Bisection Method in One Dimension

zero-finding problem:

Given a scalar function F(x) : R → R, find a point x∗ ∈ R s.t.
F(x∗) = 0.

Although not all our problems are immediately viewed in this form we can always rewrite them in this way.

Convergence criteria(标准)

Possible conditions to satisfy:
|xn+1 −x∗| < |xn −x∗|
ie. we are getting closer to the root at each step

|F(xn+1)| < |F(xn)|
ie. the function F(x) is reduced at each step

• These criteria are distinct
• one does not imply the other
• Different algorithms may satisfy one of these, rarely both, and often neither

Convergence rate

• converges linearly(线性收敛): Assume the sequence x0, x1, . . . , xn converges to x∗.

Convergence rate

if there exists 0 < α < 1 and satisfied this formula, Here α is the rate of convergence.
i.e. the error is (eventually) reduced by a constant factor of α after each iteration

If α = 1 the sequence converges sublinearly*(亚线性).

• Sequence converges superlinearly(序列超线性收敛)
  if for some q > 1 and α > 0
  
  Sequence converges superlinearly
  
  If q = 2, we say it converges quadratically(呈二次方收敛).

The Bisection Method

• condition

• 1)the function F(x) is continuous

• 2)Assume we know two points xL and xR, such that
  F(xL) F(xR) ≤ 0
  called the bracket condition for the bracket [xL , xR ]

• 3)the Intermediate Value Theorem: This implies that there is a solution x∗ ∈ [xL,xR], since the function changes sign over that interval
  

  The Bisection Method

• Bisection Method is converges linearly and its converges rate is 1/2

• Note: The convergence is not monotone(单调) in general.
  i.e. it can happen that for some steps n we have |F(xn+1)| > |F(xn)|.

• The upper bound above guarantees that eventually lim n→∞ x Cn = x∗ so that limn→∞ F(xCn) = 0.

• Pros and cons of bisection method

• Performance:
  Guaranteed to converge to x∗
  Slow (linear convergence rate)

Other issues:

• We require an initial bracket (2 values), not just an initial guess (1 value)
• In practice we may have to search for a bracket given one point
• The initial bracket [xL , xR ] may contain more than one zero and it is not clear which it will compute

The difference of Bisection Method and Newton Method

• The Bisection Method is usually stopped when |b−a| < TOLx
  for a bracket [a, b].
• Newton’s Method is usually stopped when
  |F(x)| < TOLF
  TOLx and TOLF are appropriately chosen tolerances