,\pi_{d-1})$, and the argument interval represents the cross product $[l_0,u_0)\times[l_1,u_1)\times\cdots\times[l_{d-1},u_{d-1})$, then the result represents the cross product $[l_{\pi_0},u_{\pi_0})\times[l_{\pi_1},u_{\pi_1})\times\cdots\times[l_{\pi_{d-1}},u_{\pi_{d-1}})$.

For example, if the argument interval represents $[0,4)\times[0,8)\times[0,21)\times [0,16)$ and the
permutation is `#(3 0 1 2)`

, then the result of `(interval-permute `

will be
the representation of $[0,16)\times [0,4)\times[0,8)\times[0,21)$.`interval` `permutation`)

**Procedure: **`interval-scale `

`interval` `scales`

If

is a $d$-dimensional interval $[0,u_1)\times\cdots\times[0,u_{d-1})$ with all lower bounds zero, and `interval`

is a length-$d$ vector of positive exact integers, which we'll denote by $\vec s$, then `scales``interval-scale`

returns the interval $[0,\operatorname{ceiling}(u_1/s_1))\times\cdots\times[0,\operatorname{ceiling}(u_{d-1}/s_{d-1})$.

It is an error if

and `interval`

do not satisfy this condition.`scales`

Conceptually, a storage-class is a set of functions to manage the backing store of a specialized-array. The functions allow one to make a backing store, to get values from the store and to set new values, to return the length of the store, and to specify a default value for initial elements of the backing store. Typically, a backing store is a (heterogeneous or homogeneous) vector. A storage-class has a type distinct from other Scheme types.

**Procedure: **`make-storage-class `

`getter` `setter` `checker` `maker` `length` `default`

Here we assume the following relationships between the arguments of `make-storage-class`

. Assume that the "elements" of
the backing store are of some "type", either heterogeneous (all Scheme types) or homogeneous (of some restricted type).

`(`

returns an object containing`maker n``value`)

elements of value`n`

.`value`- If

is an object created by`v``(`

and 0 <=`maker n value`)

<`i`

, then`n``(`

returns the current value of the`getter v i`)

'th element of`i`

, and`v``(`

.`checker`(`getter v i`)) => #t - If

is an object created by`v``(`

, 0 <=`maker n value`)

<`i`

, and`n``(`

, then`checker``val`) => #t`(`

sets the value of the`setter v i val`)

'th element of`i`

to`v`

.`val` - If

is an object created by`v``(`

then`maker n value`)`(`

returns`length v`)

.`n`

If the arguments do not satisfy these conditions, then it is an error to call `make-storage-class`

.

Note that we assume that

and `getter`

generally take `setter`*O*(1) time to execute.

**Procedure: **`storage-class? `

`m`

Returns `#t`

if

is a storage class, and `m``#f`

otherwise.

**Procedure: **`storage-class-getter `

`m`

**Procedure: **`storage-class-setter `

`m`

**Procedure: **`storage-class-checker `

`m`

**Procedure: **`storage-class-maker `

`m`

**Procedure: **`storage-class-length `

`m`

**Procedure: **`storage-class-default `

`m`

If

is an object created by`m`

`(make-storage-class ``getter setter checker maker length default`)

then `storage-class-getter`

returns

, `getter``storage-class-setter`

returns

, `setter``storage-class-checker`

returns

, `checker``storage-class-maker`

returns

, and `maker``storage-class-length`

returns

, and `length``storage-class-default`

returns

. Otherwise, it is an error to call any of these routines.`default`

**Variable: **`generic-storage-class`

**Variable: **`s8-storage-class`

**Variable: **`s16-storage-class`

**Variable: **`s32-storage-class`

**Variable: **`s64-storage-class`

**Variable: **`u1-storage-class`

**Variable: **`u8-storage-class`

**Variable: **`u16-storage-class`

**Variable: **`u32-storage-class`

**Variable: **`u64-storage-class`

**Variable: **`f32-storage-class`

**Variable: **`f64-storage-class`

**Variable: **`c64-storage-class`

**Variable: **`c128-storage-class`

`generic-storage-class`

is defined as if by

```
(define generic-storage-class
(make-storage-class vector-ref
vector-set!
(lambda (arg) #t)
make-vector
vector-length
#f))
```

Furthermore, `s``X`-storage-class

is defined for `X`

=8, 16, 32, and 64 (which have default values 0 and
manipulate exact integer values between -2`u``X`-storage-class

is defined for `X`

=1, 8, 16, 32, and 64 (which have default values 0 and manipulate exact integer values between 0 and
2`f``X`-storage-class

is defined for `X`

= 32 and 64 (which have default value 0.0 and manipulate 32- and 64-bit floating-point numbers), and
`c``X`-storage-class

is defined for `X`

= 64 and 128 (which have default value 0.0+0.0i and manipulate complex numbers with, respectively, 32- and 64-bit floating-point numbers as real and imaginary parts). Each of these
could be defined simply as `generic-storage-class`

, but it is assumed that implementations with homogeneous vectors will give definitions
that either save space, avoid boxing, etc., for the specialized arrays.
Arrays are a data type distinct from other Scheme data types.

**Procedure: **`make-array `

`interval` `getter` [ `setter` ]

Assume first that the optional argument `setter`

is not given.

If

is an interval and `interval`

is a function from
`getter`

to Scheme objects, then `interval``make-array`

returns an array with domain

and getter `interval`

.`getter`

It is an error to call `make-array`

if

and `interval`

do not satisfy these conditions.`getter`

If now

is specified, assume that it is a procedure such that getter and setter satisfy: If`setter`

`(`

$\neq$i_{1},...,i_{n})`(`

j_{1},...,j_{n})

are elements of

and `interval`

`(getter ``j`_{1} ... `j`_{n}) => x

then "after"

`(setter v ``i`_{1} ... `i`_{n})

we have

`(getter ``j`_{1} ... `j`_{n}) => x

and

`(getter ``i`_{1},...,`i`_{n}) => v

Then `make-array`

builds a mutable array with domain

, getter `interval`

, and
setter `getter`

. It is an error to call `setter``make-array`

if its arguments do not satisfy these conditions.

Example:

```
(define a (make-array (make-interval '#(1 1) '#(11 11))
(lambda (i j)
(if (= i j)
1
0))))
```

defines an array for which `(array-getter a)`

returns 1 when i=j and 0 otherwise.

Example:

```
(define sparse-array
(let ((domain
(make-interval '#(0 0)
'#(1000000 1000000)))
(sparse-rows
(make-vector 1000000 '())))
(make-array
domain
(lambda (i j)
(cond ((assv j (vector-ref sparse-rows i))
=> cdr)
(else
0.0)))
(lambda (v i j)
(cond ((assv j (vector-ref sparse-rows i))
=> (lambda (pair)
(set-cdr! pair v)))
(else
(vector-set!
sparse-rows
i
(cons (cons j v)
(vector-ref sparse-rows i)))))))))
((array-getter sparse-array) 12345 6789) => 0.
((array-getter sparse-array) 0 0) => 0.
((array-setter sparse-array) 1.0 0 0) => undefined
((array-getter sparse-array) 12345 6789) => 0.
((array-getter sparse-array) 0 0) => 1.
```

**Procedure: **`array? `

`obj`

Returns `#t`

if

is an array and `obj``#f`

otherwise.

**Procedure: **`array-domain `

`array`

**Procedure: **`array-getter `

`array`

If

is an array built by`array`

`(make-array ``interval` `getter` [`setter`])

(with or without the optional

argument) then `setter``array-domain`

returns

and `interval``array-getter`

returns

.
It is an error to call `getter``array-domain`

or `array-getter`

if

is not an array.`array`

Example:

```
(define a (make-array (make-interval '#(1 1) '#(11 11))
(lambda (i j)
(if (= i j)
1
0))))
((array-getter a) 3 3) => 1
((array-getter a) 2 3) => 0
((array-getter a) 11 0) => is an error
```

**Procedure: **`array-dimension `

`array`

Shorthand for `(interval-dimension (array-domain `

. It is an error to call this function if `array`))

is not an array.`array`

**Procedure: **`mutable-array? `

`obj`

Returns `#t`

if

is a mutable array and `obj``#f`

otherwise.

**Procedure: **`array-setter `

`array`

If

is an array built by`array`

`(make-array ``interval` `getter` `setter`)

then `array-setter`

returns

. It is an error to call `setter``array-setter`

if

is not a mutable array.`array`

**Procedure: **`specialized-array-default-safe? [ `

`bool` ]

With no argument, returns `#t`

if newly-constructed specialized arrays check the arguments of setters and getters by default, and `#f`

otherwise.

If

is `bool``#t`

then the next call to `specialized-array-default-safe?`

will return `#t`

;
if

is `bool``#f`

then the next call to `specialized-array-default-safe?`

will return `#f`

;
otherwise it is an error.

**Procedure: **`make-specialized-array `

`interval` [ `storage-class` generic-storage-class ] [ `safe?` (specialized-array-default-safe?) ]

Constructs a specialized-array from its arguments.

must be given as a nonempty interval. If given, `interval`

must be a storage class; if it is not given it defaults to `storage-class``generic-storage-class`

. If given,

must be a boolean; if it is not given it defaults to the current value of `safe?``(specialized-array-default-safe?)`

.

The body of the result is constructed as

```
((storage-class-maker
````storage-class`)
(interval-volume `interval`)
(storage-class-default `storage-class`))

The indexer of the resulting array is constructed as the lexicographical mapping of

onto the interval `interval``[0,(interval-volume `

.`interval`))

If

is `safe``#t`

, then the arguments of the getter and setter (including the value to be stored) of the resulting array are always checked for correctness.

After correctness checking (if needed), `(array-getter `

is defined simply as `array`)

```
(lambda multi-index
((storage-class-getter
````storage-class`)
(array-body `array`)
(apply (array-indexer `array`) multi-index)))

and `(array-setter `

is defined as `array`)

```
(lambda (val . multi-index)
((storage-class-setter
````storage-class`)
(array-body `array`)
(apply (array-indexer `array`) multi-index)
val))

It is an error if the arguments of `make-specialized-array`

do not satisfy these conditions.

**Examples. **A simple array that can hold any type of element can be defined with `(make-specialized-array (make-interval '#(0 0) '#(3 3)))`

. If you find that you're using a lot of unsafe arrays of unsigned 16-bit integers, one could define

```
(define (make-u16-array interval)
(make-specialized-array interval u16-storage-class #f))
```

and then simply call, e.g., `(make-u16-array (make-interval '#(0 0) '#(3 3)))`

.

**Procedure: **`specialized-array? `

`obj`

Returns `#t`

if

is a specialized-array, and `obj``#f`

otherwise. A specialized-array is an array.

