11—Numerical Analysis 290Problems11.1 Show that a two point extrapolation formula is
f(0)≈ 2 f(−h)−f(− 2 h) +h^2 f′′(0).
11.2 Show that a three point extrapolation formula is
f(0)≈ 3 f(−h)− 3 f(− 2 h) +f(− 3 h) +h^3 f′′′(0).
11.3 Solvex^2 −a= 0by Newton’s method, showing graphically that in this case, no matter what the
initial guess is (positive or negative), the sequence will always converge. Draw graphs. Find
√
- (This
is the basis for the library square root algorithm on some computers.)
11.4 Find all real roots ofe−x= sinxto± 10 −^4. Ans: 0. 588533 , π− 0. 045166 , 2 π+ 0. 00187 ...
11.5 The first rootr 1 ofe−ax= sinxis a function of the variablea > 0. Finddr 1 /daata= 1by
two means. (a) First findr 1 for some values ofanear 1 and use a four-point differentiation formula.
(b) Second, use analytical techniques on the equation to solve fordr 1 /daand evaluate the derivative
in terms of the known value of the root from the previous problem.
11.6 Evaluateerf(1) =√^2 π
∫ 1
0 dte
−t^2 Ans: 0.842736 (more exact: 0.842700792949715)11.7 The principal value of an integral is(a < x 0 < b)
P
∫baf(x)
x−x 0
dx= lim
→ 0[∫
x 0 −af(x)
x−x 0
dx+
∫bx 0 +f(x)
x−x 0
dx
]
.
(a) Show that an equal spaced integration scheme to evaluate such an integral is (using points 0 ,±h)
P
∫+h−hf(x)
x
dx=f(h)−f(−h)−
2
9
h^3 f′′′(0).
(b) Also, an integration scheme of the Gaussian type is
√
3[
f(h
/√
3)−f(−h
/√
3)
]
+
h^5
675
fv(0).
11.8 Devise a two point Gaussian integration with errors for the class of integrals
∫+∞−∞dxe−x
2f(x).
Find what standard polynomial has roots at the points wherefis to be evaluated.
Ans:^12
√
π
[
f(− 1 /
√
2) +f(1/
√
2)
]
11.9 Same as the preceding problem, but make it a three point method.
Ans: