Iterative improvement of the inverse matrix1
Begin with the basic set of equations
 
Mc  b .
Unfortunately the best inverse that we can find is only approximate
 

c A  M A1b
Substitute these into the original equations

 

M cA    b
 

M  b  Mc A

 
d  b  Mc A
 

  M A1d
Writing this out in a bit more detail


 
c  c A    M a1b  M a1d

 

 M a1b  M a1 b  MM a1b



 M a1  M a1 I  MM a1 b
b g
d i
d i
d
d
i
it
o
which shows that to this approximation




M M M d
I  MM i
1
1
a
1
a
1
a
Note that until the ( ) becomes zero, that each new M-1 can be taken equal to Ma-1. The matrix multiplication is such
n
that each iteration requires n3 operations. -- M i , j   Ai ,k Bk , j n operations for each of n2 elements. This should be
k 1
compared to the n3 operations required to invert the matrix originally.
1
V.N. Fadeeva, translated by C.D. Benster, Computational Methods of Linear Algebra, Dover (1959) pp 117-127