**Procedure: **`array-storage-class `

`array`

**Procedure: **`array-indexer `

`array`

**Procedure: **`array-body `

`array`

**Procedure: **`array-safe? `

`array`

`array-storage-class`

returns the storage-class of

. `array``array-safe?`

is true if and only if the arguments of `(array-getter `

and `array`)`(array-setter `

(including the value to be stored in the array) are checked for correctness.`array`)

`(array-indexer `

is assumed to be a one-to-one, but not necessarily onto, affine mapping from `array`)`(array-domain `

into `array`)`(array-body `

.`array`)

It is an error to call any of these routines if

is not a specialized-array.`array`

**Procedure: **`specialized-array-share `

`array` `new-domain` `new-domain->old-domain`

Constructs a new specialized-array that shares the body of the specialized-array

.
Returns an object that is behaviorally equivalent to a specialized array with the following fields:`array`

```
domain: new-domain
storage-class: (array-storage-class
````array`)
body: (array-body `array`)
indexer: (lambda multi-index
(call-with-values
(lambda ()
(apply `new-domain->old-domain`
multi-index))
(array-indexer `array`)))

must be an affine one-to-one mapping from `new-domain->old-domain`

to
`new-domain``(array-domain `

.`array`)

Note: It is assumed that affine structure of the composition of

and `new-domain->old-domain``(array-indexer `

will be used to simplify:`array`

```
(lambda multi-index
(call-with-values
(lambda ()
(apply
````new-domain->old-domain` multi-index))
(array-indexer `array`)))

It is an error if

is not a specialized array, or if `array`

is not an interval, or if `new-domain`

is not a one-to-one affine mapping with the appropriate domain and range.`new-domain->old-domain`

**Example: **One can apply a "shearing" operation to an array as follows:

```
(define a
(array->specialized-array
(make-array (make-interval '#(0 0) '#(5 10))
list)))
(define b
(specialized-array-share
a
(make-interval '#(0 0) '#(5 5))
(lambda (i j)
(values i (+ i j)))))
;; Print the "rows" of b
(array-for-each (lambda (row)
(pretty-print (array->list row)))
(array-curry b 1))
;; which prints
;; ((0 0) (0 1) (0 2) (0 3) (0 4))
;; ((1 1) (1 2) (1 3) (1 4) (1 5))
;; ((2 2) (2 3) (2 4) (2 5) (2 6))
;; ((3 3) (3 4) (3 5) (3 6) (3 7))
;; ((4 4) (4 5) (4 6) (4 7) (4 8))
```

This "shearing" operation cannot be achieved by combining the procedures `array-extract`

, `array-translate`

, `array-permute`

, `array-translate`

, `array-curry`

, `array-reverse`

, and `array-sample`

.

**Procedure: **`array->specialized-array `

`array` [ `result-storage-class` generic-storage-class ] [ `safe?` (specialized-array-default-safe?) ]

If

is an array whose elements can be manipulated by the storage-class
`array`

, then the specialized-array returned by `result-storage-class``array->specialized-array`

can be defined by:

```
(let* ((domain
(array-domain
````array`))
(getter
(array-getter `array`))
(result
(make-specialized-array domain
`result-storage-class`
`safe?`))
(result-setter
(array-setter result)))
(interval-for-each (lambda multi-index
(apply result-setter
(apply getter
multi-index)
multi-index))
domain)
result)

It is guaranteed that `(array-getter `

is called precisely once for each multi-index in `array`)`(array-domain `

in lexicographical order.`array`)

It is an error if

does not satisfy these conditions, or if `result-storage-class`

is not a boolean.`safe?`

**Procedure: **`array-curry `

`array` `inner-dimension`

If

is an array whose domain is an interval $[l_0,u_0)\times\cdots\times[l_{d-1},u_{d-1})$, and `array`

is an exact integer strictly between $0$ and $d$, then `inner-dimension``array-curry`

returns an immutable array with domain $[l_0,u_0)\times\cdots\times[l_{d-\text{inner-dimension}-1},u_{d-\text{inner-dimension}-1})$, each of whose entries is in itself an array with domain $[l_{d-\text{inner-dimension}},u_{d-\text{inner-dimension}})\times\cdots\times[l_{d-1},u_{d-1})$.

For example, if `A`

and `B`

are defined by

```
(define interval (make-interval '#(0 0 0 0)
'#(10 10 10 10)))
(define A (make-array interval list))
(define B (array-curry A 1))
```

so

```
((array-getter A) i j k l) => (list i j k l)
```

then `B`

is an immutable array with domain `(make-interval '#(0 0 0) '#(10 10 10))`

, each
of whose elements is itself an (immutable) array and

```
(equal?
((array-getter A) i j k l)
((array-getter ((array-getter B) i j k)) l)) => #t
```

for all multi-indices `i j k l`

in `interval`

.

The subarrays are immutable, mutable, or specialized according to whether the array argument is immutable, mutable, or specialized.

More precisely, if

`0 < ``inner-dimension` < (interval-dimension (array-domain `array`))

then `array-curry`

returns a result as follows.

If the input array is specialized, then array-curry returns

```
(call-with-values
(lambda () (interval-projections (array-domain
````array`)
`inner-dimension`))
(lambda (outer-interval inner-interval)
(make-array
outer-interval
(lambda outer-multi-index
(specialized-array-share
`array`
inner-interval
(lambda inner-multi-index
(apply values
(append outer-multi-index
inner-multi-index))))))))

Otherwise, if the input array is mutable, then array-curry returns

```
(call-with-values
(lambda () (interval-projections (array-domain
````array`)
`inner-dimension`))
(lambda (outer-interval inner-interval)
(make-array
outer-interval
(lambda outer-multi-index
(make-array
inner-interval
(lambda inner-multi-index
(apply (array-getter `array`)
(append outer-multi-index
inner-multi-index)))
(lambda (v . inner-multi-index)
(apply (array-setter `array`)
v
(append outer-multi-index
inner-multi-index))))))))

Otherwise, array-curry returns

```
(call-with-values
(lambda () (interval-projections (array-domain
````array`)
`inner-dimension`))
(lambda (outer-interval inner-interval)
(make-array
outer-interval
(lambda outer-multi-index
(make-array
inner-interval
(lambda inner-multi-index
(apply (array-getter `array`)
(append outer-multi-index
inner-multi-index))))))))

It is an error to call `array-curry`

if its arguments do not satisfy these conditions.

