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

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.

1. function [Q r] = sporth(S,varargin)

2. % Q = sporth(S)

3. % returns an (sparse) orthonormal basis for the range of S.

4. % That is, Q'*Q = I, the columns of Q span the same space as 

5. % the columns of S, and the number of columns of Q is the 

6. % rank of S.

7. %

8. % [~, r] = sporth(S)

9. % returns the rank of S

10. %

11. % If S is sparse, Q is obtained from the QR decomposition.

12. % Otherwise, Q is obtained from the SVD decomposition

13. %

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

15. % History

16. %   10-May-2010: original version

17. %   13-Dec-2012: added an optional second argument TOL, which specifies the

18. %   tolerance (Nathaniel Johnston <nathaniel@njohnston.ca>)

19. %

20. % See also SPNULL, NULL, QR, SVD, ORTH, RANK

22. if issparse(S)

23.     m = size(S,1);

24.     try

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

26.         if m > 1

27.             s = diag(R);

28.         elseif m == 1

29.             s = R(1);

30.         else

31.             s = 0;

32.         end

33.         s = abs(s);

34.         if(nargin == 1)

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

36.         else

37.             tol = varargin{1};

38.         end

39.         r = sum(s > tol);

40.         Q = Q(:,1:r);

41.     catch %#ok

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

43.         err = lasterror(); %#ok

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

45.             Q = orth(full(S));

46.         else

47.             rethrow(err);

48.         end

49.     end

50. else % Full matrix

51.     Q = orth(S);

52. end

54. r = size(Q,2);

56. end

External links

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