BUILT-IN MATHEMATICA SYMBOL

SparseArray

yields a sparse array in which values

appear at positions

.

SparseArray

yields the same sparse array.

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

SparseArray

yields a sparse array representing a

array.

SparseArray

yields a sparse array in which unspecified elements are taken to have value val.

MORE INFORMATION

By default, SparseArray takes unspecified elements to be .

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

Normal gives the ordinary array corresponding to a sparse array object.

ArrayRules gives the list of rules .

The elements in SparseArray need not be numeric.

The position specifications can contain patterns.

SparseArray gives a d×d identity matrix.

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

With rules the are evaluated separately for each set of indices that match .

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.

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 nondefault 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:

Construct a sparse matrix with exact integer values:

N[s] is the same as

:

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

:

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: