Consider the linear system *\bold{A} x = b* where *\bold{A} \in R\textsuperscript{m x n}*, *x \in R\textsuperscript{n}*, and *b \in R\textsuperscript{m}*. The technique of least squares proposes to compute *x* so that the sum of squared residuals is minimized. `NNLS`

solves the least squares problem *\min{||\bold{A} x = b||\textsuperscript{2}}* subject to the constraint *x ≥ 0*. This implementation of the Sequential Coordinate-wise Algorithm uses a sparse input matrix *\bold{A}*, which makes it efficient for large sparse problems.

1 2 3 4 5 |

`A` |
List representing the sparse matrix with integer components i and j, numeric component x. The fourth component, dimnames, is a list of two components that contains the names for every row (component 1) and column (component 2). |

`b` |
Numeric matrix for the set of observed values. (See details section below.) |

`precision` |
The desired accuracy. |

`processors` |
The number of processors to use, or |

`verbose` |
Logical indicating whether to display progress. |

The input *b* can be either a matrix or a vector of numerics. If it is a matrix then it is assumed that each column contains a set of observations, and the output *x* will have the same number of columns. This allows multiple NNLS problems using the same *\bold{A}* matrix to be solved simultaneously, and greatly accelerates computation relative to solving each sequentially.

A list of two components:

`x` |
The matrix of non-negative values that best explains the observed values given by |

`res` |
A matrix of residuals given by |

Franc, V., et al. (2005). Sequential coordinate-wise algorithm for the non-negative least squares problem. Computer Analysis of Images and Patterns, 407-414.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
# unconstrained least squares:
A <- matrix(c(1, -3, 2, -3, 10, -5, 2, -5, 6), ncol=3)
b <- matrix(c(27, -78, 64), ncol=1)
x <- solve(crossprod(A), crossprod(A, b))
# Non-negative least squares:
w <- which(A > 0, arr.ind=TRUE)
A <- list(i=w[,"row"], j=w[,"col"], x=A[w],
dimnames=list(1:dim(A)[1], 1:dim(A)[2]))
x_nonneg <- NNLS(A, b)
# compare the unconstrained and constrained solutions:
cbind(x, x_nonneg$x)
# the input value "b" can also be a matrix:
b2 <- matrix(b, nrow=length(b), ncol=2) # repeat b in two columns
x_nonneg <- NNLS(A, b2) # solution is repeated in two output columns
``` |

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