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Chapter 1 Introductory Concepts and Calculus Review 1 Introduction The subjects The derivation of the algorithms The implementation of the algorithms Analyze the algorithms mathematically Accuracy, efficiency, and stability 2 1.1 Basic Tools of Calculus 1.1.1 Taylor’s Theorem Integral mean value theorem 3 Three particular expansions of Taylor’s Theorem where x0= ? 0 1 2 3 ( x 0 ) ( x 0 ) ( x 0 ) ( x 0 ) ex e0 e0 e0 e0 ... 0! 1! 2! 3! 4 Three particular expansions of Taylor’s Theorem where x0= 0 where x0= 0 5 Example : ex Let x [1,1] x0= 0 If we want then Finally, n can be found! (here n = 9) 6 Example : ex p9 (x) p2 (x) exp(x) 7 Example : ex 8 Example : ex The result tells us We can approximate the exponential function to within 10-6 accuracy using a specific polynomial, and this accuracy holds for all x in a specified interval. 9 Example 1.1 Let f (x) = (x+1)1/2, then the second-order Taylor polynomial (computed about x0= 0) is computed as follows: 2 10 Example 1.2: sin Function: Accuracy: 11 Example 1.3: arctan Function: http://zh.wikipedia.org/wiki/File:Atan_acot_plot.svg 12 Example 1.3: arctan Function: and Let Error term 13 Example 1.3 : arctan Please determine the error in a ninth-degree Taylor approximation to the arctangent function. Since 2n +1 = 9 implies that n = 4, we have 14 Taylor’s Theorem Expansion Let x x + h and x0 x 15 1.1.2 Mean Value and Extreme Value Theorems http://en.wikipedia.org/wiki/Mean_value_theorem 16 1.1.2 Mean Value and Extreme Value Theorems W 17 1.1.2 Mean Value and Extreme Value Theorems Critical point Critical point M m 18 1.1.2 Mean Value and Extreme Value Theorems The integral mean value theorem (a corollary of the intermediate value theorem) states that a function continuous on an interval takes on its average value somewhere in the interval. More exactly, if is continuous on , then there exists in such that . 19 1.1.2 Mean Value and Extreme Value Theorems 20 1.2 Error, Approximate Equality, and Asymptotic Order Notation 1.2.1 Error A : a quantity(數量) we want to compute Ah: an approximation(近似值) to that quantity Relative error (相對誤差) is better. These errors are both computational errors. 21 1.2.2 Notation: Approximate Equality Approximate equality It is an equivalence relation, and satisfy the following properties: Transitive(遞移性): Symmetric(對稱性): Reflexive(反身性): 22 1.2.3 Notation: Asymptotic Order (Big O) 23 Example 1.4 Let Simple calculus shows that so that we have Here 24 1.2.3 Notation: Asymptotic Order (Big O) 25 Example 1.6 26 27 1.3 A Primer on Computer Arithmetic Computer arithmetic is generally inexact. While the errors are very small, they can accumulate and dominate the calculation. Example: floating-point arithmetic Reference: An Introduction to Computer Science, Chapter 3, Excess System (Excess_127 or Excess_1023) is the sign of the number, f is the fraction (0 <= f <= 1), is the base of the internal number system 28 IEEE standards for floating-point representation （底數 尾數） Example Show the representation of the normalized number + 26 x 1.01000111001 Solution The sign is positive. The Excess_127 representation of the exponent is 133. You add extra 0s on the right to make it 23 bits. The number in memory is stored as: 0 10000101 01000111001000000000000 Errors Rounding error v.s. chopping error Rounding: 四捨五入 Chopping: 無條件捨去 Discussion: Rounding is more accurate but chopping is faster. The chopping error is indeed lager than the rounding error. 31 Example Rounding error Chopping error 32 Subtractive Cancellation If a and b are accurate to 16 decimal digits. What about their difference c = a - b ? Example: a e 0.9999000049998333 (1/100) 2 b e(1/1000) 0.9999990000005000 2 The result c is accurate to 12 digits. This is because we were subtracting two nearly equal numbers. 33 Example Function : We know that : Taylor’s Theorem : 34 …… 35 1.5 Simple Approximations Error function: (probability theory) It is not possible to evaluate this integral by means of the fundamental theorem of calculus. Use Taylor’s Theorem to approximate. where 36 Substitution: Define So that we have Set where c depends on t and 37 Apply the Integral Mean Value Theorem: The structured form: where 38 Use the big O notation: Use the approximate equality notation: Simplify: if the values of x between 0 and 2 if k >=1 thus 39 Fundamental Idea When confronted with a computation that cannot be done exactly, we often replace that relevant function with something simpler that approximates it, and carry out the computation exactly on the simple approximation. 40