15c challenge: accuracy of a complex equation system

[J-F Garnier]

Thanks Albert for your HP-71B solution.

My solution is based on considerations on the physical system but is not so different as we will see later.

As Werner noted, the condition number of the problem is very high, and is due to the presence of the R2=1E6 values in two rows of the matrix:



And here starts a kind of paradoxical situation:
- from a physical point of view, when R2 becomes higher and higher, it is more and more negligible in the electrical circuit,
- but from a mathematical point of view, the matrix becomes more and more ill-conditioned because the two row 2 and 3 becomes closer and closer.

This lead me to think that the mathematical description of the problem was not relevant for high values of R2.
If R2 becomes negligible , then the two central loops can be replaced by a single one (in red below).
So to get the better representation for high R2 values, I replaced one of these loops with this one in red, and keep another one of the two.
The system description becomes:



and the problem can be expressed numerically by the complex matrix equation:

| (100,-116.6666667) (0,266.6666667)    (0,0)             (0,0)              |   | I1 |   | 10 |
| (0,266.6666667)    (0,-116.6666667)   (0,147.3333333)   (0,-150)           |   | I2 |   | 0  |
| (0,0)              (-1E6,0)           (1E6,147.3333333) (0,-150)           | x | I3 | = | 0  |
| (0,0)              (0,0)              (0,-150)          (1E5,147.3333333)  |   | I4 |   | 0  |

This system can be easily solved, and the 15c results become:

I1 =  1.995795134 e-4  ,  4.096399080 e-3
I2 = -1.448833618 e-3  , -3.563298307 e-2
I3 = -1.454083172 e-3  , -3.563276083 e-2
I4 =  5.344581171 e-5  , -2.259868254 e-6

now very close to the Free42 reference solution:

I1 =  1.995795134 e-4  ,  4.096399075 e-3
I2 = -1.448833616 e-3  , -3.563298308 e-2
I3 = -1.454083170 e-3  , -3.563276083 e-2
I4 =  5.344581171 e-5  , -2.259868251 e-6

The problem can be solved on a 41C + Advantage Pac too.
Here are the corresponding results:

I1 =  1.995795132 e-4  ,  4.096399077 e-3
I2 = -1.448833617 e-3  , -3.563298309 e-2
I3 = -1.454083171 e-3  , -3.563276084 e-2
I4 =  5.344581172 e-5  , -2.259868252 e-6

that are globally as good as on the 15C, but different.
Indeed, the matrix functions of the Advantage Pac are based on code from the german CCD module, not on HP code from the 15C.

A last remark, some of you surely already noticed that all my considerations on the physical system just lead to add the row 2 and 3 of the initial system matrix, and replace one of the rows with it.
Albert added two columns and I added two rows. The math and physical points of view meet at the end!

J-F