Construct Sparse Banded Matrix from (Sup-/Super-) Diagonals

Construct a sparse banded matrix by specifying its non-zero sup- and super-diagonals.

bandSparse(n, m = n, k, diagonals, symmetric = FALSE,
           repr = "C", giveCsparse = (repr == "C"))

Arguments

n,m

the matrix dimension \((n,m) = (nrow, ncol)\).

k

integer vector of “diagonal numbers”, with identical meaning as in band(*, k), i.e., relative to the main diagonal, which is k=0.

diagonals

optional list of sub-/super- diagonals; if missing, the result will be a pattern matrix, i.e., inheriting from class nMatrix.

diagonals can also be \(n' \times d\) matrix, where d <- length(k) and \(n' >= min(n,m)\). In that case, the sub-/super- diagonals are taken from the columns of diagonals, where only the first several rows will be used (typically) for off-diagonals.

symmetric

logical; if true the result will be symmetric (inheriting from class symmetricMatrix) and only the upper or lower triangle must be specified (via k and diagonals).

repr

character string, one of "C", "T", or "R", specifying the sparse representation to be used for the result, i.e., one from the super classes CsparseMatrix, TsparseMatrix, or RsparseMatrix.

giveCsparse

(deprecated, replaced with repr): logical indicating if the result should be a CsparseMatrix or a TsparseMatrix, where the default was TRUE, and now is determined from repr; very often Csparse matrices are more efficient subsequently, but not always.

Value

a sparse matrix (of class CsparseMatrix) of dimension \(n \times m\) with diagonal “bands” as specified.

See also

band, for extraction of matrix bands; bdiag, diag, sparseMatrix, Matrix.

Examples

diags <- list(1:30, 10*(1:20), 100*(1:20))
s1 <- bandSparse(13, k = -c(0:2, 6), diag = c(diags, diags[2]), symm=TRUE)
s1
#> 13 x 13 sparse Matrix of class "dsCMatrix"
#>                                                              
#>  [1,]   1  10 100   .   .   .  10   .   .    .    .    .    .
#>  [2,]  10   2  20 200   .   .   .  20   .    .    .    .    .
#>  [3,] 100  20   3  30 300   .   .   .  30    .    .    .    .
#>  [4,]   . 200  30   4  40 400   .   .   .   40    .    .    .
#>  [5,]   .   . 300  40   5  50 500   .   .    .   50    .    .
#>  [6,]   .   .   . 400  50   6  60 600   .    .    .   60    .
#>  [7,]  10   .   .   . 500  60   7  70 700    .    .    .   70
#>  [8,]   .  20   .   .   . 600  70   8  80  800    .    .    .
#>  [9,]   .   .  30   .   .   . 700  80   9   90  900    .    .
#> [10,]   .   .   .  40   .   .   . 800  90   10  100 1000    .
#> [11,]   .   .   .   .  50   .   .   . 900  100   11  110 1100
#> [12,]   .   .   .   .   .  60   .   .   . 1000  110   12  120
#> [13,]   .   .   .   .   .   .  70   .   .    . 1100  120   13
s2 <- bandSparse(13, k =  c(0:2, 6), diag = c(diags, diags[2]), symm=TRUE)
stopifnot(identical(s1, t(s2)), is(s1,"dsCMatrix"))

## a pattern Matrix of *full* (sub-)diagonals:
bk <- c(0:4, 7,9)
(s3 <- bandSparse(30, k = bk, symm = TRUE))
#> 30 x 30 sparse Matrix of class "nsCMatrix"
#>                                                                  
#>  [1,] | | | | | . . | . | . . . . . . . . . . . . . . . . . . . .
#>  [2,] | | | | | | . . | . | . . . . . . . . . . . . . . . . . . .
#>  [3,] | | | | | | | . . | . | . . . . . . . . . . . . . . . . . .
#>  [4,] | | | | | | | | . . | . | . . . . . . . . . . . . . . . . .
#>  [5,] | | | | | | | | | . . | . | . . . . . . . . . . . . . . . .
#>  [6,] . | | | | | | | | | . . | . | . . . . . . . . . . . . . . .
#>  [7,] . . | | | | | | | | | . . | . | . . . . . . . . . . . . . .
#>  [8,] | . . | | | | | | | | | . . | . | . . . . . . . . . . . . .
#>  [9,] . | . . | | | | | | | | | . . | . | . . . . . . . . . . . .
#> [10,] | . | . . | | | | | | | | | . . | . | . . . . . . . . . . .
#> [11,] . | . | . . | | | | | | | | | . . | . | . . . . . . . . . .
#> [12,] . . | . | . . | | | | | | | | | . . | . | . . . . . . . . .
#> [13,] . . . | . | . . | | | | | | | | | . . | . | . . . . . . . .
#> [14,] . . . . | . | . . | | | | | | | | | . . | . | . . . . . . .
#> [15,] . . . . . | . | . . | | | | | | | | | . . | . | . . . . . .
#> [16,] . . . . . . | . | . . | | | | | | | | | . . | . | . . . . .
#> [17,] . . . . . . . | . | . . | | | | | | | | | . . | . | . . . .
#> [18,] . . . . . . . . | . | . . | | | | | | | | | . . | . | . . .
#> [19,] . . . . . . . . . | . | . . | | | | | | | | | . . | . | . .
#> [20,] . . . . . . . . . . | . | . . | | | | | | | | | . . | . | .
#> [21,] . . . . . . . . . . . | . | . . | | | | | | | | | . . | . |
#> [22,] . . . . . . . . . . . . | . | . . | | | | | | | | | . . | .
#> [23,] . . . . . . . . . . . . . | . | . . | | | | | | | | | . . |
#> [24,] . . . . . . . . . . . . . . | . | . . | | | | | | | | | . .
#> [25,] . . . . . . . . . . . . . . . | . | . . | | | | | | | | | .
#> [26,] . . . . . . . . . . . . . . . . | . | . . | | | | | | | | |
#> [27,] . . . . . . . . . . . . . . . . . | . | . . | | | | | | | |
#> [28,] . . . . . . . . . . . . . . . . . . | . | . . | | | | | | |
#> [29,] . . . . . . . . . . . . . . . . . . . | . | . . | | | | | |
#> [30,] . . . . . . . . . . . . . . . . . . . . | . | . . | | | | |