Example:

```
(define a (make-array (make-interval '#(0 0)
'#(10 10))
list))
((array-getter a) 3 4) => (3 4)
(define curried-a (array-curry a 1))
((array-getter ((array-getter curried-a) 3)) 4)
=> (3 4)
```

**Procedure: **`array-extract `

`array` `new-domain`

Returns a new array with the same getter (and setter, if appropriate) of the first argument, defined on the second argument.

Assumes that

is an array and `array`

is an interval that is a sub-interval of `new-domain``(array-domain `

. If `array`)

is a specialized array, then returns `array`

```
(specialized-array-share
````array`
`new-domain`
values)

Otherwise, if

is a mutable array, then `array``array-extract`

returns

```
(make-array
````new-domain`
(array-getter `array`)
(array-setter `array`))

Finally, if

is an immutable array, then `array``array-extract`

returns

```
(make-array
````new-domain`
(array-getter `array`))

It is an error if the arguments of `array-extract`

do not satisfy these conditions.

**Procedure: **`array-translate `

`array` `translation`

Assumes that

is a valid array, `array`

is a valid translation, and that the dimensions of the array and the translation are the same. The resulting array will have domain `translation``(interval-translate (array-domain array) translation)`

.

If

is a specialized array, returns a new specialized array`array`

```
(specialized-array-share
````array`
(interval-translate (array-domain `array`)
`translation`)
(lambda multi-index
(apply values
(map -
multi-index
(vector->list `translation`)))))

that shares the body of

.`array`

If

is not a specialized array but is a mutable array, returns a new mutable array`array`

```
(make-array
(interval-translate (array-domain
````array`)
`translation`)
(lambda multi-index
(apply (array-getter `array`)
(map -
multi-index
(vector->list `translation`))))
(lambda (val . multi-index)
(apply (array-setter `array`)
val
(map -
multi-index
(vector->list `translation`)))))

that employs the same getter and setter as the original array argument.

If

is not a mutable array, returns a new array`array`

```
(make-array
(interval-translate (array-domain
````array`)
`translation`)
(lambda multi-index
(apply (array-getter `array`)
(map - multi-index (vector->list `translation`)))))

that employs the same getter as the original array.

It is an error if the arguments do not satisfy these conditions.

**Procedure: **`array-permute `

`array` `permutation`

Assumes that

is a valid array, `array`

is a valid permutation, and that the dimensions of the array and the permutation are the same. The resulting array will have domain `permutation``(interval-permute (array-domain array) permutation)`

.

We begin with an example. Assume that the domain of

is represented by the interval $[0,4)\times[0,8)\times[0,21)\times [0,16)$, as in the example for `array``interval-permute`

, and the permutation is `#(3 0 1 2)`

. Then the domain of the new array is the interval $[0,16)\times [0,4)\times[0,8)\times[0,21)$.

So the multi-index argument of the `getter`

of the result of `array-permute`

must lie in the new domain of the array, the interval $[0,16)\times [0,4)\times[0,8)\times[0,21)$. So if we define

as `old-getter``(array-getter `

, the definition of the new array must be in fact`array`)

```
(make-array (interval-permute (array-domain
````array`)
'#(3 0 1 2))
(lambda (l i j k)
(old-getter i j k l)))

So you see that if the first argument if the new getter is in $[0,16)$, then indeed the fourth argument of

is also in $[0,16)$, as it should be. This is a subtlety that I don't see how to overcome. It is the listing of the arguments of the new getter, the `old-getter``lambda`

, that must be permuted.

Mathematically, we can define $\pi^{-1}$, the inverse of a permutation $\pi$, such that $\pi^{-1}$ composed with $\pi$ gives the identity permutation. Then the getter of the new array is, in pseudo-code, `(lambda multi-index (apply `

$\pi^{-1}$`old-getter` (` multi-index)))`

. We have assumed that $\pi^{-1}$ takes a list as an argument and returns a list as a result.

Employing this same pseudo-code, if

is a specialized-array and we denote the permutation by $\pi$, then `array``array-permute`

returns the new specialized array

```
(specialized-array-share
````array`
(interval-permute (array-domain `array`) π)
(lambda multi-index
(apply values (π^{-1} multi-index))))

The result array shares `(array-body `

with the argument.`array`)

Again employing this same pseudo-code, if

is not a specialized array, but is
a mutable-array, then `array``array-permute`

returns the new mutable

```
(make-array (interval-permute (array-domain
````array`) π)
(lambda multi-index
(apply (array-getter `array`)
(π^{-1} multi-index)))
(lambda (val . multi-index)
(apply (array-setter `array`)
val
(π^{-1} multi-index))))

which employs the setter and the getter of the argument to `array-permute`

.

Finally, if

is not a mutable array, then `array``array-permute`

returns

```
(make-array (interval-permute (array-domain
````array`) π)
(lambda multi-index
(apply (array-getter `array`)
(π^{-1} multi-index))))

It is an error to call `array-permute`

if its arguments do not satisfy these conditions.

