11—Numerical Analysis 35011.12 The same phenomenon caused by roundoff errors occurs in integration. For any of the integration schemes
discussed here, analyze the dependence on the number of significant figures kept and determine the most accurate
interval. (Surprise?)
11.13 Compute the solution ofy′= 1 +y^2 and check the numbers in the table where that example was given,
( 37 ).
11.14 If in the least square fit to a linear combination of functions, the result is constrained to pass through one
point, so that
∑
αμfμ(x 0 ) =Kis a requirement on theα’s, show that the result becomesa=C−^1[
b+λf 0]
,
wheref 0 is the vectorfμ(x 0 )andλsatisfies
λ〈
f 0 ,C−^1 f 0〉
=K−
〈
f 0 ,C−^1 b〉
.
11.15 Find the variances in the formulas ( 8 ) and ( 10 ) forf′, assuming noisy data. Ans:σ^2 / 2 h^2 , 65 σ^2 / 72 h^2
11.16 Derive Eqs. ( 56 ), ( 57 ), and ( 58 ).
11.17 The Van der Pol equation arises in (among other places) nonlinear circuits and leads to self-exciting
oscillations as in multi-vibrators
d^2 x
dt^2
−(1−x^2 )dx
dt+x= 0.Take=. 3 and solve subject to any non-zero initial conditions. Solve over many periods to demonstrate the
development of the oscillations.
11.18 Find a formula for the numerical third derivative. Cf. (2.13)
11.19 The equation resulting from the secant method, Eq. ( 7 ), can be simplified by placing everything over a
common denominator,
(
f(x 2 )−f(x 1 ))
. Explain why this is a bad thing to do, how it can lead to inaccuracies.