3.1 Shipley-Coleman Matrix Inversion.
3.1 Shipley-Coleman Matrix Inversion
3.2 Mathematical Derivation
3.3 Procedure for Computer Implementation
3.4 Computer Program (SHIPLEY.C)
3.5 Example Calculation Output

This inversion does not augment a matrix by a unity as in Gauss-Jordan. Inversion is being done utilizing the matrix itself. It is an "in-place" inversion, just like the "Gauss-Jordan In-place Inversion". Unlike the latter, the formula derivation is simple. We will use this inversion routine in full matrix even for production grade software because matrix size is just 20x20. This routine is used to invert the primitive coupling impedance matrix [z]. See Ref. 9.

3.2 Mathematical derivation.
From A*x=b, we would like to manipulate A to arrive on ,
where A is a n x n matrix, and b and x are vectors of n size.
Consider a size 3 x 3.

Loop 1: From Eq. 1 transfer x1 to RHS (right hand side)
--- (4)
substitute Eq. 4 to Eq. 2, i.e.,

collecting terms,
--- (5)
Also substitute Eq. 4 to Eq. 3,

collecting terms,
--- (6)
Rewriting Eqs. 4,5,6 in matrix form;

Multiply column 1 by -1 and by -1 does not alter the matrix. The reason for such will be made clear during the programming aspect.
After the necessary calculations, each element in the A matrix would change as indicated by the matrix above. It can be again rewritten as;

where the elements Ai,j are now different from Eqs. 1,2,3.

Loop 2: From Eq. 8, solve for x2,



Substitute Eq. 10 to Eq. 7,

Collecting terms,

Substitute Eq. 10 to Eq. 9,

Collecting terms,

Writing in matrix form and multiplying column 2 by -1 and b2 by -1,

--- Eq. (11)

Doing the same for Loop 3 will result into the form,
--- Eq. (12)

where from , Eq. 12 can be transformed as [-C]*[b] =[x] by multiplying [b] by -1 and[C] by -1, so .

3.3 Procedure for Computer Implementation.
Generally, with Shipley inversion you can search the largest diagonal to act as a pivot, similar to Gauss-Jordan in-place inversion. However, in short circuit applications, there is no need for such diagonal search. In power system analysis, diagonal elements of any matrix is said to be dominant. Its value is greater than the off diagonal elements. The code to implement the method is given below. Note on the minus sign as previously done in the derivation.

/**********************/ /*Shipley_Inversion();*/ CODE SNIPPET /**********************/ 1: for (k=1;k<=z_size;k++) 2: // the pivot element 3: { A[k][k]= -1/A[k][k]; 4: 5: //the pivot column 6: for (i=1;i<=z_size;i++) 7: if (i!=k) A[i][k]*=A[k][k]; 8: 9: //elements not in a pivot row or column 10: for (i=1;i<=z_size;i++) 11: if (i!=k) 12: for (j=1;j<=z_size;j++) 13: if (j!=k) 14: A[i][j]+=A[i][k]*A[k][j]; 15: 16: //elements in a pivot row 17: for (i=1;i<=z_size;i++) 18: if (i!=k) 19: A[k][i]*=A[k][k]; 20: } 21: 22: //change sign 23: for (i=1;i<=z_size;i++) /*reverse sign*/ 24: for (j=1;j<=z_size;j++) 25: A[i][j]=-A[i][j];

We can understand the code segment by comparing Eq. (11) with the pictorial representation and the code. The inversion has the outermost k-loop. To minimize the amount of arithmetic operations, the pivot has its negative reciprocal stored at line_3. Then computed next is the pivot column except of course the pivot A[k][k], that is the test on line_7. The pivot row is not computed next, because its value without the denominator A[k][k] is needed. Otherwise, we would have the denominator twice for the elements not in the pivot-rows and columns. In line_14 pivot-col A[i][k] had been an updated value and pivot-row A[k][j]) is not. While Eq. (11) says it is a divide for pivot row, Line_19 is a multiplication and also with Lines_14 and _7 because we have done the division already in Line_3.

3.4 Computer Program (SHIPLEY.C), Input/Output