**Procedure: **`array-reverse `

`array` `flip?`

We assume that

is an array and `array`

is a vector of booleans whose length is the same as the dimension of `flip?`

.`array`

`array-reverse`

returns a new array that is specialized, mutable, or immutable according to whether

is specialized, mutable, or immutable, respectively. Informally, if `array``(vector-ref `

is true, then the ordering of multi-indices in the k'th coordinate direction is reversed, and is left undisturbed otherwise.`flip?` k)

More formally, we introduce the function

```
(define flip-multi-index
(let* ((domain (array-domain
````array`

))
(lowers (interval-lower-bounds->list domain))
(uppers (interval-upper-bounds->list domain)))
(lambda (multi-index)
(map (lambda (i_k flip?_k l_k u_k)
(if flip?
(- (+ l_k u_k -1) i_k)
i_k))
multi-index
(vector->list `flip?`)
lowers
uppers))))

Then if

is specialized, then `array``array-reverse`

returns

```
(specialized-array-share
````array`

domain
(lambda multi-index
(apply values
(flip-multi-index multi-index))))

Otherwise, if

is mutable, then `array``array-reverse`

returns

```
(make-array
domain
(lambda multi-index
(apply (array-getter
````array`

)
(flip-multi-index multi-index)))
(lambda (v . multi-index)
(apply (array-setter `array`

)
v
(flip-multi-index multi-index)))))

Finally, if

is immutable, then `array``array-reverse`

returns

```
(make-array
domain
(lambda multi-index
(apply (array-getter
````array`

)
(flip-multi-index multi-index)))))

It is an error if

and `array`

don't satisfy these requirements.`flip?`

**Procedure: **`array-sample `

`array` `scales`

We assume that

is an array all of whose lower bounds are zero, and `array`

is a vector of positive exact integers whose length is the same as the dimension of `scales`

.`array`

`array-sample`

returns a new array that is specialized, mutable, or immutable according to whether

is specialized, mutable, or immutable, respectively. Informally, if we construct a new matrix $S$ with the entries of `array`

on the main diagonal, then the $\vec i$th element of `scales``(array-sample `

is the $S\vec i$th element of `array` `scales`)

.`array`

More formally, if

is specialized, then `array``array-sample`

returns

```
(specialized-array-share
````array`

(interval-scale (array-domain `array`

)
`scales`

)
(lambda multi-index
(apply values
(map * multi-index (vector->list `scales`

)))))

Otherwise, if

is mutable, then `array``array-sample`

returns

```
(make-array
(interval-scale (array-domain
````array`

)
`scales`

)
(lambda multi-index
(apply (array-getter `array`

)
(map * multi-index (vector->list `scales`

))))
(lambda (v . multi-index)
(apply (array-setter `array`

)
v
(map * multi-index (vector->list `scales`

)))))

Finally, if

is immutable, then `array``array-sample`

returns

```
(make-array
(interval-scale (array-domain
````array`

)
`scales`

)
(lambda multi-index
(apply (array-getter `array`

)
(map * multi-index (vector->list `scales`

)))))

It is an error if

and `array`

don't satisfy these requirements.`scales`

**Procedure: **`array-map `

`f` `array` . `arrays`

If

, `array``(car `

, ... all have the same domain and `arrays`)

is a procedure, then `f``array-map`

returns a new array with the same domain and getter

```
(lambda multi-index
(apply
````f`
(map (lambda (g)
(apply g multi-index))
(map array-getter
(cons `array` `arrays`)))))

It is assumed that

is appropriately defined to be evaluated in this context.`f`

It is an error to call `array-map`

if its arguments do not satisfy these conditions.

**Procedure: **`array-for-each `

`f` `array` . `arrays`

If

, `array``(car `

, ... all have the same domain and `arrays`)

is an appropriate procedure, then `f``array-for-each`

calls

```
(interval-for-each
(lambda multi-index
(apply
````f`
(map (lambda (g)
(apply g multi-index))
(map array-getter
(cons `array`
`arrays`)))))
(array-domain `array`))

In particular, `array-for-each`

always walks the indices of the arrays in lexicographical order.

It is expected that `array-map`

and `array-for-each`

will specialize the construction of

```
(lambda multi-index
(apply
````f`
(map (lambda (g)
(apply g multi-index))
(map array-getter
(cons `array`
`arrays`)))))

It is an error to call `array-for-each`

if its arguments do not satisfy these conditions.

**Procedure: **`array-fold `

`kons` `knil` `array`

If we use the defining relations for fold over lists from SRFI-1:

```
(fold kons knil lis)
= (fold kons (kons (car lis) knil) (cdr lis))
(fold kons knil '())
= knil
```

then `(array-fold `

returns `kons` `knil` `array`)

```
(fold
````kons` `knil` (array->list `array`))

It is an error if

is not an array, or if `array`

is not a procedure.`kons`

**Procedure: **`array-fold-right `

`kons` `knil` `array`

If we use the defining relations for fold-right over lists from SRFI-1:

```
(fold-right kons knil lis)
= (kons (car lis) (fold-right kons knil (cdr lis)))
(fold-right kons knil '())
= knil
```

then `(array-fold-right `

returns `kons` `knil` `array`)

```
(fold-right
````kons` `knil` (array->list `array`))

It is an error if

is not an array, or if `array`

is not a procedure.`kons`

**Procedure: **`array-any `

`pred` `array1` `array2` ...

Assumes that

, `array1`

, etc., are arrays, all with the same domain, which we'll call `array2``interval`

. Also assumes that

is a procedure that takes as many arguments as there are arrays and returns a single value.`pred`

`array-any`

first applies `(array-getter `

, etc., to the first element of `array1`)`interval`

in lexicographical order, to which values it then applies

.`pred`

If the result of

is not `pred``#f`

, then that result is returned by `array-any`

. If the result of

is `pred``#f`

, then `array-any`

continues with the second element of `interval`

, etc., returning the first nonfalse value of

.`pred`

If

always returns `pred``#f`

, then `array-any`

returns `#f`

.

