MATLAB Function Reference

nnls

Nonnegative least squares

Note    The nnls function was replaced by lsqnonneg in Release 11 (MATLAB 5.3). In Release 12 (MATLAB 6.0), nnls displays a warning message and calls lsqnonneg.

Syntax

• x = nnls(A,b)
  x = nnls(A,b,tol)
  [x,w] = nnls(A,b)
  [x,w] = nnls(A,b,tol)
  

Description

x = nnls(A,b) solves the system of equations in a least squares sense, subject to the constraint that the solution vector x has nonnegative elements . The solution x minimizes subject to x >= 0.

x = nnls(A,b,tol) solves the system of equations, and specifies a tolerance tol. By default, tol is: max(size(A))*norm(A,1)*eps.

[x,w] = nnls(A,b) also returns the dual vector w, where when and when .

[x,w] = nnls(A,b,tol) solves the system of equations, returns the dual vector w, and specifies a tolerance tol.

Examples

Compare the unconstrained least squares solution to the nnls solution for a 4-by-2 problem:

• A =
      0.0372    0.2869
      0.6861    0.7071
      0.6233    0.6245
      0.6344    0.6170
  b =
      0.8587
      0.1781
      0.0747
      0.8405
  [A\b nnls(A,b)] =
      -2.5627         0
       3.1108   0.6929
  
  [norm(A*(a\b)-b) norm(A*nnls(a,b)-b)] =
  
       0.6674 0.9118
  

The solution from nnls does not fit as well, but has no negative components.

Algorithm

The nnls function uses the algorithm described in [1], Chapter 23. The algorithm starts with a set of possible basis vectors, computes the associated dual vector w, and selects the basis vector corresponding to the maximum value in w to swap out of the basis in exchange for another possible candidate, until w <= 0.

See Also

\ Matrix left division (backslash)

References

[1]  Lawson, C. L. and R. J. Hanson, Solving Least Squares Problems, Prentice-Hall, 1974, Chapter 23.

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