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- For example, the array 3, 2, 1, 0 represents the permutation that maps the element at index 0 to index 3, the element at index 1 to index 2, the element at index 2 to index 1 and the element at index 3 to index 0.
- Given a collection of distinct integers, return all possible permutations. Example: Input: 1,2,3 Output: 1,2,3, 1,3,2, 2,1,3, 2,3,1, 3,1,2, 3,2,1.
Julia has support for sparse vectors and sparse matrices in the 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.
In Julia, sparse matrices are stored in the Compressed Sparse Column (CSC) format. Julia sparse matrices have the type 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:
The compressed sparse column storage makes it easy and quick to access the elements in the column of a sparse matrix, whereas accessing the sparse matrix by rows is considerably slower. Operations such as insertion of previously unstored entries one at a time in the CSC structure tend to be slow. This is because all elements of the sparse matrix that are beyond the point of insertion have to be moved one place over.
All operations on sparse matrices are carefully implemented to exploit the CSC data structure for performance, and to avoid expensive operations. Stellar mail converter 2 0 1 download free.
If you have data in CSC format from a different application or library, and wish to import it in Julia, make sure that you use 1-based indexing. The row indices in every column need to be sorted. If your SparseMatrixCSC
object contains unsorted row indices, one quick way to sort them is by doing a double transpose.
In some applications, it is convenient to store explicit zero values in a 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.
Sparse vectors are stored in a close analog to compressed sparse column format for sparse matrices. In Julia, sparse vectors have the type 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:
As for SparseMatrixCSC
, the SparseVector
type can also contain explicitly stored zeros. (See Sparse Matrix Storage.).
The simplest way to create a sparse array is to use a function equivalent to the 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: Keykey 2 6 – typing tutor mangal.
The 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]
.
The inverse of the 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).
Another way to create a sparse array is to convert a dense array into a sparse array using the sparse
function:
You can go in the other direction using the Array
constructor. https://teens-software.mystrikingly.com/blog/waves-crack-os-x. The issparse
function can be used to query if a matrix is sparse.
Arithmetic operations on sparse matrices also work as they do on dense matrices. Indexing of, assignment into, and concatenation of sparse matrices work in the same way as dense matrices. Indexing operations, especially assignment, are expensive, when carried out one element at a time. In many cases it may be better to convert the sparse matrix into (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.
Arithmetic operations on sparse matrices also work as they do on dense matrices. Indexing of, assignment into, and concatenation of sparse matrices work in the same way as dense matrices. Indexing operations, especially assignment, are expensive, when carried out one element at a time. In many cases it may be better to convert the sparse matrix into (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.
The following table gives a correspondence between built-in methods on sparse matrices and their corresponding methods on dense matrix types. In general, methods that generate sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given 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.
Details can be found in the Sparse Vectors and Matrices section of the standard library reference.
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 |
Supertype for N
-dimensional sparse arrays (or array-like types) with elements of type Tv
and index type Ti
. SparseMatrixCSC
, SparseVector
and SuiteSparse.CHOLMOD.Sparse
are subtypes of this.
Last Updated on October 12, 2020 by
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Here are some key features:
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– Versatile – Videos, audio, images – Permute can handle it all. Want to create a DVD? No problem, Permute can do that too. With Permute 3.2 or later, you can also convert images into text!
– Customizations – Customize the presets, or just that particular conversion. Increase volume, rotate video, change its resolution, flip it and so on!
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