Mathematical Tools for Physics

11—Numerical Analysis 348

Problems

11.1 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. Find

√

2. (This is the basis for the library
   square root algorithm on some computers.)

11.4 Find all real roots ofe−x= sinxto± 10 −^4.

11.5 The first rootr 1 ofe−ax= sinxis a function of the variablea > 0. Finddr 1 /daata= 1by two means.
First findr 1 for some values ofanear 1 and use a four-point differentiation formula. 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

11.7 The principal value of an integral is(a < x 0 < b)

P

∫b

a

f(x)

x−x 0

dx= lim

→ 0

[∫

x 0 −

a

f(x)

x−x 0

dx+

∫b

x 0 +

f(x)

x−x 0

dx

]

.