If it happens that

is applied to the results of applying `pred``(array-getter `

, etc., to the last element of `array1`)`interval`

, then this last call to

is in tail position.`pred`

The functions `(array-getter `

, etc., are applied only to those values of `array1`)`interval`

necessary to determine the result of `array-any`

.

It is an error if the arguments do not satisfy these assumptions.

**Procedure: **`array-every `

`pred` `array1` `array2` ...

Assumes that

, `array1`

, etc., are arrays, all with the same domain, which we'll call `array2``interval`

. Also assumes that

is a procedure that takes as many arguments as there are arrays and returns a single value.`pred`

`array-every`

first applies `(array-getter `

, etc., to the first element of `array1`)`interval`

in lexicographical order, to which values it then applies

.`pred`

If the result of

is `pred``#f`

, then that result is returned by `array-every`

. If the result of

is nonfalse, then `pred``array-every`

continues with the second element of `interval`

, etc., returning the first value of

that is `pred``#f`

.

If

always returns a nonfalse value, then the last nonfalse value returned by `pred`

is also returned by `pred``array-every`

.

If it happens that

is applied to the results of applying `pred``(array-getter `

, etc., to the last element of `array1`)`interval`

, then this last call to

is in tail position.`pred`

The functions `(array-getter `

, etc., are applied only to those values of `array1`)`interval`

necessary to determine the result of `array-every`

.

It is an error if the arguments do not satisfy these assumptions.

**Procedure: **`array->list `

`array`

Stores the elements of

into a newly-allocated list in lexicographical order. It is an error if `array`

is not an array.`array`

**Procedure: **`list->specialized-array `

`l` `interval` [ `result-storage-class` generic-storage-class ] [ `safe?` (specialized-array-default-safe?) ]

Returns a specialized-array with domain

whose elements are the elements of the list `interval`

stored in lexicographical order. It is an error if `l`

is not a list, if `l`

is not an interval, if the length of `interval`

is not the same as the volume of `l`

, if `interval`

(when given) is not a storage class, if `result-storage-class`

(when given) is not a boolean, or if any element of `safe?`

cannot be stored in the body of `l`

, and this last error shall be detected and raised if `result-storage-class`

is `safe``#t`

.

We provide an implementation in Gambit-C; the nonstandard techniques used
in the implementation are: DSSSL-style optional and keyword arguments; a
unique object to indicate absent arguments; `define-structure`

;
and `define-macro`

.

Final SRFIs 25, 47, 58, and 63 deal with "Multi-dimensional Array Primitives", "Array", "Array Notation", and "Homogeneous and Heterogeneous Arrays", respectively. Each of these previous SRFIs deal with what we call in this SRFI specialized-arrays. Many of the functions in these previous SRFIs have corresponding forms in this SRFI. For example, from SRFI 63, we can translate:

`(array? obj)`

`(array? obj)`

`(array-rank a)`

`(array-dimension obj)`

`(make-array prototype k1 ...)`

`(make-specialized-array (make-interval (vector 0 ...) (vector k1 ...)) storage-class)`

.`(make-shared-array array mapper k1 ...)`

`(specialized-array-share array (make-interval (vector 0 ...) (vector k1 ...)) mapper)`

`(array-in-bounds? array index1 ...)`

`(interval-contains-multi-index? (array-domain array) index1 ...)`

`(array-ref array k1 ...)`

`((array-getter array) k1 ...)`

`(array-set! array obj k1 ...)`

`((array-setter array) obj k1 ...)`

At the same time, this SRFI has some special features:

- Intervals, used as the domains of arrays in this SRFI, are useful objects in their own rights, with their own procedures. We make a sharp distinction between the domains of arrays and the arrays themselves.
- Intervals can have nonzero lower bounds in each dimension.
- Intervals cannot be empty.
- Arrays must have a getter, but may have no setter.

Image processing applications provided significant motivation for this SRFI.

**Reading an image file in PGM format. **On a system with eight-bit chars, one
can write a function to read greyscale images in the PGM format of the netpbm package as follows. The lexicographical
order in array->specialized-array guarantees the the correct order of execution of the input procedures:

```
(define make-pgm cons)
(define pgm-greys car)
(define pgm-pixels cdr)
(define (read-pgm file)
(define (read-pgm-object port)
(skip-white-space port)
(let ((o (read port)))
;; to skip the newline or next whitespace
(read-char port)
(if (eof-object? o)
(error "reached end of pgm file")
o)))
(define (skip-to-end-of-line port)
(let loop ((ch (read-char port)))
(if (not (eq? ch #\newline))
(loop (read-char port)))))
(define (white-space? ch)
(case ch
((#\newline #\space #\tab) #t)
(else #f)))
(define (skip-white-space port)
(let ((ch (peek-char port)))
(cond ((white-space? ch)
(read-char port)
(skip-white-space port))
((eq? ch #\#)
(skip-to-end-of-line port)
(skip-white-space port))
(else #f))))
;; The image file formats defined in netpbm
;; are problematical, because they read the data
;; in the header as variable-length ISO-8859-1 text,
;; including arbitrary whitespace and comments,
;; and then they may read the rest of the file
;; as binary data.
;; So we give here a solution of how to deal
;; with these subtleties in Gambit Scheme.
(call-with-input-file
(list path: file
char-encoding: 'ISO-8859-1
eol-encoding: 'lf)
(lambda (port)
;; We're going to read text for a while,
;; then switch to binary.
;; So we need to turn off buffering until
;; we switch to binary.
(port-settings-set! port '(buffering: #f))
(let* ((header (read-pgm-object port))
(columns (read-pgm-object port))
(rows (read-pgm-object port))
(greys (read-pgm-object port)))
;; now we switch back to buffering
;; to speed things up
(port-settings-set! port '(buffering: #t))
(make-pgm
greys
(array->specialized-array
(make-array
(make-interval '#(0 0)
(vector rows columns))
(cond ((or (eq? header 'p5)
(eq? header 'P5))
;; pgm binary
(if (< greys 256)
;; one byte/pixel
(lambda (i j)
(char->integer
(read-char port)))
;; two bytes/pixel,
;;little-endian
(lambda (i j)
(let* ((first-byte
(char->integer
(read-char port)))
(second-byte
(char->integer
(read-char port))))
(+ (* second-byte 256)
first-byte)))))
;; pgm ascii
((or (eq? header 'p2)
(eq? header 'P2))
(lambda (i j)
(read port)))
(else
(error "not a pgm file"))))
(if (< greys 256)
u8-storage-class
u16-storage-class)))))))
```

