The LA_LEAST_SQUARE_EQUALITY function is used to solve the linear least-squares problem:
Minimizex ||Ax - c||2 with constraint Bx = d
where A is an n-column by m-row array, B is an n-column by p-row array, c is an m-element input vector, and d is an p-element input vector with p ≤ n ≤ m+p. If B has
full row rank p and the array has full column rank n, then a unique solution exists.
LA_ LEAST_SQUARE_EQUALITY is based on the following LAPACK routines:
Given the following system of equations:
2t + 5u + 3v + 4w = 9
7t + u + 3v + 5w = 1
4t + 3u + 6v + 2w = 2
-3t + u + 2v + 4w = -4
2t + 5u + 9v + 1w = 4
find the solution using the following code:
; Define the coefficient array:
a = [[2, 5, 3, 4], $
[7, 1, 3, 5], $
[4, 3, 6, 2]]
; Define the constraint array:
b = [[-3, 1, 2, 4], $
[2, 5, 9, 1]]
; Define the right-hand side vector c:
c = [9, 1, 2]
; Define the constraint right-hand side d:
d = [-4, 4]
; Find and print the minimum norm solution of a:
x = LA_LEAST_SQUARE_EQUALITY(a, b, c, d)
PRINT, 'LA_LEAST_SQUARE_EQUALITY solution:'
0.651349 2.72695 -1.14638 -0.620036
The result (x) is an n-element vector.
The n-by-m array used in the least-squares minimization.
The n-by-p array used in the equality constraint.
An m-element input vector containing the right-hand side of the least-squares system.
A p-element input vector containing the right-hand side of the equality constraint.
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.
Set this keyword to a named variable in which to return a scalar giving the residual sum-of-squares for Result. If n = m + p then RESIDUAL will be zero.
Resources and References
For details see Anderson et al., LAPACK Users' Guide, 3rd ed., SIAM, 1999.