The LA_LEAST_SQUARES function is used to solve the linear least-squares problem:

Minimize_{x }||*Ax - b||*_{2}

where *A* is a (possibly rank-deficient) *n*-column by *m*-row array, *b* is an *m*-element input vector, and *x* is the* n*-element solution vector. There are three possible cases:

- If
*m*≥*n*and the rank of*A*is*n*, then the system is overdetermined and a unique solution may be found, known as the least-squares solution. - If
*m*<*n*and the rank of*A*is*m*, then the system is under determined and an infinite number of solutions satisfy*Ax - b = 0*. In this case, the solution is found which minimizes ||x||_{2}, known as the minimum norm solution. - If
*A*is rank deficient, such that the rank of*A*is less than MIN(*m*,*n*), then the solution is found which minimizes both |*|Ax - b||*_{2}and*||x||*_{2}, known as the minimum-norm least-squares solution.

The LA_LEAST_SQUARES function may also be used to solve for multiple systems of least squares, with each column of *b* representing a different set of equations. In this case, the result is a* k*-by-*n* array where each of the *k* columns represents the solution vector for that set of equations.

LA_ LEAST_SQUARES is based on the following LAPACK routines:

Output Type |
LAPACK Routines |

Float |
sgels, sgelsy, sgelss, sgelsd |

Double |
dgels, dgelsy, dgelss, dgelsd |

Complex |
cgels, cgelsy, cgelss, cgelsd |

Double complex |
zgels, zgelsy, zgelss, zgelsd |

## Examples

Given the following under determined system of equations:

2t + 5u + 3v + 4w = 3

7t + u + 3v + 5w = 1

4t + 3u + 6v + 2w = 6

The following program can be used to find the solution:

`; Define the coefficient array:`

a = [[2, 5, 3, 4], $

[7, 1, 3, 5], $

[4, 3, 6, 2]]

`; Define the right-hand side vector b:`

b = [3, 1, 6]

`; Find and print the minimum norm solution of a:`

`x = LA_LEAST_SQUARES(a, b)`

PRINT, 'LA_LEAST_SQUARES solution:', x

IDL prints:

LA_LEAST_SQUARES solution:

-0.0376844 0.350628 0.986164 -0.409066

## Syntax

*Result* = LA_LEAST_SQUARES( *A*, *B* [, /DOUBLE] [, METHOD=*value*] [, RANK=*variable*] [, RCONDITION=*value*] [, RESIDUAL=*variable*] [, STATUS=*variable*] )

## Return Value

The result is an *n*-element vector or *k*-by-*n* array.

## Arguments

### A

The *n*-by-*m* array used in the least-squares system.

### B

An *m*-element input vector containing the right-hand side of the linear least-squares system, or a *k*-by-*m* array, where each of the *k* columns represents a different least-squares system.

## Keywords

### DOUBLE

Set this keyword to use double-precision for computations and to return a double-precision (real or complex) result. Set DOUBLE = 0 to use single-precision for computations and to return a single-precision (real or complex) result. The default is /DOUBLE if* A* is double precision, otherwise the default is DOUBLE = 0.

### METHOD

Set this keyword to indicate which computation method to use. Possible values are:

- METHOD = 0 (the default): Assume that array
*A*has full rank equal to min(*m*,*n*). If*m*≥*n*, find the least-squares solution to the overdetermined system. If*m*<*n*, find the minimum norm solution to the under determined system. Both cases use QR or LQ factorization of*A*. - METHOD = 1: Assume that array
*A*may be rank deficient; use a complete orthogonal factorization of*A*to find the minimum norm least-squares solution. - METHOD = 2: Assume that array
*A*may be rank deficient; use singular value decomposition (SVD) to find the minimum norm least-squares solution. - METHOD = 3: Assume that array
*A*may be rank deficient; use SVD with a divide-and-conquer algorithm to find the minimum norm least-squares solution. The divide-and-conquer method is faster than regular SVD, but may require more memory.

### RANK

Set this keyword to a named variable in which to return the effective rank of *A*. If METHOD = 0 or the array is full rank, then RANK will have the value MIN(*m*,* n*).

### RCONDITION

Set this keyword to the reciprocal condition number used as a cutoff value in determining the effective rank of *A*. Arrays with condition numbers larger than 1/RCONDITION are assumed to be rank deficient. If RCONDITION is set to zero or omitted, then array *A* is assumed to be of full rank. This keyword is ignored for METHOD = 0.

### RESIDUAL

If *m* > *n* and the rank of *A* is *n* (the system is overdetermined), then set this keyword to a named variable in which to return the residual sum-of-squares for *Result*. If *B* is an *m*-element vector then RESIDUAL will be a scalar; if *B* is a *k*-by-*m *array then RESIDUAL will be a *k*-element vector containing the residual sum-of-squares for each system of equations. If *m* ≤* n* or* A* is rank deficient (rank < *n*) then the values in RESIDUAL will be zero.

### STATUS

Set this keyword to a named variable that will contain the status of the computation. Possible values are:

- STATUS = 0: The computation was successful.
- STATUS > 0: For METHOD=2 or METHOD=3, this indicates that the SVD algorithm failed to converge, and STATUS off-diagonal elements of an intermediate bidiagonal form did not converge to zero. For METHOD=0 or METHOD=1 the STATUS will always be zero.

## Version History

5.6 |
Introduced |

## Resources and References

For details see Anderson et al., *LAPACK Users' Guide*, 3rd ed., SIAM, 1999.

## See Also

LA_GM_LINEAR_MODEL, LA_LEAST_SQUARE_EQUALITY