Theorem 2.2.1: Convergence of bisection method

Suppose f is continuous on [a,b] and f(a)*f(b) < 0. Then the method generates a sequence {pn} approximating a zero of p of f(x) with |pn-p| < b-a \ 2^n

Bisection method has linear convergence

Theorem 2.3.1: Convergence of newton method

Let f be 2 differentiable on [a,b] & assume f(p) = 0, & f'(p) not If p0 is chosen close to p, then it generates a sequence that converges to p with the limit as n approaches infinity of p-pn+1 / (p-pn)^2 = f''(p)/2f'(p) with quadratic convergence.

Theorem 2.4.1: Convergence of secant method

F is 2 differentiable, f(p) = 0, f'(p) not 0. Initial guesses po & p1 sufficiently close to p, then method converges to p with |f''(p)/2f'(p)|^r1

Theorem 2.6.1: Existence & Uniqueness of fixed point

G is continuous [a,b], g(x) exists for all x, then g has at least one fixed point in [a,b]. If |g(x) - g(y)| < lamda|x-y| where 0<lamda<1, point is unique

Theorem 2.6.2: Convergence of fixed point iteration

If there is existence and uniqueness of fixed point then the iteration Pn = g(pn-1) converges to p, the unique fixed point.

Theorem 2.6.3: Linear convergence of fixed point

If p is a solution of g(x) = x, & g is continuously differentiable with |g'(p)| < 1, Then the iteration converges to p, assuming p0 chosen sufficiently close to p. Convergence is linear as long is g'(p) is not 0

Theorem 2.7.1: Higher order fixed point iteration

P is a solution, g is alpha continuously differentiable, sequence g'(p) = g''(p) = ... galpha(p) then iteration will have order of convergence alpha and = galpha(p)/alpha!

Intermediate value theorem

If f is continuous on [a,b], either f(a) k > f(b), then there must be a c in [a,b] such that f(c) = k

F is continuations and differentiable on [a,b], then there is a point c in [a,b] such that f'(c) = f(b) - f(a) / b-a

Attracting fixed point

Slope of tangent line is flatter than y = x or if |g'(p)| < 1

Slope of tangent line is stepper than y = x or if |g'(p)| > 1

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Math 441 Midterm

Question	Answer
Theorem 2.2.1: Convergence of bisection method	Suppose f is continuous on [a,b] and f(a)*f(b) < 0. Then the method generates a sequence {pn} approximating a zero of p of f(x) with \|pn-p\| < b-a \ 2^n
Corollary 2.2.1	Bisection method has linear convergence
Theorem 2.3.1: Convergence of newton method	Let f be 2 differentiable on [a,b] & assume f(p) = 0, & f'(p) not If p0 is chosen close to p, then it generates a sequence that converges to p with the limit as n approaches infinity of p-pn+1 / (p-pn)^2 = f''(p)/2f'(p) with quadratic convergence.
Theorem 2.4.1: Convergence of secant method	F is 2 differentiable, f(p) = 0, f'(p) not 0. Initial guesses po & p1 sufficiently close to p, then method converges to p with \|f''(p)/2f'(p)\|^r1
Theorem 2.6.1: Existence & Uniqueness of fixed point	G is continuous [a,b], g(x) exists for all x, then g has at least one fixed point in [a,b]. If \|g(x) - g(y)\| < lamda\|x-y\| where 0<lamda<1, point is unique
Theorem 2.6.2: Convergence of fixed point iteration	If there is existence and uniqueness of fixed point then the iteration Pn = g(pn-1) converges to p, the unique fixed point.
Theorem 2.6.3: Linear convergence of fixed point	If p is a solution of g(x) = x, & g is continuously differentiable with \|g'(p)\| < 1, Then the iteration converges to p, assuming p0 chosen sufficiently close to p. Convergence is linear as long is g'(p) is not 0
Theorem 2.7.1: Higher order fixed point iteration	P is a solution, g is alpha continuously differentiable, sequence g'(p) = g''(p) = ... galpha(p) then iteration will have order of convergence alpha and = galpha(p)/alpha!
Intermediate value theorem	If f is continuous on [a,b], either f(a) < k < f(b) or f(a) > k > f(b), then there must be a c in [a,b] such that f(c) = k
Mean value theorem	F is continuations and differentiable on [a,b], then there is a point c in [a,b] such that f'(c) = f(b) - f(a) / b-a
Attracting fixed point	Slope of tangent line is flatter than y = x or if \|g'(p)\| < 1
Repelling fixed pint	Slope of tangent line is stepper than y = x or if \|g'(p)\| > 1

Created by: slpause

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