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Material Type: Notes; Class: INTRO NUM ANALYS & COMPUT; Subject: Mathematics; University: San Diego State University; Term: Fall 2002;
Typology: Study notes
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Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Lecture Notes #02 — Calculus Review; Computer Artihmetic and Finite Precision; Algorithms and Convergence; Solutions of Equations of One Variable
Peter Blomgren, 〈[email protected]〉
Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182- http://terminus.sdsu.edu/
Fall 2009 Peter Blomgren, 〈[email protected]〉 Lecture Notes #02 — (1/63)
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Outline
1 Calculus Review Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem (^2) Computer Arithmetic & Finite Precision Binary Representation, IEEE 754- Something’s Missing... Roundoff and Truncation, Errors, Digits Cancellation (^3) Algorithms Algorithms, Pseudo-Code Fundamental Concepts 4 Solutions of Equations of One Variable f (x) = 0, “Root Finding” The Bisection Method When do we stop?! *** Homework #1 ***
Peter Blomgren, 〈[email protected]〉 Lecture Notes #02 — (2/63)
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Background Material — A Crash Course in Calculus
Key concepts from Calculus
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Limit / Continuity
Definition (Limit) A function f defined on a set X of real numbers X ⊂ R has the limit L at x 0 , written xlim→x 0 f^ (x) =^ L if given any real number ǫ > 0 (∀ǫ > 0), there exists a real number δ > 0 (∃δ > 0) such that |f (x) − L| < ǫ, whenever x ∈ X and 0 < |x − x 0 | < δ.
Definition (Continuity (at a point)) Let f be a function defined on a set X of real numbers, and x 0 ∈ X. Then f is continuous at x 0 if
xlim→x 0
f (x) = f (x 0 ).
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Examples: Jump Discontinuity
(^00) 0.2 0.4 0.6 0.8 1
1
2
The function
f (x) =
x + 12 sin(2πx) x < 0. 5 x + 12 sin(2πx) + 1 x > 0. 5
has a jump discontinuity at x 0 = 0.5.
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Examples: “Spike” Discontinuity
0 0.2 0.4 0.6 0.8 1
0
1
The function
f (x) =
1 x = 0. 5 0 x 6 = 0. 5
has a discontinuity at x 0 = 0.5.
The limit exists, but
lim x→ 0. 5
f (x) = 0 6 = 1
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Illustration: Convergence of a Complex Sequence
2 1 3 4
k>=N
N−
N−
A sequence in z = {zk }∞ k=1 converges to z 0 ∈ C (the black dot) if for any ǫ (the radius of the circle), there is a value N (which depends on ǫ) so that the “tail” of the sequence zt = {zk }∞ k=N is inside the circle.
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Differentiability
Theorem If f is a function defined on a set X of real numbers and x 0 ∈ X , the the following statements are equivalent: (a) f is continuous at x 0 (b) If {xn}∞ n=1 is any sequence in X converging to x 0 , then limn→∞ f (xn) = f (x 0 ).
Definition (Differentiability (at a point)) Let f be a function defined on an open interval containing x 0 (a < x 0 < b). f is differentiable at x 0 if
f ′(x 0 ) = (^) xlim→x 0 f^ (x x) −−^ fx^ (x^0 ) 0
exists.
If the limit exists, f ′(x 0 ) is the derivative at x 0.
Definition (Differentiability (in an interval)) If f ′(x 0 ) exists ∀x 0 ∈ X , then f is differentiable on X.
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Continuity / Rolle’s Theorem
Theorem (Differentiability ⇒ Continuity) If f is differentiable at x 0 , then f is continuous at x 0.
Theorem (Rolle’s Theorem Wiki-Link^ ) Suppose f ∈ C [a, b] and that f is differentiable on (a, b). If f (a) = f (b), then ∃c ∈ (a, b): f ′(c) = 0.
a c b
f’(c)=
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Mean Value Theorem
Theorem (Mean Value Theorem Wiki-Link^ ) If f ∈ C [a, b] and f is differentiable on (a, b), then ∃c ∈ (a, b): f ′(c) =
f (b) − f (a) b − a
a c b
f’(c)=[f(b)−f(a)] / [b−a]
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Intermediate Value Theorem
Theorem (Intermediate Value Theorem Wiki-Link^ ) if f ∈ C [a, b] and K is any number between f (a) and f (b), then there exists a number c in (a, b) for which f (c) = K.
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Taylor’s Theorem
Theorem (Taylor’s Theorem Wiki-Link^ ) Suppose f ∈ C n[a, b], f (n+1)∃ on [a, b], and x 0 ∈ [a, b]. Then ∀x ∈ (a, b), ∃ξ(x) ∈ (x 0 , x) with f (x) = Pn(x) + Rn(x) where
Pn(x) =
∑^ n
k=
f (k)(x 0 ) k!
(x−x 0 )k^ , Rn(x) =
f (n+1)(ξ(x)) (n + 1)!
(x−x 0 )(n+1).
Pn(x) is called the Taylor polynomial of degree n, and Rn(x) is the remainder term (truncation error).
This theorem is extremely important for numerical analysis; Taylor expansion is a fundamental step in the derivation of many of the algorithms we see in this class (and in Math 693ab).
Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Limits, Continuity, and Convergence Differentiability, Rolle’s, and the Mean Value Theorem Extreme Value, Intermediate Value, and Taylor’s Theorem
Taylor Expansions — Matlab
A Taylor polynomial of degree n requires all derivatives up to order n, and order n + 1 for the remainder. Derivatives may be [more] complicated expression [than the original function]. Matlab can compute derivatives for you:
Matlab: Symbolic Computations Try this!!!
syms x diff(sin(2x)) diff(sin(2x),3) taylor(exp(x),5) taylor(exp(x),5,1)
Calculus Review Computer Arithmetic & Finite PrecisionAlgorithms Solutions of Equations of One Variable
Binary Representation, IEEE 754- Something’s Missing...Roundoff and Truncation, Errors, Digits Cancellation
Computer Arithmetic and Finite Precision
Peter Blomgren, 〈[email protected]〉 Computer Arithmetic & Finite Precision — (20/63)
