Definition. A Taylor series of a function f is said to represent function f, iff the error term converges to 0 for n going to infinity.

Transcription

1 Definition A Taylor series of a function f is said to represent function f, iff the error term converges to 0 for n going to infinity : ESM4A - Numerical Methods 32

2 f(x) = e x at point c = 0. Taylor s theorem: Example 2 Let x s. Then, ζ(x) s and e ζ(x) e s. Thus, and : ESM4A - Numerical Methods 33

3 Example 3 Polynomial function f(x) = 4x 2 + 5x + 7 at c=2. As the Taylor series is finite, the error term is 0 from some n on. Hence, the Taylor series represents the function : ESM4A - Numerical Methods 34

4 Example 4 f(x) = ln (1+x) at point c=0 f (x) = (1+x) -1 f (x) = -(1+x) -2 f (k) (x) = (-1) k-1 (k-1)! (1+x) -k f (k) (0) = (-1) k-1 (k-1)! : ESM4A - Numerical Methods 35

5 Example 4 The Taylor series represents the function, if x є [0,1]. For other values of x, the error term may not converge to 0. Hence, for x > 1, we cannot use the Taylor series. Conclusion: We have to compute the so-called range of convergence before we apply Taylor expansion : ESM4A - Numerical Methods 36

6 Putting it into practice Use Taylor series to approximate function values. Example 1: cos (0.1) Actual value: Taylor series at c=0: Approximate values for cos (0.1) using truncated Taylor series: Conclusion: We can quickly get good approximations : ESM4A - Numerical Methods 37

7 Speed of convergence We have observed that the Taylor expansion does not have to converge to the actual solution. Question: If it does converge, how fast does it converge? In practice: How many terms of the truncated Taylor series do we need for a good approximation? : ESM4A - Numerical Methods 38

8 Observation Compute ln (2): First solution: Determine Taylor series for ln (1+x) at c=0 and evaluate Taylor series for x=1. Truncating after 8 terms delivers ln (2) Second solution: Determine Taylor series for at c=0 and evaluate Taylor series for x=1/3. Truncating after 4 terms delivers ln (2) The actual value is The second solution converges much faster : ESM4A - Numerical Methods 39

9 Proximity of x to c The closer x is to c, the higher the accuracy of our approximation. Note that this error is in addition to the truncation error : ESM4A - Numerical Methods 40

10 Taylor s theorem for f(x+h) Let. Then, we get for that truncated Taylor series error term with : ESM4A - Numerical Methods 41

11 Remarks This second theorem follows directly from the first one for x old = x+h and c = x. If h->0, the error term converges to 0 with at least the speed of h n+1, if the (n+1)-st derivative is bounded on the interval [x,x+h]. We write error term = O(h n+1 ). The O-notation means (there exists a C such that for sufficiently large values of h) In our case, C > : ESM4A - Numerical Methods 42

12 Example Evaluation of interest: Use f(z) = ln (z) and expand at e. Derivatives: Expansion: Range of convergence: sufficient. Convergence rate is O(h n ). We can find n such that error is below given threshold : ESM4A - Numerical Methods 43

13 Summary: Taylor series approximation Given problem: evaluate f(x) with error bound e. Known: f(c) for c close to x (and its derivatives). Requirement: for. Check: Taylor series represents function f on [a,b]. Estimate maximum error when computing f(x) using a truncated Taylor series with n terms. Choose n such that the estimated maximum error is smaller than error bound e. Evaluate the truncated Taylor series with n terms to approximate f(x) : ESM4A - Numerical Methods 44

14 Generalization: Numerical approach Given: hard problem. Solution: Find an algorithmic approach to solve the problem approximately. Caveat: Check the limitations/constraints of the applicability of the approach. Approximation error: Compare the maximum error to the error threshold determined by the application. Convergence: Numerical methods often improve when executing more computations. Does the approximation converge towards the actual solution? I.e., does the error go to 0? Convergence rate: How fast does the error go to 0? : ESM4A - Numerical Methods 45

15 Goals revisited In this course, we will: Discuss algorithmic approaches to solve standard mathematical problems with applications in engineering and science. Discuss the approaches with respect to their applicability (constraints, convergence). Discuss the approaches with respect to the practicability (approximation error, convergence rate) : ESM4A - Numerical Methods 46

16 1.2 Number Representations : ESM4A - Numerical Methods 47

17 Definition Let b є N\{1}. Every number x є N 0 can be written in a unique representation with respect to base b by with a i є N 0 and a i < b : ESM4A - Numerical Methods 48

18 b=10 Base 10: Notation: Fractions: Real numbers: : ESM4A - Numerical Methods 49

19 Infinite representations For irrational numbers (such as e or π) an infinite number of coefficients b i is required. But: not every infinite representation implies irrationality. Counter-example: 1/ : ESM4A - Numerical Methods 50

20 Simple base representations A number with a simple base representation with respect to one base may have a complicated base representation (many coefficients, maybe even infinite) with respect to another base : ESM4A - Numerical Methods 51

21 Base representations in computers Computer systems are using base 2 (binary) base8(octal) base 16 (hexadecimal) : ESM4A - Numerical Methods 52

22 Base conversion How do we get from one base representation to another? In particular, how can we switch between bases 2, 8, 16, and 10? : ESM4A - Numerical Methods 53

23 Conversion b->10 (a n a n-1 a 0 ) b = a n b n + a n-1 b n-1 + a 0 b 0 Then, just do the math Example: (42) 8 = 4x x8 0 = (34) : ESM4A - Numerical Methods 54

24 Conversion 2 <-> 8 and 2 <-> 16 2 <-> 8: Three consecutive bits represent one octal digit. Example: 2 <-> 16: Four consecutive bits represent one hexadecimal digit : ESM4A - Numerical Methods 55

25 Conversion 10 -> b The only somewhat more sophisticated part is the conversion from basis 10 to basis b. Two approaches: Algorithm by Euclid ( b.c.) Algorithm using Horner s scheme ( ) : ESM4A - Numerical Methods 56

26 Algorithm by Euclid Input: (x) 10 Output: (x) b. 1. Determine (smallest) exponent n with x < b n+1 2. For i = n to 0, compute a i := x div b i // integer division x := x mod b i // modulo operation 3. Return a n a n-1 a : ESM4A - Numerical Methods 57

27 Example Conversion 10->8: (34) < 34 < 8 2 -> n=1. 2. Iteration: a 1 := 34 div 8 1 = 4 x := 34 mod 8 1 = 2 a 0 := 2 div 8 0 = 2 x := 2 mod 8 0 = 0 3. Output (42) : ESM4A - Numerical Methods 58

28 Remark The algorithm can be easily extended to rational numbers. Only the stopping criteria needs to be changed. The algorithm is intuitive, but the first step is inefficient for a computer implementation : ESM4A - Numerical Methods 59