# spray v1.0-7

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## Sparse Arrays and Multivariate Polynomials

Sparse arrays interpreted as multivariate polynomials.

## Readme

# The spray package: sparse arrays in R

# Overview

The `spray`

package provides functionality for sparse arrays.

In a sparse arrays, nonzero elements are stored along with an index
vector describing their coordinates. The `spray`

package provides
functionality for sparse arrays and interprets them as multivariate
polynomials.

# Installation

You can install the released version of `spray`

from
CRAN with:

```
# install.packages("spray") # uncomment this to install the package
library("spray")
```

# The `spray`

package in use

Base R has extensive support for multidimensional arrays. Consider

```
a <- array(0,dim=4:12)
a[2,2,2,2,2,2,2,2,2] <- 17
a[3,4,2,2,7,2,3,2,3] <- 18
```

Handling `a`

requires storage of
floating point numbers (of which two are
nonzero), represented in an elegant format amenable to extraction and
replacement. Arrays such as this in which many of the elements are zero
are common and in this case storing only the nonzero elements and their
positions would be a more compact and efficient representation. To
create a sparse array object in the `spray`

package, one specifies a
matrix of indices with each row corresponding to the position of a
nonzero element, and a numeric vector of values:

```
library("spray")
M <- rbind(
c(2,2,2,2,2,2,2,2,2),
c(3,4,2,2,7,2,3,2,3))
S1 <- spray(M,7:8)
S1
#> val
#> 3 4 2 2 7 2 3 2 3 = 8
#> 2 2 2 2 2 2 2 2 2 = 7
```

Note that object ``S1`

is rather compact by comparison with plain array
`a`

, as it needs to record only a 18-element index array of integers and
two double-precision entries. The order in which the elements are stored
is implementation-specific (see the vignette for details and an extended
discussion).

Basic arithmetic is implemented where appropriate. If we define

```
S2 <-spray(rbind(
c(1,2,3,1,3,3,1,4,1),
c(3,4,2,2,7,2,3,2,3)), c(100,-8))
S2
#> val
#> 3 4 2 2 7 2 3 2 3 = -8
#> 1 2 3 1 3 3 1 4 1 = 100
```

then

```
S1+S2
#> val
#> 2 2 2 2 2 2 2 2 2 = 7
#> 1 2 3 1 3 3 1 4 1 = 100
```

(the entry with value `8`

has cancelled out).

# The spray package and multivariate polynomials

One natural application for `spray`

objects is multivariate polynomials.
Defining

```
S1 <- spray(matrix(c(0,0,0,1,0,0,1,1,1,2,0,3),ncol=3),1:4)
S2 <- spray(matrix(c(6,-7,8,0,0,2,1,1,3),byrow=TRUE,ncol=3),c(17,11,-4))
S1
#> val
#> 1 1 3 = 4
#> 0 1 0 = 3
#> 0 0 1 = 1
#> 0 0 2 = 2
S2
#> val
#> 1 1 3 = -4
#> 6 -7 8 = 17
#> 0 0 2 = 11
```

it is natural to interpret the rows of the index matrix as powers of
different variables of a multivariate polynomial, and the values as
being the coefficients. This is realised in the package using the
`polyform`

print option, which if set to `TRUE`

, modifies the print
method:

```
options(polyform = TRUE)
S1
#> +4*x*y*z^3 +3*y +z +2*z^2
S2
#> -4*x*y*z^3 +17*x^6*y^-7*z^8 +11*z^2
```

(only the print method has changed; the objects themselves are unaltered). The print method interprets, by default, the three columns as variables although this behaviour is user-definable. With this interpretation, multiplication and addition have natural definitions as multivariate polynomial multiplication and addition:

```
S1+S2
#> +3*y +13*z^2 +z +17*x^6*y^-7*z^8
S1*S2
#> +17*x^6*y^-7*z^9 -4*x*y*z^4 +68*x^7*y^-6*z^11 +11*z^3
#> +34*x^6*y^-7*z^10 +33*y*z^2 -12*x*y^2*z^3 -16*x^2*y^2*z^6
#> +51*x^6*y^-6*z^8 +22*z^4 +36*x*y*z^5
S1^2+4*S2
#> +9*y^2 -16*x*y*z^3 +16*x^2*y^2*z^6 +24*x*y^2*z^3 +4*z^4
#> +68*x^6*y^-7*z^8 +45*z^2 +12*y*z^2 +16*x*y*z^5 +8*x*y*z^4 +4*z^3
#> +6*y*z
```

It is possible to introduce an element of symbolic calculation,
exhibiting familiar algebraic identities. Consider the `lone()`

function, which creates a sparse array whose multivariate polynomial
interpretation is a single variable:

```
x <- lone(1, 3)
y <- lone(2, 3)
z <- lone(3, 3)
(x + y) * (y + z) * (x + z) - (x + y + z) * (x*y + x*z + y*z)
#> -x*y*z
```

thus illustrating the identity .

Spray objects can be coerced to functions:

```
S4 <- spray(cbind(1:3, 3:1), 1:3)
f <- as.function(S4)
f(c(1, 2))
#> X
#> 22
```

Differentiation is also straightforward. Suppose we wish to calculate the multivariate polynomial corresponding to

This would be

```
aderiv((xyz(3) + linear(1:3))^3, 1:3)
#> +216*x +108*x^2*y
```

The package vignette offers a detailed discussion of the package design
philosophy; also, the `mvp`

package provdies a further interpretation of
the concept of “sparse” in the context of multivariate polynomials.

## Functions in spray

Name | Description | |

subs | Substitute values into a spray object | |

as.array | Coerce spray objects to arrays | |

arity | The arity of a spray object | |

rspray | Random spray objects | |

spray | Create sparse array objects | |

spray_cpp | Low-level functions that call C++ source code | |

spray-package | Sparse arrays and multivariate polynomials | |

pmax | Parallel maxima and minima for sprays | |

asum | Sum over dimension margins | |

print.spray | Print methods for spray objects | |

zero | The zero polynomial | |

homog | Various functions to create simple spray objects | |

knight | Generating function for a chess knight and king | |

constant | Get or set the constant term of a spray object | |

deriv | Partial differentiation of spray objects | |

nterms | Number of nonzero terms in a spray object | |

ooom | One-over-one-minus for spray objects | |

Ops.spray | Arithmetic Ops Group Methods for sprays | |

Extract.spray | Extract or Replace Parts of a spray | |

as.function.spray | Coerce a spray object to a function | |

spray_missing_accessor | Discussion document | |

No Results! |

## Vignettes of spray

Name | ||

four_dimensional_knight.pdf | ||

four_dimensional_knight.svg | ||

spray.Rnw | ||

spray.bib | ||

No Results! |

## Last month downloads

## Details

Type | Package |

License | GPL (>= 2) |

LinkingTo | Rcpp |

SystemRequirements | C++11 |

URL | https://github.com/RobinHankin/spray.git |

BugReports | https://github.com/RobinHankin/spray/issues |

NeedsCompilation | yes |

Packaged | 2019-09-02 01:12:41 UTC; rhankin |

Repository | CRAN |

Date/Publication | 2019-09-02 22:30:02 UTC |

imports | magic , partitions , Rcpp (>= 0.12.3) |

depends | methods |

suggests | polynom |

Contributors |

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