User Manual

170 Appendix: Accuracy of Numerical Calculations
170
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000,52000
03.000,5000002.000
4503.000,50000,500
4503.000,5000050000020 ,.
X
and
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
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692300001923070000
95192076480005000
980770764803000500000200
0005000050
1
.
.,,
.,.,.
qp,,
-
X
Ideally, p = q = 0, but the HP-15C's approximation to X
-1
, namely ⁄ (X), has
q = 9.643.269231 instead, a relative error
,0964.0
)(
1
1

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X
XX
Nearly 10 percent. On the other hand, if X + ΔX differs from X only in its second column
where −50,000 and 50,000 are replaced respectively by −50,000.000002 and 49,999.999998
(altered in the 11th significant digit), then (X + ΔX)
-1
differs significantly from X
-1
only
insofar as p = 0 and q = 0 must be replaced by p = 10,000.00600 ... and q = 9,615.396154 ....
Hence,
 
;196.0
1
1
1

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X
ΔXXX
the relative error in (X + ΔX)
-1
is nearly twice that in ⁄ (X). Do not try to calculate (X +
ΔX)
-1
directly. but use instead the formula
(X − cb
T
)
-1
= X
-1
+ X
-1
cb
T
X
-1
/ (1 − b
T
X
-1
c),
which is valid for any column vector c and row vector b
T
, and specifically for
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0
0
1
1
c
and
 
00000002.00
T
b
.
Despite that
||X
-1
− ⁄ (X)|| < ||X
-1
− (X+ ΔX)
-1
|| ,
it can be shown that no very small end-figure perturbation δX exists for which (X + δX)
-1
matches ⁄ (X) to more than five significant digits in norm.
⁄