LA_GM_LINEAR_MODEL

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

minimizex ||y||2 with 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, minimizex ||B -1(Ax - d)||2.
• If B is the identity matrix then the problem reduces to the ordinary linear least-squares problem, minimizex ||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

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:

```PRO ExLA_GM_LINEAR_MODEL
; 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
END
```

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
```

Introduced 5.6