A multidimensional array in MATLAB® is an array with more than two dimensions. In a matrix, the two dimensions are represented by rows and columns.
= 7. 6. 5. 4. 3. 2. 1 = 5,040. To calculate permutations, we use the equation n P r, where n is the total number of choices and r is the amount of items being selected. Permutation Problem 1. Choose 3 horses from group of 4 horses. In a race of 15 horses you beleive that you know the best 4 horses and that 3 of them will finish in the top spots: win, place and show (1st, 2nd and 3rd). So out of that set of 4 horses you want to pick the subset of 3 winners and the order in which they finish.
Each element is defined by two subscripts, the row index and the column index. Multidimensional arrays are an extension of 2-D matrices and use additional subscripts for indexing. A 3-D array, for example, uses three subscripts. The first two are just like a matrix, but the third dimension represents pages or sheets of elements.
Creating Multidimensional Arrays
You can create a multidimensional array by creating a 2-D matrix first, and then extending it. For example, first define a 3-by-3 matrix as the first page in a 3-D array.
Permute 2 2 1 7 X 2
Now add a second page. To do this, assign another 3-by-3 matrix to the index value 2 in the third dimension. The syntax
A(:,:,2)
uses a colon in the first and second dimensions to include all rows and all columns from the right-hand side of the assignment. The
cat
function can be a useful tool for building multidimensional arrays. For example, create a new 3-D array B
by concatenating A
with a third page. The first argument indicates which dimension to concatenate along.Another way to quickly expand a multidimensional array is by assigning a single element to an entire page. For example, add a fourth page to
B
that contains all zeros. ![Permute 2 2 1 7 Permute 2 2 1 7](https://i.ytimg.com/vi/_vRoDF69oYs/maxresdefault.jpg)
Accessing Elements
To access elements in a multidimensional array, use integer subscripts just as you would for vectors and matrices. For example, find the 1,2,2 element of
A
, which is in the first row, second column, and second page of A
. Use the index vector
[1 3]
in the second dimension to access only the first and last columns of each page of A
. Splayerx 4 1 5 x 2.To find the second and third rows of each page, use the colon operator to create your index vector.
Manipulating Arrays
Elements of multidimensional arrays can be moved around in many ways, similar to vectors and matrices.
reshape
, permute
, and squeeze
are useful functions for rearranging elements. Consider a 3-D array with two pages. Reshaping a multidimensional array can be useful for performing certain operations or visualizing the data. Use the
reshape
function to rearrange the elements of the 3-D array into a 6-by-5 matrix.reshape
operates columnwise, creating the new matrix by taking consecutive elements down each column of A
, starting with the first page then moving to the second page. Permutations are used to rearrange the order of the dimensions of an array. Consider a 3-D array
M
.Permute 2 2 1 7 Exe
Use the
permute
function to interchange row and column subscripts on each page by specifying the order of dimensions in the second argument. The original rows of M
are now columns, and the columns are now rows.Similarly, interchange row and page subscripts of
M
.When working with multidimensional arrays, you might encounter one that has an unnecessary dimension of length 1. The
squeeze
function performs another type of manipulation that eliminates dimensions of length 1. For example, use the repmat
function to create a 2-by-3-by-1-by-4 array whose elements are each 5, and whose third dimension has length 1. Use the
squeeze
function to remove the third dimension, resulting in a 3-D array.Related Topics
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.
Atv flash (black) 2 5 download free. All operations on sparse matrices are carefully implemented to exploit the CSC data structure for performance, and to avoid expensive operations.
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:![7th 7th](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Symmetric_group_4%3B_permutation_list%3B_Steinhaus%E2%80%93Johnson%E2%80%93Trotter.svg/360px-Symmetric_group_4%3B_permutation_list%3B_Steinhaus%E2%80%93Johnson%E2%80%93Trotter.svg.png)
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: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:Permute 2 2 1 7th
You can go in the other direction using the
Array
constructor. 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.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.