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:
For details see Anderson et al., LAPACK Users' Guide, 3rd ed., SIAM, 1999.
Result = LA_LEAST_SQUARE_EQUALITY( A, B, C, D [, /DOUBLE] [, RESIDUAL=variable] )
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.
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 program:
PRO ExLA_LEAST_SQUARE_EQUALITY ; 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:' PRINT, x END
When this program is compiled and run, IDL prints:
LA_LEAST_SQUARE_EQUALITY solution: 0.651349 2.72695 -1.14638 -0.620036