LA_GM_LINEAR_MODEL

The LA_GM_LINEAR_MODEL function is used to solve a general Gauss-Markov linear model problem:

minimize_x ||y||₂with constraint d = Ax + By

where A is an m-column by n-row array, B is a p-column by n-row array, and d is an n-element input vector with m ≤ n ≤ m+p.

The following items should be noted:

If A has full column rank m and the array (A B) has full row rank n, then there is a unique solution x and a minimal 2-norm solution y.

If B is square and nonsingular then the problem is equivalent to a weighted linear least-squares problem, minimize_x ||B ^-1(Ax - d)||₂.

If B is the identity matrix then the problem reduces to the ordinary linear least-squares problem, minimize_x ||Ax - d||₂.

LA_ GM_LINEAR_MODEL is based on the following LAPACK routines:

Table 14-8: LAPACK Routine Basis for LA_GM_LINEAR_MODEL
Output Type	LAPACK Routine
Float	sggglm
Double	dggglm
Complex	cggglm
Double complex	zggglm

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

Syntax

Result = LA_GM_LINEAR_MODEL( A, B, D, Y [, /DOUBLE] )

Return Value

The result (x) is an m-element vector whose type is identical to A.

Arguments

A

The m-by-n array used in the constraint equation.

B

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

D

An n-element input vector used in the constraint equation.

Y

Set this argument to a named variable, which will contain the p-element output vector.

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.

Examples

Given the constraint equation d = Ax + By, (where A, B, and d are defined in the program below) the following example program solves the general Gauss-Markov problem:

; Define some example coefficient arrays:
a = [[2, 7, 4], $
     [5, 1, 3], $
     [3, 3, 6], $
     [4, 5, 2]]
b = [[-3, 2], $
     [1, 5], $
     [2, 9], $
     [4, 1]]

; Define a sample left-hand side vector D:
d = [-1, 2, -3, 4]

; Find and print the solution x:
x = LA_GM_LINEAR_MODEL(a, b, d, y)
PRINT, 'LA_GM_LINEAR_MODEL solution:'
PRINT, X
PRINT, 'LA_GM_LINEAR_MODEL 2-norm solution:'
PRINT, Y

When this program is compiled and run, IDL prints:

LA_GM_LINEAR_MODEL solution: 
      1.04668     0.350346     -1.28445 
LA_GM_LINEAR_MODEL 2-norm solution: 
     0.151716    0.0235733

Version History


5.6	Introduced