# Sporth

sporth | |

Returns a sparse orthonormal basis for the range | |

Other toolboxes required | none |
---|---|

Related functions | spnull |

Function category | Helper functions |

License | license_sporth.txt |

This is a helper function that only exists to aid other functions in QETLAB. If you are an end-user of QETLAB, you likely will never have a reason to use this function. |

` sporth` is a function that computes an orthonormal basis for the range of a full or sparse matrix. When the matrix is sparse, this computation is performed via the QR decomposition and is typically much faster than using

`orth(full(S))`. This function is useful in particular for computing the rank of a sparse matrix without having to use the (slow)

`rank(full(S))`or the (often inaccurate)

`sprank(S)`.

## Syntax

`Q = sporth(S)``Q = sporth(S,TOL)``[Q,r] = sporth(S,TOL)`

## Argument descriptions

### Input arguments

`S`: The matrix to have its range computed.`TOL`(optional, default`norm(S,'fro') * eps(class(S))`): The numerical tolerance used.

### Output arguments

`Q`: A matrix whose columns form an orthonormal basis for the range of`S`.`r`(optional): The rank of`S`.

## Examples

The following example gives a 4-by-4 matrix whose range is spanned by the two vectors $[1,0,0,0]^T$ and $[0,0,1,0]^T$:

```
>> S = sparse(4,4);
>> S(1,1) = 1; S(3,2) = 1;
>> sporth(S)
ans =
(1,1) 1
(3,2) 1
```

Note that the output is sparse because `S` is sparse. If the input is full then the output will be full as well:

```
>> sporth(full(S))
ans =
1 0
0 0
0 -1
0 0
```

If we just want to compute the rank of `S`, we can do the following:

```
>> [~,r] = sporth(S)
r =
2
```

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.

## External links

- Sparse null space and orthogonal: The source of this file on MATLAB File Exchange