Dyadic-class: The class to represent a dyadic matrix

In DyadiCarma: Dyadic Matrices and their Algebra using 'Rcpp' and 'RcppArmadillo'

The class to represent a dyadic matrix

Description

The main class in the Dyadic-package used for representing three types of dyadic matrices: horizontal, vertical, symmetric, and asymmetric.

Value

running new("Dyadic") return an object that belongs to the class Dyadic, with the initialization of the default values for the fields.

Slots

height

positive integer, the number of dyadic levels;

breadth

positive integer, the breadth of the dyadic structure;

type

string, one of the following character strings: horiz,vert,symm, asymm which indicates the type of dyadic matrix

• horiz horizontal,

• vert vertical,

• symm symmetric,

• asymm asymmetric,

where the last two types distinguish symmetrically dyadic matrices (they both have symmetric dyadic structure) that correspond to symmetric or not symmetric matrices.

entries

list (of matrices); a list of the length height containing (2^(l)-1)*breadth x 2^(height-l)*breadth matrices, where l is the index running through the list. Each matrix in the list includes the entries corresponding to 2^(height-l) (2^l-1)*breadth x breadth-matrices put side by side columnwise in the lth level of a dyadic structure. In the 'symm'- and 'asymm'-cases, the terms below diagonal on the diagonal blocks are set to zero.

aentries

list (of matrices); a list which is either empty if the slot type is not 'asymm' or of the length height otherwise, in which the case it contains (2^(l)-1)*breadth x 2^(height-l)*breadth matrices, where l is the index running through the list. Each matrix in the list includes the entries corresponding to 2^(height-l). (2^l-1)*breadth x breadth-matrices put side by side columnwise in the lth horizontal level of an asymmetric dyadic structure. The terms above and on the diagonal in the diagonal blocks are set to zero because they are accounted in the slot entries.

References

Kos, M., Podgórski, K., & Wu, H. (2025). Dyadic Factorization and Efficient Inversion of Sparse Positive Definite Matrices. arXiv. https://arxiv.org/abs/2505.08144

Examples

#-------------------------------------------------------------#
#------- Generating an object from the 'Dyadic' class --------#
#-------------------------------------------------------------#

# The most generic generation of an object of class 'Dyadic':
D <- new("Dyadic") # a generic format for 'Splinets' object
D
# The SLOTs of 'Dyadic' - the default values
D@height
D@breadth
D@type
D@entries[[1]]
D@aentries

N <- 4
k <- 3 # the height and breadth of a dyadic matrix

# The construction of a horizontally dyadic matrix with height 4 and breadth 3.

E <- list()
for (i in 1:4) {
    E[[i]] <- matrix(1, nrow = (2^(i) - 1) * 3, ncol = 2^(4 - i) * 3)
}


DD <- new("Dyadic", height = N, breadth = k, type = "horiz", entries = E)

DD

# The classic R matrix representation of DD.
mat_DD <- as.matrix(DD)
mat_DD

DyadiCarma documentation built on June 8, 2025, 12:45 p.m.