Built-in Mathematica Symbol

SparseArray

SparseArray[{pos₁->val₁, pos₂->val₂, ...}]
yields a sparse array in which values val_i appear at positions pos_i.

SparseArray[{pos₁, pos₂, ...}->{val₁, val₂, ...}]
yields the same sparse array.

SparseArray[list]
yields a sparse array version of list.

SparseArray[data, {d₁, d₂, ...}]
yields a sparse array representing a d₁×d₂×... array.

SparseArray[data, dims, val]
yields a sparse array in which unspecified elements are taken to have value val.

MORE INFORMATION

By default, SparseArray takes unspecified elements to be 0.

SparseArray[data, ...] is always converted to an optimized standard form with structure SparseArray[Automatic, dims, val, ...].

Normal[SparseArray[...]] gives the ordinary array corresponding to a sparse array object.

ArrayRules[SparseArray[...]] gives the list of rules {pos₁->val₁, pos₂->val₂, ...}.

The elements in SparseArray need not be numeric.

The position specifications pos_i can contain patterns.

SparseArray[{{i_, i_}->1}, {d, d}] gives a dd identity matrix.

Rules of the form Band[...]->vals specify values on bands in the sparse array.

With rules pos_i:>val_i the val_i are evaluated separately for each set of indices that match pos_i.

SparseArray[list] requires that list be a full array, with all parts at a particular level being lists of the same length.

The individual elements of a sparse array cannot themselves be lists.

SparseArray[rules] yields a sparse array with dimensions exactly large enough to include elements whose positions have been explicitly specified.

SparseArray[rules, Automatic, val] takes unspecified elements to have value val.

List and matrix operations are typically set up to work as they do on Normal[SparseArray[...]].

Functions with attribute Listable are automatically threaded over the individual elements of the ordinary arrays represented by SparseArray objects.

Part extracts specified parts of the array represented by a SparseArray object, rather than parts of the SparseArray expression itself.

Functions like Map are automatically applied to components in a SparseArray object.

SparseArray is treated as a raw object by functions like AtomQ, and for purposes of pattern matching.

Dimensions gives the dimensions of a sparse array.

The standard output format for a sparse array indicates the number of non-default elements and the total dimensions.

EXAMPLES

CLOSE ALL

Basic Examples (1)

Construct a sparse matrix with values at only a few specified positions:

View it as a matrix:

Convert it to an ordinary dense matrix:

Construct a sparse matrix with values at only a few specified positions:

In[1]:=

Out[1]=

View it as a matrix:

In[2]:=

Out[2]//MatrixForm=

Convert it to an ordinary dense matrix:

In[3]:=

Out[3]=

Scope (7)

Make a large sparse vector, matrix and depth-3 array:

Construct a tridiagonal matrix using patterns for indices:

Construct a 10,000 by 10,000 version:

Make a sparse diagonal matrix:

This is equivalent to DiagonalMatrix:

Except that as a sparse matrix, it uses much less memory:

Construct a block diagonal matrix using rules with Band:

Convert an ordinary matrix into a sparse matrix:

Make a rank-4 sparse tensor with values at random positions:

ArrayRules produces the minimal list of rules needed to specify the SparseArray:

Many typical operations work with SparseArray objects as they would for equivalent lists:

Arithmetic works element-wise just as it does for lists:

Matrix products are done with Dot:

Many linear algebra functions are done efficiently with the sparse form:

Many other list commands work automatically:

Generalizations & Extensions (2)

The unspecified elements can have any value:

Symbolic values can be replaced with local definitions:

Construct a sparse matrix with all machine-number values:

This is the same as N[s]:

Applications (4)

Create a list with a single nonzero element:

Plot a list of rules:

Represent a network with an adjacency matrix:

Solve a boundary-value problem

using finite differences:

Properties & Relations (3)

A SparseArray object is Equal to the corresponding ordinary list:

They are not SameQ because the expression structure is different:

For functions f that work with SparseArray objects, typically f[s]

f[Normal[s]]:

This includes all functions with the attribute Listable:

Convert linear expressions to SparseArray objects using CoefficientArrays:

Convert from SparseArray to expressions using Dot:

Possible Issues (6)

If a position is repeated in the rule list for SparseArray, the first instance is used:

SparseArray objects can represent data too large to represent in normal form:

Using Normal will give a SystemException:

Sparse operations do not by default check for cancellation:

Use SparseArray to recompute the sparse structure:

Operations with side effects may give different values when iterating over SparseArray:

With Reap and Sow you can see what elements are accessed:

For a SparseArray object, Part gives parts of the represented list:

The FullForm is a way of reconstructing the object from basic storage information:

It should not be necessary, but if you want, you can get pieces of the full form with patterns:

A SparseArray object is treated as atomic for functions that do not work on the representation:

Cases does not work on the represented matrix:

You can often use the result of ArrayRules to get the information without expanding:

Neat Examples (1)

The game of life: