SparseArrays
stdlib module. Sparse arrays are arrays that contain enough zeros that storing them in a special data structure leads to savings in space and execution time, compared to dense arrays.SparseMatrixCSC{Tv,Ti}
, where Tv
is the type of the stored values, and Ti
is the integer type for storing column pointers and row indices. The internal representation of SparseMatrixCSC
is as follows:SparseMatrixCSC
object contains unsorted row indices, one quick way to sort them is by doing a double transpose.SparseMatrixCSC
. These are accepted by functions in Base
(but there is no guarantee that they will be preserved in mutating operations). Such explicitly stored zeros are treated as structural nonzeros by many routines. The nnz
function returns the number of elements explicitly stored in the sparse data structure, including structural nonzeros. In order to count the exact number of numerical nonzeros, use count(!iszero, x)
, which inspects every stored element of a sparse matrix. dropzeros
, and the in-place dropzeros!
, can be used to remove stored zeros from the sparse matrix.SparseVector{Tv,Ti}
where Tv
is the type of the stored values and Ti
the integer type for the indices. The internal representation is as follows:SparseMatrixCSC
, the SparseVector
type can also contain explicitly stored zeros. (See Sparse Matrix Storage.).zeros
function that Julia provides for working with dense arrays. To produce a sparse array instead, you can use the same name with an sp
prefix:sparse
function is often a handy way to construct sparse arrays. For example, to construct a sparse matrix we can input a vector I
of row indices, a vector J
of column indices, and a vector V
of stored values (this is also known as the COO (coordinate) format). sparse(I,J,V)
then constructs a sparse matrix such that S[I[k], J[k]] = V[k]
. The equivalent sparse vector constructor is sparsevec
, which takes the (row) index vector I
and the vector V
with the stored values and constructs a sparse vector R
such that R[I[k]] = V[k]
.sparse
and sparsevec
functions is findnz
, which retrieves the inputs used to create the sparse array. findall(!iszero, x)
returns the cartesian indices of non-zero entries in x
(including stored entries equal to zero).sparse
function:Array
constructor. The issparse
function can be used to query if a matrix is sparse.(I,J,V)
format using findnz
, manipulate the values or the structure in the dense vectors (I,J,V)
, and then reconstruct the sparse matrix.S
, or that the resulting sparse matrix has density d
, i.e. each matrix element has a probability d
of being non-zero.Sparse | Dense | Description |
---|---|---|
spzeros(m,n) | zeros(m,n) | Creates a m-by-n matrix of zeros. (spzeros(m,n) is empty.) |
sparse(I, n, n) | Matrix(I,n,n) | Creates a n-by-n identity matrix. |
Array(S) | sparse(A) | Interconverts between dense and sparse formats. |
sprand(m,n,d) | rand(m,n) | Creates a m-by-n random matrix (of density d) with iid non-zero elements distributed uniformly on the half-open interval $[0, 1)$. |
sprandn(m,n,d) | randn(m,n) | Creates a m-by-n random matrix (of density d) with iid non-zero elements distributed according to the standard normal (Gaussian) distribution. |
sprandn(rng,m,n,d) | randn(rng,m,n) | Creates a m-by-n random matrix (of density d) with iid non-zero elements generated with the rng random number generator |
N
-dimensional sparse arrays (or array-like types) with elements of type Tv
and index type Ti
. SparseMatrixCSC
, SparseVector
and SuiteSparse.CHOLMOD.Sparse
Perfectly clear complete 3 5 6 1148 download free. are subtypes of this.Ln = Rn-1
Rn = Ln-1 ⊕ f(Rn-1,Kn)
Kn ⊕ E(Rn-1)
Kn⊕E(Rn-1) = B1B2B3B4B5B6B7B8,
S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8)
f =
R16L16