## If you want a pattern matrix, but with "sparse"-diagonals,
## you currently need to go via logical sparse:
lLis <- lapply(list(rpois(20, 2), rpois(20, 1), rpois(20, 3))[c(1:3, 2:3, 3:2)],
               as.logical)
(s4 <- bandSparse(20, k = bk, symm = TRUE, diag = lLis))
#> 20 x 20 sparse Matrix of class "lsCMatrix"
#>                                              
#>  [1,] | | | | | . . | . | . . . . . . . . . .
#>  [2,] | | : | : | . . | . : . . . . . . . . .
#>  [3,] | : | : | : | . . | . : . . . . . . . .
#>  [4,] | | : | : | : | . . | . : . . . . . . .
#>  [5,] | : | : | | | | | . . | . | . . . . . .
#>  [6,] . | : | | | | : | : . . : . | . . . . .
#>  [7,] . . | : | | | | | | | . . | . | . . . .
#>  [8,] | . . | | : | | | | | | . . | . | . . .
#>  [9,] . | . . | | | | | | | | | . . | . | . .
#> [10,] | . | . . : | | | | | | | | . . | . | .
#> [11,] . : . | . . | | | | | | | | | . . | . |
#> [12,] . . : . | . . | | | | | | | | | . . | .
#> [13,] . . . : . : . . | | | | | | | | | . . |
#> [14,] . . . . | . | . . | | | | | | | | | . .
#> [15,] . . . . . | . | . . | | | | | | | | | .
#> [16,] . . . . . . | . | . . | | | | | : | : |
#> [17,] . . . . . . . | . | . . | | | : | | | |
#> [18,] . . . . . . . . | . | . . | | | | : : |
#> [19,] . . . . . . . . . | . | . . | : | : | |
#> [20,] . . . . . . . . . . | . | . . | | | | :
(s4. <- as(drop0(s4), "nsparseMatrix"))
#> 20 x 20 sparse Matrix of class "nsCMatrix"
#>                                              
#>  [1,] | | | | | . . | . | . . . . . . . . . .
#>  [2,] | | . | . | . . | . . . . . . . . . . .
#>  [3,] | . | . | . | . . | . . . . . . . . . .
#>  [4,] | | . | . | . | . . | . . . . . . . . .
#>  [5,] | . | . | | | | | . . | . | . . . . . .
#>  [6,] . | . | | | | . | . . . . . | . . . . .
#>  [7,] . . | . | | | | | | | . . | . | . . . .
#>  [8,] | . . | | . | | | | | | . . | . | . . .
#>  [9,] . | . . | | | | | | | | | . . | . | . .
#> [10,] | . | . . . | | | | | | | | . . | . | .
#> [11,] . . . | . . | | | | | | | | | . . | . |
#> [12,] . . . . | . . | | | | | | | | | . . | .
#> [13,] . . . . . . . . | | | | | | | | | . . |
#> [14,] . . . . | . | . . | | | | | | | | | . .
#> [15,] . . . . . | . | . . | | | | | | | | | .
#> [16,] . . . . . . | . | . . | | | | | . | . |
#> [17,] . . . . . . . | . | . . | | | . | | | |
#> [18,] . . . . . . . . | . | . . | | | | . . |
#> [19,] . . . . . . . . . | . | . . | . | . | |
#> [20,] . . . . . . . . . . | . | . . | | | | .

n <- 1e4
bk <- c(0:5, 7,11)
bMat <- matrix(1:8, n, 8, byrow=TRUE)
bLis <- as.data.frame(bMat)
B  <- bandSparse(n, k = bk, diag = bLis)
Bs <- bandSparse(n, k = bk, diag = bLis, symmetric=TRUE)
B [1:15, 1:30]
#> 15 x 30 sparse Matrix of class "dgCMatrix"
#>                                                                  
#>  [1,] 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . . . . . .
#>  [2,] . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . . . . .
#>  [3,] . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . . . .
#>  [4,] . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . . .
#>  [5,] . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . .
#>  [6,] . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . .
#>  [7,] . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . .
#>  [8,] . . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . .
#>  [9,] . . . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . .
#> [10,] . . . . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . .
#> [11,] . . . . . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . .
#> [12,] . . . . . . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . . .
#> [13,] . . . . . . . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . . .
#> [14,] . . . . . . . . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . . .
#> [15,] . . . . . . . . . . . . . . 1 2 3 4 5 6 . 7 . . . 8 . . . .
Bs[1:15, 1:30]
#> 15 x 30 sparse Matrix of class "dgCMatrix"
#>                                                                  
#>  [1,] 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . . . . . .
#>  [2,] 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . . . . .
#>  [3,] 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . . . .
#>  [4,] 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . . .
#>  [5,] 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . . .
#>  [6,] 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . . .
#>  [7,] . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . . .
#>  [8,] 7 . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . . .
#>  [9,] . 7 . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . . .
#> [10,] . . 7 . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . . .
#> [11,] . . . 7 . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . . .
#> [12,] 8 . . . 7 . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . . .
#> [13,] . 8 . . . 7 . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . . .
#> [14,] . . 8 . . . 7 . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . . .
#> [15,] . . . 8 . . . 7 . 6 5 4 3 2 1 2 3 4 5 6 . 7 . . . 8 . . . .
## can use a list *or* a matrix for specifying the diagonals:
stopifnot(identical(B,  bandSparse(n, k = bk, diag = bMat)),
    identical(Bs, bandSparse(n, k = bk, diag = bMat, symmetric=TRUE))
          , inherits(B, "dtCMatrix") # triangular!
)