**Viewing two-dimensional slices of three-dimensional data. **One example might be viewing two-dimensional slices of three-dimensional data in different ways. If one has a $1024 \times 512\times 512$ 3D image of the body stored as a variable

, then one could get 1024 axial views, each $512\times512$, of this 3D body by `body``(array-curry `

; or 512 median views, each $1024\times512$, by `body` 2)`(array-curry (array-permute `

; or finally 512 frontal views, each again $1024\times512$ pixels, by `body` '#(1 0 2)) 2)`(array-curry (array-permute `

; see Anatomical plane.`body` '#(2 0 1)) 2)

**Calculating second differences of images. **For another example, if a real-valued function is defined
on a two-dimensional interval $I$, its second difference in the direction $d$ at the point $x$ is defined as $\Delta^2_df(x)=f(x+2d)-2f(x+d)+f(x)$,
and this function is defined only for those $x$ for which $x$, $x+d$, and $x+2d$ are all in $I$. See the beginning of the section on "Moduli of smoothness" in these notes on wavelets and approximation theory for more details.

Using this definition, the following code computes all second-order forward differences of an image in the directions $d,2 d,3 d,\ldots$, defined only on the domains where this makes sense:

```
(define (all-second-differences image direction)
(let ((image-domain (array-domain image)))
(let loop ((i 1)
(result '()))
(let ((negative-scaled-direction
(vector-map (lambda (j)
(* -1 j i))
direction))
(twice-negative-scaled-direction
(vector-map (lambda (j)
(* -2 j i))
direction)))
(cond ((interval-intersect
image-domain
(interval-translate
image-domain
negative-scaled-direction)
(interval-translate
image-domain
twice-negative-scaled-direction))
=>
(lambda (subdomain)
(loop
(+ i 1)
(cons
(array->specialized-array
(array-map
(lambda (f_i f_i+d f_i+2d)
(+ f_i+2d
(* -2. f_i+d)
f_i))
(array-extract
image
subdomain)
(array-extract
(array-translate
image
negative-scaled-direction)
subdomain)
(array-extract
(array-translate
image
twice-negative-scaled-direction)
subdomain)))
result))))
(else
(reverse result)))))))
```

We can define a small synthetic image of size 8x8 pixels and compute its second differences in various directions:

```
(define image
(array->specialized-array
(make-array (make-interval '#(0 0) '#(8 8))
(lambda (i j)
(exact->inexact (+ (* i i) (* j j)))))))
(define (expose difference-images)
(pretty-print (map (lambda (difference-image)
(list (array-domain difference-image)
(array->list difference-image)))
difference-images)))
(begin
(display
"\nSecond-differences in the direction $k\times (1,0)$:\n")
(expose (all-second-differences image '#(1 0)))
(display
"\nSecond-differences in the direction $k\times (1,1)$:\n")
(expose (all-second-differences image '#(1 1)))
(display
"\nSecond-differences in the direction $k\times (1,-1)$:\n")
(expose (all-second-differences image '#(1 -1))))
```

On Gambit 4.8.5, this yields (after some hand editing):

Second-differences in the direction $k\times (1,0)$: ((#<##interval #2 lower-bounds: #(0 0) upper-bounds: #(6 8)> (2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2. 2.)) (#<##interval #3 lower-bounds: #(0 0) upper-bounds: #(4 8)> (8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8.)) (#<##interval #4 lower-bounds: #(0 0) upper-bounds: #(2 8)> (18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18.))) Second-differences in the direction $k\times (1,1)$: ((#<##interval #5 lower-bounds: #(0 0) upper-bounds: #(6 6)> (4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4.)) (#<##interval #6 lower-bounds: #(0 0) upper-bounds: #(4 4)> (16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16.)) (#<##interval #7 lower-bounds: #(0 0) upper-bounds: #(2 2)> (36. 36. 36. 36.))) Second-differences in the direction $k\times (1,-1)$: ((#<##interval #8 lower-bounds: #(0 2) upper-bounds: #(6 8)> (4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4. 4.)) (#<##interval #9 lower-bounds: #(0 4) upper-bounds: #(4 8)> (16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16.)) (#<##interval #10 lower-bounds: #(0 6) upper-bounds: #(2 8)> (36. 36. 36. 36.)))

You can see that with differences in the direction of only the first coordinate, the domains of the difference arrays get smaller in the first coordinate while staying the same in the second coordinate, and with differences in the diagonal directions, the domains of the difference arrays get smaller in both coordinates.

