date
                on Mon Oct 25 02:08:57 EDT 2010

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## We've empirically measured a matrix, M, but there is some uncertainty
in some of its entries.  For example, we /think/ the (1,3) entry of M 
is 1, but we allow for some uncertainty by writing it as "1+c", 
where c lies between, say, -1/100 and 1/100.

Consider a SoE (system of equations) with M as the coeff-matrix; a system
OTForm M*X = T, where each of X and T is a 3x1 column vector.  The
target T is known, and the X is what we solve for.  However, even in the
"known" T, there is some uncertainty in its "-1" entry, so we write that
as "-1 + y".

GOAL: We'd like to estimate how the uncertainty in M and T affect the entries of X.

> M := <<1 | 0 | 1 + c>, <d - 1 |  -2 | 1>, <7 | -1  | h>>; T := << -1 + y , 3, 0>>;

       [  1       0    1 + c]          [-1 + y]
       [                    ]          [      ]
  M := [d - 1    -2      1  ] ,   T := [  3   ]
       [                    ]          [      ]
       [  7      -1      h  ]          [  0   ]

DetM := Det(M); 
                     DetM := -2 h + 16 - d - d c + 15 c

> MT1 := < T | M[1..3,2] | M[1..3,3] >   ;   x1 := Det(MT1) / DetM ;

         [-1 + y     0    1 + c]
         [                     ]             2 h - 4 - 2 y h + y - 3 c
  MT1 := [  3       -2      1  ] ,     x1 = --------------------------
         [                     ]            -2 h + 16 - d - d c + 15 c
         [  0       -1      h  ]

> MT2 := <  M[1..3,1] | T | M[1..3,3] >   ;   x2 := Det(MT2) / DetM ;

       [1  -1+y  1+c]
       [            ]        2 h + d h - h d y + y h - 28 + 7 y - 21 c
 MT2 = [d-1  3     1],  x2 = -----------------------------------------
       [            ]                -2 h + 16 - d - d c + 15 c       
       [7   0     h]

> MT3 := <  M[1..3, 1] | M[1..3, 2] | T >   ;   x3 := Det(MT3) / DetM ;

	[  1       0    -1 + y]
	[                     ]              -12 + d - d y + 15 y   
  MT3 = [d - 1    -2      3   ],    x3 := --------------------------
	[                     ]	          -2 h + 16 - d - d c + 15 c
	[  7      -1      0   ]

## So each x_j is a rational-function of "uncertainty tuple" (c,d,h,y),
and we can apply the techniques of differential calculus to estimate the
resulting uncertainty in x1, x2 and x3.

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