CISE-301: Numerical Methods Topic 1:

Transcription

1 CISE-3: Numerical Methods Topic : Introduction to Numerical Methods and Taylor Series Lectures -4: KFUPM Term 9 Section 8 CISE3_Topic KFUPM - T9 - Section 8

2 Lecture Introduction to Numerical Methods What are NUMERICAL METHODS? Why do we need them? Topics covered in CISE3. Reading Assignment: Pages 3- o tetboo CISE3_Topic KFUPM - T9 - Section 8

3 Numerical Methods Numerical Methods: Algorithms that are used to obtain numerical solutions o a mathematical problem. Why do we need them?. No analytical solution eists,. An analytical solution is diicult to obtain or not practical. CISE3_Topic KFUPM - T9 - Section 8 3

4 What do we need? Basic Needs in the Numerical Methods: Practical: Can be computed in a reasonable amount o time. Accurate: Good approimate to the true value, Inormation about the approimation error Bounds, error order,. CISE3_Topic KFUPM - T9 - Section 8 4

5 Outlines o the Course Taylor Theorem Number Representation Solution o nonlinear Equations Interpolation Numerical Dierentiation Numerical Integration Solution o linear Equations Least Squares curve itting Solution o ordinary dierential equations Solution o Partial dierential equations CISE3_Topic KFUPM - T9 - Section 8 5

6 Solution o Nonlinear Equations Some simple equations can be solved analytically: 4 3 Analyticsolution roots and 3 4 ± 4 43 Many other equations have no analytical solution: 9 5 No analyticsolution e CISE3_Topic KFUPM - T9 - Section 8 6

7 Methods or Solving Nonlinear Equations o Bisection Method o Newton-Raphson Method o Secant Method CISE3_Topic KFUPM - T9 - Section 8 7

8 CISE3_Topic KFUPM - T9 - Section 8 8 Solution o Systems o Linear Equations equations in unnowns. we have What to do i 3, 5 3, 3 We can solve it as : 5 3

9 Cramer s Rule is Not Practical Cramer's Rule can be used to solve the system : 3 5, 3 5 But Cramer's Rule is not practical or large problems. Tosolve N equations with N unnowns, we need N N N! multiplications. To solve a 3 by 3 system,.3 A super computer needs more than 35 multiplications are needed. years to compute this. CISE3_Topic KFUPM - T9 - Section 8 9

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11 Curve Fitting Given a set o data: y Select a curve that best its the data. One choice is to ind the curve so that the sum o the square o the error is minimized. CISE3_Topic KFUPM - T9 - Section 8

12 Interpolation Given a set o data: i y i Find a polynomial P whose graph passes through all tabulated points. y i P i i i is in the table CISE3_Topic KFUPM - T9 - Section 8

13 Methods or Curve Fitting o Least Squares o Linear Regression o Nonlinear Least Squares Problems o Interpolation o Newton Polynomial Interpolation o Lagrange Interpolation CISE3_Topic KFUPM - T9 - Section 8 3

14 Integration Some unctions can be integrated analytically: 3 But many unctions a d e d? have no analyticalsolutions : CISE3_Topic KFUPM - T9 - Section 8 4

15 Methods or Numerical Integration o Upper and Lower Sums o Trapezoid Method o Romberg Method o Gauss Quadrature CISE3_Topic KFUPM - T9 - Section 8 5

16 Solution o Ordinary Dierential Equations A solution to the dierential equation : t 3 t 3 t ; is a unction t that satisies the equations. * Analyticalsolutions are available or special cases only. CISE3_Topic KFUPM - T9 - Section 8 6

17 Solution o Partial Dierential Equations Partial Dierential Equations are more diicult to solve than ordinary dierential equations: u t u u, t u, t, u, sin π CISE3_Topic KFUPM - T9 - Section 8 7

18 Summary Numerical Methods: Algorithms that are used to obtain numerical solution o a mathematical problem. We need them when No analytical solution eists or it is diicult to obtain it. Topics Covered in the Course Solution o Nonlinear Equations Solution o Linear Equations Curve Fitting Least Squares Interpolation Numerical Integration Numerical Dierentiation Solution o Ordinary Dierential Equations Solution o Partial Dierential Equations CISE3_Topic KFUPM - T9 - Section 8 8

19 Lecture Number Representation and Accuracy Number Representation Normalized Floating Point Representation Signiicant Digits Accuracy and Precision Rounding and Chopping Reading Assignment: Chapter 3 CISE3_Topic KFUPM - T9 - Section 8 9

