LA_GM_LINEAR

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:

Output Type	LAPACK Routine
Float	sggglm
Double	dggglm
Complex	cggglm
Double complex	zggglm

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

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.

Version History

5.6	Introduced

Resources and References

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