**Separable operators. **Many multi-dimensional transforms in signal processing are *separable*, in that that the multi-dimensional transform can be computed by applying one-dimensional transforms in each of the coordinate directions. Examples of such transforms include the Fast Fourier Transform and the Fast Wavelet Transform. Each one-dimensional subdomain of the complete domain is called a *pencil*, and the same one-dimensional transform is applied to all pencils in a given direction. Given the one-dimensional array transform, one can compute the multidimensional transform as follows:

```
(define (make-separable-transform 1D-transform)
(lambda (array)
;; Works on arrays of any dimension.
(let* ((n
(array-dimension array))
(permutation
;; we start with the identity permutation
(let ((result (make-vector n)))
(do ((i 0 (fx+ i 1)))
((fx= i n) result)
(vector-set! result i i)))))
;; We apply the one-dimensional transform
;; in each coordinate direction.
(do ((d 0 (fx+ d 1)))
((fx= d n))
;; Swap the d'th and n-1'st coordinates
(vector-set! permutation (fx- n 1) d)
(vector-set! permutation d (fx- n 1))
;; Apply the transform in the d'th coordinate
;; direction to all "pencils" in that direction.
;; array-permute re-orders the coordinates to
;; put the d'th coordinate at the end, array-curry
;; returns an $n-1$-dimensional array of
;; one-dimensional subarrays, and 1D-transform
;; is applied to each of those sub-arrays.
(array-for-each
1D-transform
(array-curry (array-permute array permutation)
1))
;; return the permutation to the identity
(vector-set! permutation d d)
(vector-set! permutation (fx- n 1) (fx- n 1))))))
```

We can test this by turning a one-dimensional Haar wavelet transform into a multi-dimensional Haar transform:

```
(define (1D-Haar-loop a)
(let ((getter (array-getter a))
(setter (array-setter a))
(n (interval-upper-bound (array-domain a) 0)))
(do ((i 0 (fx+ i 2)))
((fx= i n))
(let* ((a_i
(getter i))
(a_i+1
(getter (fx+ i 1)))
(scaled-sum
(fl/ (fl+ a_i a_i+1) (flsqrt 2.0)))
(scaled-difference
(fl/ (fl- a_i a_i+1) (flsqrt 2.0))))
(setter scaled-sum i)
(setter scaled-difference (fx+ i 1))))))
(define (1D-Haar-transform a)
;; works only on mutable arrays with domains
;; $[0, 2^k)$ for some $k$
(let ((n (interval-upper-bound (array-domain a) 0)))
(if (fx< 1 n)
(begin
;; calculate the scaled sums and differences
(1D-Haar-loop a)
;; Apply the transform to the
;; sub-array of scaled sums
(1D-Haar-transform (array-sample a '#(2)))))))
(define (1D-Haar-inverse-transform a)
;; works only on mutable arrays with domains
;; $[0, 2^k)$ for some $k$
(let* ((n (interval-upper-bound (array-domain a) 0)))
(if (fx< 1 n)
(begin
;; Apply the inverse transform to
;; get the array of scaled sums
(1D-Haar-inverse-transform
(array-sample a '#(2)))
;; reconstruct the array values from
;; the scaled sums and differences
(1D-Haar-loop a)))))
(define Haar-transform
(make-separable-transform 1D-Haar-transform))
(define Haar-inverse-transform
(make-separable-transform 1D-Haar-inverse-transform))
```

We then define an image that is a multiple of a single, two-dimensional Haar wavelet, compute its transform (which should be nonzero for only a single Haar coefficient), and then the inverse transform:

```
(let ((image
(array->specialized-array
(make-array (make-interval '#(0 0) '#(4 4))
(lambda (i j)
(if (fx< i 2) 1. -1.))))))
(display "\nInitial image: \n")
(pretty-print (list (array-domain image)
(array->list image)))
(Haar-transform image)
(display "\nArray of Haar wavelet coefficients: \n")
(pretty-print (list (array-domain image)
(array->list image)))
(Haar-inverse-transform image)
(display "\nReconstructed image: \n")
(pretty-print (list (array-domain image)
(array->list image))))
```

This yields:

Initial image: (#<##interval #11 lower-bounds: #(0 0) upper-bounds: #(4 4)> (1. 1. 1. 1. 1. 1. 1. 1. -1. -1. -1. -1. -1. -1. -1. -1.)) Array of Haar wavelet coefficients: (#<##interval #11 lower-bounds: #(0 0) upper-bounds: #(4 4)> (0. 0. 0. 0. 0. 0. 0. 0. 3.9999999999999987 0. 0. 0. 0. 0. 0. 0.)) Reconstructed image: (#<##interval #11 lower-bounds: #(0 0) upper-bounds: #(4 4)> (.9999999999999993 .9999999999999993 .9999999999999993 .9999999999999993 .9999999999999993 .9999999999999993 .9999999999999993 .9999999999999993 -.9999999999999993 -.9999999999999993 -.9999999999999993 -.9999999999999993 -.9999999999999993 -.9999999999999993 -.9999999999999993 -.9999999999999993))

In perfect arithmetic, this Haar transform is *orthonormal*, in that the sum of the squares of the elements of the image is the same as the sum of the squares of the Haar coefficients of the image. We can see that this is approximately true here.

The SRFI author thanks Edinah K Gnang, John Cowan, Sudarshan S Chawathe, Jamison Hope, and Per Bothner for their comments and suggestions, and Arthur A Gleckler, SRFI Editor, for his guidance and patience.

- "multi-dimensional arrays in R5RS?", by Alan Bawden.
- SRFI 4: Homogeneous Numeric Vector Datatypes, by Marc Feeley.
- SRFI 25: Multi-dimensional Array Primitives, by Jussi Piitulainen.
- SRFI 47: Array, by Aubrey Jaffer.
- SRFI 58: Array Notation, by Aubrey Jaffer.
- SRFI 63: Homogeneous and Heterogeneous Arrays, by Aubrey Jaffer.

© 2016 Bradley J Lucier. All Rights Reserved.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

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