# Sparse matrices (`scipy.sparse`)#

SciPy 2-D sparse array package for numeric data.

Note

This package is switching to an array interface, compatible with NumPy arrays, from the older matrix interface. We recommend that you use the array objects (`bsr_array`, `coo_array`, etc.) for all new work.

When using the array interface, please note that:

• `x * y` no longer performs matrix multiplication, but element-wise multiplication (just like with NumPy arrays). To make code work with both arrays and matrices, use `x @ y` for matrix multiplication.

• Operations such as sum, that used to produce dense matrices, now produce arrays, whose multiplication behavior differs similarly.

• Sparse arrays currently must be two-dimensional. This also means that all slicing operations on these objects must produce two-dimensional results, or they will result in an error. This will be addressed in a future version.

The construction utilities (`eye`, `kron`, `random`, `diags`, etc.) have not yet been ported, but their results can be wrapped into arrays:

```A = csr_array(eye(3))
```

## Contents#

### Sparse array classes#

 `bsr_array`(arg1[, shape, dtype, copy, blocksize]) Block Sparse Row format sparse array. `coo_array`(arg1[, shape, dtype, copy]) A sparse array in COOrdinate format. `csc_array`(arg1[, shape, dtype, copy]) Compressed Sparse Column array. `csr_array`(arg1[, shape, dtype, copy]) Compressed Sparse Row array. `dia_array`(arg1[, shape, dtype, copy]) Sparse array with DIAgonal storage. `dok_array`(arg1[, shape, dtype, copy]) Dictionary Of Keys based sparse array. `lil_array`(arg1[, shape, dtype, copy]) Row-based LIst of Lists sparse array. This class provides a base class for all sparse arrays.

### Sparse matrix classes#

 `bsr_matrix`(arg1[, shape, dtype, copy, blocksize]) Block Sparse Row format sparse matrix. `coo_matrix`(arg1[, shape, dtype, copy]) A sparse matrix in COOrdinate format. `csc_matrix`(arg1[, shape, dtype, copy]) Compressed Sparse Column matrix. `csr_matrix`(arg1[, shape, dtype, copy]) Compressed Sparse Row matrix. `dia_matrix`(arg1[, shape, dtype, copy]) Sparse matrix with DIAgonal storage. `dok_matrix`(arg1[, shape, dtype, copy]) Dictionary Of Keys based sparse matrix. `lil_matrix`(arg1[, shape, dtype, copy]) Row-based LIst of Lists sparse matrix. This class provides a base class for all sparse matrix classes.

### Functions#

Building sparse arrays:

 `diags_array`(diagonals, /, *[, offsets, ...]) Construct a sparse array from diagonals.

Building sparse matrices:

 `eye`(m[, n, k, dtype, format]) Sparse matrix with ones on diagonal `identity`(n[, dtype, format]) Identity matrix in sparse format `kron`(A, B[, format]) kronecker product of sparse matrices A and B `kronsum`(A, B[, format]) kronecker sum of sparse matrices A and B `diags`(diagonals[, offsets, shape, format, dtype]) Construct a sparse matrix from diagonals. `spdiags`(data, diags[, m, n, format]) Return a sparse matrix from diagonals. `block_diag`(mats[, format, dtype]) Build a block diagonal sparse matrix or array from provided matrices. `tril`(A[, k, format]) Return the lower triangular portion of a matrix in sparse format `triu`(A[, k, format]) Return the upper triangular portion of a matrix in sparse format `block`(blocks, *[, format, dtype]) Build a sparse array from sparse sub-blocks `bmat`(blocks[, format, dtype]) Build a sparse array or matrix from sparse sub-blocks `hstack`(blocks[, format, dtype]) Stack sparse matrices horizontally (column wise) `vstack`(blocks[, format, dtype]) Stack sparse arrays vertically (row wise) `rand`(m, n[, density, format, dtype, ...]) Generate a sparse matrix of the given shape and density with uniformly distributed values. `random`(m, n[, density, format, dtype, ...]) Generate a sparse matrix of the given shape and density with randomly distributed values.

 `save_npz`(file, matrix[, compressed]) Save a sparse matrix to a file using `.npz` format. `load_npz`(file) Load a sparse matrix from a file using `.npz` format.

