date
on Mon Oct 25 02:08:57 EDT 2010
|\^/| Maple 10 (SUN SPARC SOLARIS)
._|\| |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2005
\ MAPLE / All rights reserved. Maple is a trademark of
<____ ____> Waterloo Maple Inc.
| Type ? for help.
## 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>, , <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.
================================================================