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inverse_matrix

Computes the inverse of a general matrix using LU factorization.

Prototype

	function inverse_matrix (
		A [*][*] : numeric   
	)

	return_val [dimsizes(A)] :  float or double

Arguments

A two-dimensional (N, M) array.

Return value

An array of the same size, shape and type A.

Description

Using the LAPACK routines DGETRF and DGETRI, this function computes the inverse of a general matrix A(N, M) using the LU factorization method. This method inverts U and then computes inv(A) by solving the system inv(A) * L = inv(U) for inv(A).

The returned inverse provides one way of solving the set of equations A * x = b (i.e. x = inv(A) * b). Thus, if the inverse is known, the vector x can be solved for any b. However, if efficiency is a concern, this is not the preferred method (see solve_linsys).

Examples

Example 1:

Compute the inverse of a 3 x 3 matrix:

                 1  -1  2
           A =   3   0  1
                 1   0  2

In NCL:

   a = (/(/1.0, -1.0, 2.0/), (/ 3.0, 0.0, 1.0/), (/1.0, 0.0, 2.0/)/)

Then:

   aInv = inverse_matrix(a)

      ;
      ;           0   0.4  -0.2
      ;  aInv =  -1    0     1
      ;           0  -0.2   0.6 

   print(aInv)

Variable: aInv
Type: float
Total Size: 36 bytes
            9 values
Number of Dimensions: 2
Dimensions and sizes:   [3] x [3]
Coordinates:
(0,0)      0
(0,1)    0.4
(0,2)   -0.2
(1,0)     -1
(1,1)      0
(1,2)      1
(2,0)      0
(2,1)   -0.2
(2,2)    0.6

The identity matrix (I) is found via I = aInv # a. The "#" operator computes the dot product of two arrays.

Errors

info: = 0: successful exit; < 0: if info = -i, the i-th argument had an illegal value; > 0: if info = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.