Sparse matrix tools:

 Return the indices and values of the nonzero elements of a matrix

Identifying sparse matrices:

 Is x of a sparse array or sparse matrix type? Is x of a sparse matrix type? Is x of csc_matrix type? Is x of csr_matrix type? Is x of a bsr_matrix type? Is x of lil_matrix type? Is x of dok_array type? Is x of coo_matrix type? Is x of dia_matrix type?

### Submodules#

 `csgraph` Compressed sparse graph routines (scipy.sparse.csgraph) `linalg` Sparse linear algebra (scipy.sparse.linalg)

## Usage information#

There are seven available sparse array types:

1. `csc_array`: Compressed Sparse Column format

2. `csr_array`: Compressed Sparse Row format

3. `bsr_array`: Block Sparse Row format

4. `lil_array`: List of Lists format

5. `dok_array`: Dictionary of Keys format

6. `coo_array`: COOrdinate format (aka IJV, triplet format)

7. `dia_array`: DIAgonal format

To construct an array efficiently, use either `dok_array` or `lil_array`. The `lil_array` class supports basic slicing and fancy indexing with a similar syntax to NumPy arrays. As illustrated below, the COO format may also be used to efficiently construct arrays. Despite their similarity to NumPy arrays, it is strongly discouraged to use NumPy functions directly on these arrays because NumPy may not properly convert them for computations, leading to unexpected (and incorrect) results. If you do want to apply a NumPy function to these arrays, first check if SciPy has its own implementation for the given sparse array class, or convert the sparse array to a NumPy array (e.g., using the `toarray` method of the class) first before applying the method.

To perform manipulations such as multiplication or inversion, first convert the array to either CSC or CSR format. The `lil_array` format is row-based, so conversion to CSR is efficient, whereas conversion to CSC is less so.

All conversions among the CSR, CSC, and COO formats are efficient, linear-time operations.

### Matrix vector product#

To do a vector product between a sparse array and a vector simply use the array `dot` method, as described in its docstring:

```>>> import numpy as np
>>> from scipy.sparse import csr_array
>>> A = csr_array([[1, 2, 0], [0, 0, 3], [4, 0, 5]])
>>> v = np.array([1, 0, -1])
>>> A.dot(v)
array([ 1, -3, -1], dtype=int64)
```

Warning

As of NumPy 1.7, `np.dot` is not aware of sparse arrays, therefore using it will result on unexpected results or errors. The corresponding dense array should be obtained first instead:

```>>> np.dot(A.toarray(), v)
array([ 1, -3, -1], dtype=int64)
```

but then all the performance advantages would be lost.

The CSR format is especially suitable for fast matrix vector products.

### Example 1#

Construct a 1000x1000 `lil_array` and add some values to it:

```>>> from scipy.sparse import lil_array
>>> from scipy.sparse.linalg import spsolve
>>> from numpy.linalg import solve, norm
>>> from numpy.random import rand
```
```>>> A = lil_array((1000, 1000))
>>> A[0, :100] = rand(100)
>>> A[1, 100:200] = A[0, :100]
>>> A.setdiag(rand(1000))
```

Now convert it to CSR format and solve A x = b for x:

```>>> A = A.tocsr()
>>> b = rand(1000)
>>> x = spsolve(A, b)
```

Convert it to a dense array and solve, and check that the result is the same:

```>>> x_ = solve(A.toarray(), b)
```

Now we can compute norm of the error with:

```>>> err = norm(x-x_)
>>> err < 1e-10
True
```

It should be small :)

### Example 2#

Construct an array in COO format:

```>>> from scipy import sparse
>>> from numpy import array
>>> I = array([0,3,1,0])
>>> J = array([0,3,1,2])
>>> V = array([4,5,7,9])
>>> A = sparse.coo_array((V,(I,J)),shape=(4,4))
```

Notice that the indices do not need to be sorted.

Duplicate (i,j) entries are summed when converting to CSR or CSC.

```>>> I = array([0,0,1,3,1,0,0])
>>> J = array([0,2,1,3,1,0,0])
>>> V = array([1,1,1,1,1,1,1])
>>> B = sparse.coo_array((V,(I,J)),shape=(4,4)).tocsr()
```

This is useful for constructing finite-element stiffness and mass matrices.

### Further details#

CSR column indices are not necessarily sorted. Likewise for CSC row indices. Use the `.sorted_indices()` and `.sort_indices()` methods when sorted indices are required (e.g., when passing data to other libraries).