gmp (version 0.7-5)

matrix: Matrix manipulation with gmp

Description

Overload of “all” standard tools useful for matrix manipulation adapted to large numbers.

Usage

# S3 method for bigz
matrix(data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL, mod = NA,...)
is.matrixZQ(x)
# S3 method for bigz
%*%(x, y)
# S3 method for bigq
%*%(x, y)
# S3 method for bigq
crossprod(x, y=NULL,...)
# S3 method for bigz
tcrossprod(x, y=NULL,...)
# S3 method for bigz
cbind(..., deparse.level=1)
# S3 method for bigq
rbind(..., deparse.level=1)
## ..... etc

Value

matrix(): A matrix of class "bigz" or "bigq".

is.matrixZQ(): TRUE or FALSE.

dim(), ncol(), etc: integer or NULL, as for simple matrices.

cbind(x,y,...) and rbind(x,y,...) now (2024-01, since

gmp version 0.9-5), do drop deparse.level=. instead of wrongly creating an extra column or row and the "bigz"

method takes all arguments into account and calls the "bigq"

method in case of arguments inheriting from "bigq".

Arguments

data

an optional data vector

nrow

the desired number of rows

ncol

the desired number of columns

byrow

logical. If FALSE (the default), the matrix is filled by columns, otherwise the matrix is filled by rows.

dimnames

not implemented for "bigz" or "bigq" matrices.

mod

optional modulus (when data is "bigz").

x,y

numeric, bigz, or bigq matrices or vectors.

..., deparse.level

arguments from the generic; not made use of, i.e., disregarded here.

Author

Antoine Lucas and Martin Maechler

Details

The extract function ("[") is the same use for vector or matrix. Hence, x[i] returns the same values as x[i,]. This is not considered a feature and may be changed in the future (with warnings).

All matrix multiplications should work as with numeric matrices.

Special features concerning the "bigz" class: the modulus can be

Unset:

Just play with large numbers

Set with a vector of size 1:

Example: matrix.bigz(1:6,nrow=2,ncol=3,mod=7) This means you work in \(Z/nZ\), for the whole matrix. It is the only case where the %*% and solve functions will work in \(Z/nZ\).

Set with a vector smaller than data:

Example: matrix.bigz(1:6,nrow=2,ncol=3,mod=1:5). Then, the modulus is repeated to the end of data. This can be used to define a matrix with a different modulus at each row.

Set with same size as data:

Modulus is defined for each cell

See Also

Solving a linear system: solve.bigz. matrix

Examples

V <- as.bigz(v <- 3:7)
crossprod(V)# scalar product
(C <- t(V))
stopifnot(dim(C) == dim(t(v)), C == v,
          dim(t(C)) == c(length(v), 1),
          crossprod(V) == sum(V * V),
         tcrossprod(V) == outer(v,v),
          identical(C, t(t(C))),
          is.matrixZQ(C), !is.matrixZQ(V), !is.matrixZQ(5)
	)

## a matrix
x <- diag(1:4)
## invert this matrix
(xI <- solve(x))

## matrix in Z/7Z
y <- as.bigz(x,7)
## invert this matrix (result is *different* from solve(x)):
(yI <- solve(y))
stopifnot(yI %*% y == diag(4),
          y %*% yI == diag(4))

## matrix in Q
z  <- as.bigq(x)
## invert this matrix (result is the same as solve(x))
(zI <- solve(z))

stopifnot(abs(zI - xI) <= 1e-13,
          z %*% zI == diag(4),
          identical(crossprod(zI), zI %*% t(zI))
         )

A <- matrix(2^as.bigz(1:12), 3,4)
for(a in list(A, as.bigq(A, 16), factorialZ(20), as.bigq(2:9, 3:4))) {
  a.a <- crossprod(a)
  aa. <- tcrossprod(a)
  stopifnot(identical(a.a, crossprod(a,a)),
 	    identical(a.a, t(a) %*% a)
            ,
            identical(aa., tcrossprod(a,a)),
	    identical(aa., a %*% t(a))
 	   )
}# {for}