Nonnegative least squares
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 = 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 and .
[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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