Matrix manipulation with gmp

Description

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

Usage

## S3 method for class 'bigz'
matrix(data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL, mod = NA,...)

is.matrixZQ(x)

## S3 method for class 'bigz'
x %*% y
## S3 method for class 'bigq'
x %*% y
## S3 method for class 'bigq'
crossprod(x, y=NULL,...)
## S3 method for class 'bigz'
tcrossprod(x, y=NULL,...)

## S3 method for class 'bigz'
cbind(..., deparse.level=1)
## S3 method for class 'bigq'
rbind(..., deparse.level=1)
## ..... etc

Arguments

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

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".

Author(s)

Antoine Lucas and Martin Maechler

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}