IDL Reference Guide: Procedures and Functions

LA_LEAST_SQUARE_EQUALITY

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

Minimize_x ||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.

Syntax

Result = LA_LEAST_SQUARE_EQUALITY( A, B, C, D [, /DOUBLE] [, RESIDUAL=variable] )

Return Value

The result (x) is an n-element vector.

Arguments

A

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

B

The n-by-p array used in the equality constraint.

C

An m-element input vector containing the right-hand side of the least-squares system.

D

A p-element input vector containing the right-hand side of the equality constraint.

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.

RESIDUAL

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.

Examples

Given the following system of equations:

2t + 5u + 3v + 4w = 9  
7t +  u + 3v + 5w = 1   
4t + 3u + 6v + 2w = 2  

with constraints,

-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