# Matrix manipulation with gmp Overload of all standard tools useful for matrix manipulation adapted to large numbers. ```r ## 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 - `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. ## 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 ## Returns `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 ```r 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} ```

matrix function