20 Representing Real Numbers You are amiliar with the decimal system: Decimal System: Base, Digits,,,9 Standard Representations: ± sign integral raction part part CISE3_Topic KFUPM - T9 - Section 8

21 Normalized Floating Point Representation Normalized Floating Point Representation: ±. d d d 3 sign mantissa eponent d 4 n d, n : integer No integral part, Advantage: Eicient in representing very small or very large numbers. CISE3_Topic KFUPM - T9 - Section 8

22 Calculator Eample Suppose you want to compute: *.39 using a calculator with two-digit ractions 3.57 * True answer: CISE3_Topic KFUPM - T9 - Section 8

23 Binary System Binary System: Base, Digits {,} ±. b b3 b4 sign mantissa eponent n b b CISE3_Topic KFUPM - T9 - Section 8 3

24 7-Bit Representation sign: bit, mantissa: 3bits, eponent: 3 bits CISE3_Topic KFUPM - T9 - Section 8 4

25 Fact Numbers that have a inite epansion in one numbering system may have an ininite epansion in another numbering system:.... You can never represent. eactly in any computer. CISE3_Topic KFUPM - T9 - Section 8 5

26 Representation Hypothetical Machine real computers use 3 bit mantissa Mantissa: 3 bits Eponent: bits Sign: bit Possible positive machine numbers: CISE3_Topic KFUPM - T9 - Section 8 6

27 Representation Gap near zero CISE3_Topic KFUPM - T9 - Section 7 7

28 Remars Numbers that can be eactly represented are called machine numbers. Dierence between machine numbers is not uniorm Sum o machine numbers is not necessarily a machine number: not a machine number CISE3_Topic KFUPM - T9 - Section 8 8

29 Signiicant Digits Signiicant digits are those digits that can be used with conidence. CISE3_Topic KFUPM - T9 - Section 8 9

30 Signiicant Digits - Eample 48.9 CISE3_Topic KFUPM - T9 - Section 8 3

31 Accuracy and Precision Accuracy is related to the closeness to the true value. Precision is related to the closeness to other estimated values. CISE3_Topic KFUPM - T9 - Section 8 3

32 CISE3_Topic KFUPM - T9 - Section 8 3

33 Rounding and Chopping Rounding: Replace the number by the nearest machine number. Chopping: Throw all etra digits. CISE3_Topic KFUPM - T9 - Section 8 33

34 Rounding and Chopping - Eample CISE3_Topic KFUPM - T9 - Section 8 34

35 Error Deinitions True Error Can be computed i the true value is nown: Absolute True Error E t true value approimation Absolute Percent RelativeError ε t true value approimation true value * CISE3_Topic KFUPM - T9 - Section 8 35

36 Error Deinitions Estimated Error When the true value is not nown: Estimated Absolute Error E a current estimate previous estimate Estimated Absolute Percent Relative Error ε a current estimate previous estimate current estimate * CISE3_Topic KFUPM - T9 - Section 8 36

37 Notation We say that the estimate is correct to n decimal digits i: Error n We say that the estimate is correct to n decimal digits rounded i: Error n CISE3_Topic KFUPM - T9 - Section 8 37

38 Summary Number Representation Numbers that have a inite epansion in one numbering system may have an ininite epansion in another numbering system. Normalized Floating Point Representation Eicient in representing very small or very large numbers, Dierence between machine numbers is not uniorm, Representation error depends on the number o bits used in the mantissa. CISE3_Topic KFUPM - T9 - Section 8 38

39 Lectures 3-4 Taylor Theorem Motivation Taylor Theorem Eamples Reading assignment: Chapter 4 CISE3_Topic KFUPM - T9 - Section 8 39

40 Motivation We can easily compute epressions lie: 3 4 But, How do you compute 4., sin.6? We can use the deinition to compute sin.6? b a is this a practical way?.6 CISE3_Topic KFUPM - T9 - Section 8 4

41 CISE3_Topic KFUPM - T9 - Section 8 4 Taylor Series 3 3 '! can write : we converge, series the I!... 3!! : about epansion o series Taylor The Series Taylor or

42 CISE3_Topic KFUPM - T9 - Section 8 4 Taylor Series Eample. The series converges or!! ' ' < e or e e e e : about o epansion series Taylor Obtain e

43 Taylor Series 3 Eample.5 ep CISE3_Topic KFUPM - T9 - Section 8 43

44 Taylor Series Eample Obtain Taylor series epansion o sin about : sin ' cos ' sin 3 cos 3 sin! The series converges or <. 3 3! 5 5! 7 7!... CISE3_Topic KFUPM - T9 - Section 8 44

