Spnull

spnull
Returns a sparse orthonormal basis for the null space
Other toolboxes required	none
Related functions	sporth
Function category	Helper functions
License	license_spnull.txt

spnull is a function that computes an orthonormal basis for the null space 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 null(full(S)).

Syntax

• Z = spnull(S)
• Z = spnull(S,varargin)

Argument descriptions

• S: The matrix to have its null space computed.
• varargin (optional): Extra arguments that, if S is full, will be passed to MATLAB's null function.

Examples

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

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

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

>> spnull(full(S))
 
ans =
 
     0     0
     0     0
     1     0
     0     1

Source code

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

1. function Z = spnull(S, varargin)

2. % Z = SPNULL(S)

3. % returns a sparse orthonormal basis for the null space of S, that is,

4. % S*Z has negligible elements, and Z'*Z = I

5. %

6. % If S is sparse, Z is obtained from the QR decomposition.

7. % Otherwise, Z is obtained from the SVD decomposition

8. %

9. % Bruno Luong <brunoluong@yahoo.com>

10. % History

11. %   10-May-2010: original version

12. %

13. % See also SPORTH, NULL, QR, SVD, ORTH, RANK

15. if issparse(S)

16.     [m n] = size(S);

17.     try

18.         [Q R E] = qr(S.'); %#ok %full QR

19.         if m > 1

20.             s = diag(R);

21.         elseif m == 1

22.             s = R(1);

23.         else

24.             s = 0;

25.         end

26.         s = abs(s);

27.         tol = norm(S,'fro') * eps(class(S));

28.         r = sum(s > tol);

29.         Z = Q(:,r+1:n);

30.     catch %#ok

31.         % sparse QR is not available on old Matlab versions

32.         err = lasterror(); %#ok

33.         if strcmp(err.identifier, 'MATLAB:maxlhs')

34.             Z = null(full(S), varargin{:});

35.         else

36.             rethrow(err);

37.         end

38.     end

39. else % Full matrix

40.     Z = null(S, varargin{:});

41. end

43. end

External links

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