45 4 3-3 /3! 5 /5! sin /3! CISE3_Topic KFUPM - T9 - Section 8 45

46 Convergence o Taylor Series Observations, Eample The Taylor series converges ast ew terms are needed when is near the point o epansion. I -c is large then more terms are needed to get a good approimation. CISE3_Topic KFUPM - T9 - Section 8 46

47 Taylor Series Eample 3 Obtain Taylor series epansion o ' Taylor Series Epansion o : about ' : CISE3_Topic KFUPM - T9 - Section 8 47

48 Eample 3 Remars Can we apply Taylor series or >?? How many terms are needed to get a good approimation??? These questions will be answered using Taylor s Theorem. CISE3_Topic KFUPM - T9 - Section 8 48

49 Taylor s Theorem I a unction possesses o orders,,..., n in a then or any c [a,b]: continuous closed interval derivatives [a,b], n terms Truncated Taylor Series where : E n n n n! c! c c ξ n E n and ξ is Remainder between c and. CISE3_Topic KFUPM - T9 - Section 8 49

50 Taylor s Theorem We can apply Taylor's theorem or : with the point o epansion c i <. I [ a, b] includes, then the unction and its derivatives are not deined. Taylor Theorem is not applicable. CISE3_Topic KFUPM - T9 - Section 8 5

51 Error Term To get an idea about the approimation we can derive an upper bound on : E ξ n n c n n! or all values o ξ between c and. error, CISE3_Topic KFUPM - T9 - Section 8 5

52 Error Term Eample 4 How large is the error i we replaced the irst 4 terms n 3o its Taylor series epansion about when.? E n E n e n n ξ!. e n! c n ξ n or. E 8.468E 5 CISE3_Topic KFUPM - T9 - Section e. e by

53 Alternative orm o Taylor s Theorem Let have continuous derivatives o orders,,..., n on an interval [a,b], and [a,b] and h [a,b], then : h n! h E n E n n ξ n! h n where ξ is between and h CISE3_Topic KFUPM - T9 - Section 8 53

54 CISE3_Topic KFUPM - T9 - Section 8 54 Taylor s Theorem Alternative orms. and between is!!,. and between is!! h where h n h h c h c where c n c c n n n n n n ξ ξ ξ ξ

55 Mean Value Theorem I is a continuous unction on a closed interval[a, b] and its derivative is deined on the open interval a,b then there eists ξ [ a, b] dξ d b a b-a Proo : Use Taylor's Theorem or n, a, h b b a dξ d b-a CISE3_Topic KFUPM - T9 - Section 8 55

56 Alternating Series Theorem Consider the alternating series : S a I a a lim n a and a a n 3 a 3 a 4 a 4 then The series converges and S Sn an S a n n : : Partial sum sum o First omitted term the irst n terms CISE3_Topic KFUPM - T9 - Section 8 56

57 Alternating Series Eample 5 sin can be computed using : sin 3! This is a convergent alternating series since : a a Then : a 3 sin sin 3! 3! a 4 5! 5! and 7! lim n a n 5! 7! CISE3_Topic KFUPM - T9 - Section 8 57

58 Eample 6 Obtain the o e How large can the error be when n termsare used to approimate Taylor series epansion about c.5the center o epansion e with? CISE3_Topic KFUPM - T9 - Section 8 58

59 CISE3_Topic KFUPM - T9 - Section 8 59 Eample 6 Taylor Series...!.5...! ! '.5 '.5 e e e e e e e e e e e e e.5, o epansion series Taylor Obtain c e

60 CISE3_Topic KFUPM - T9 - Section 8 6 Eample 6 Error Term! ma!.5!.5.5! 3 [.5,] n e Error e n Error n e Error n Error e n n n n n n ξ ξ ξ ξ

61 Remar In this course, all angles are assumed to be in radian unless you are told otherwise. CISE3_Topic KFUPM - T9 - Section 8 6

62 Maclaurin Series Find Maclaurin series epansion o cos. Maclaurin series is a special case o Taylor series with the center o epansion c. CISE3_Topic KFUPM - T9 - Section 8 6

63 Maclaurin Series Eample 7 Obtain Maclaurin series epansion o : cos cos ' sin ' cos 3 sin 3 cos! The series converges or! <. 4 4! 6 6!... CISE3_Topic KFUPM - T9 - Section 8 63