Lecture Notes Course ÃMA 190 Numerical Mathematics, First ...

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Definition 2.2.1 We distinguish between the following classes of errors:• Round-off error, due to the approximation of a number to its computerrepresentation and the fact that arithmetic operations are not carried outexactly.• Data error if given parameters are not exactly known.• Truncation error: The replacement of an infinite sequence of operationsby a finite one.Remark 2.2.2 It is desirable to organise the computations such that the effectsof round-offs and truncations are negligible in comparison to the effects of theuncertainties in given data. If e.g. data are given with an uncertainty of 1%, thecalculations should be carried out such that the contributions of the round-offsand truncations are certainly less then 0.1%.2.3 Representing numbers in a computerWhen working on a computer, several modifications to the presentation aboveare called for. The number s of decimal places is fixed for a given computer andsoft-ware. Also the range of the exponent x 2 is limited such that there are twointegers m, M such thatm ≤ x 2 ≤ M.The number 10 which is the basis of the decimal system, is replaced by a positiveinteger B. The choices B = 2, 8 or 16 are most common. Thus for an integerN we havem∑N = sign(N)(b m b m−1 · b 0 ) B = b r B r ,where the integers b r satisfy 0 ≤ b r < B. Thus the common decimal systemhas B = 10. Fixed-point and floating point numbers are introduced as before.Thus the result of correctly rounding to s B-mals gives the bound|a − ā| ≤ 0.5 · B −s .Example 2.3.1 a = 1/3, B = 8, s = 4.We write13 = b 18 + b 28 2 + b 38 3 + b 48 4 + ɛ,where ɛ ≤ 0.5 · 8 −4 . Multiplying both sides with 8 we get the relation:Here we take b 1 = 2 and obtain:r=083 = b 1 + b 28 + b 38 2 + b 48 3 + ɛ · 8 .23 = b 28 + b 38 2 + b 48 3 + ɛ · 8 .23
This relation is again multiplied by 8 giving:Now we put b 2 = 5 and arrive atMultiplying by 8 we findThus b 3 = 2 andFinally we findThis gives:163 = b 2 + b 38 + b 48 2 + ɛ · 82 .13 = b 38 + b 48 2 + ɛ · 82 .83 = b 3 + b 48 + ɛ · 83 .23 = b 48 + ɛ · 83 .163 = b 4 + ɛ · 8 4 .b 4 = 5, ɛ = 1 3 · 8−4 .We next establish:We have namely:1/3 = (0.252525 . . .) 8 .(0.252525 . . .) = 2 8 + 5 8 2 + 2 8 3 + 5 8 4 + . . .= 2 8 (1 + 8−2 + 8 −4 + . . .) + 5 8 2 (1 + 8−2 + 8 −4 + . . .)= (2/8 + 5/8 2 )(1/(1 − 1/64)) = 1/3 .Remark 2.3.2 If B = B p 0 with p an integer, then one digit in the B systemcorresponds to p numbers in the B 0 -system.Example 2.3.3 B = 1000, p = 3, B 0 = 10π = 3.141 592 654 = 3 · B 0 + 141 · B −1 + 592 · B −2 + 654 · B −3 .We also find that one digit in the system with B = 16 corresponds to 4 digitsfor B = 2. We now introduce:Definition 2.3.4 Two real numbers a and b are said to be computationallyequivalent, if they have the same representation in a given computer.We realise that whether or not two given numbers are computationally equivalentdepends on the working accuracy of the computer used. The definition ofcomputational equivalence is extended to vectors, matrices and functions in anobvious way.24
Page 1: Lecture NotesCourse ÅMA 190Numeric
Page 6 and 7: 13.8.4 Inverse linear interpolation
Page 8 and 9: AbstractNumerical analysts study th
Page 10 and 11: 1.1.1 Two standard inequalitiesLet
Page 12 and 13: Example 1.1.3 Examples with s = 3:N
Page 14 and 15: and the relative condition number b
Page 16 and 17: This general example may be special
Page 18 and 19: Theorem 1.5.2 Let A be a square mat
Page 20 and 21: 1.6.2 Arithmetic sequencen(n − 1)
Page 22 and 23: 1.6.7 Taylor’s formulaWe derive T
Page 24 and 25: )c)becauselimx→0x 2e x − 1 −
Page 26 and 27: 1.7.7 Locating rootsPutShow that th
Page 28 and 29: We next construct the general solut
Page 32 and 33: Example 2.3.5 Consider a computer w
Page 34 and 35: We now determine c such that 2 + 3c
Page 36 and 37: e recorded) If there is no such col
Page 38 and 39: the left hand side and the right ha
Page 40 and 41: Chapter 4Iterative methods for Ax =
Page 42 and 43: or, in matrix form with the coeffic
Page 44 and 45: Chapter 5Least squares fit5.1 Norma
Page 46 and 47: c 1,4 is the product of the first a
Page 48 and 49: PutandL i (t) =P (t) =5∏(t − t
Page 50 and 51: 6.2 On the choice of nodesIf t i in
Page 52 and 53: k arguments:f[t i1 , . . . , t ik ]
Page 54 and 55: Q(x) = 4.6 − 0.8(x + 2) + 0.6x(x
Page 56 and 57: a relative error bound if x ≠ 0.
Page 58 and 59: 7.2.2 MultiplicationPutwheres = x 1
Page 60 and 61: We now assume that f has a continuo
Page 62 and 63: functional values. Let s ∗ be an
Page 64 and 65: 7.5.2 Example: additionLetWe then h
Page 66 and 67: Chapter 8Numerical treatment ofnonl
Page 68 and 69: 8.3 Determination of approximate ro
Page 70 and 71: Proof:The case n = 0 is the stateme
Page 72 and 73: 8.6 The fixed-point methodTo use th
Page 74 and 75: Chapter 9On approximation offunctio
Page 76 and 77: 9.1.3 The square rootTo find √ a,
Page 78 and 79: which is achieved forTo gett i = 1/
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whereF (h, w) = h∞∑n=−∞e iw
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Proof PutThen we haveandM n = supr>
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10.3 Numerical treatment of rapidly
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n a n−1 s n1 1.00000 1.000002 −
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phenomenon we needDefinition 10.4.1
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a r and b r are given bya r = exp(
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11.2 Constructing the trapezoidal r
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In the case of (11.2) it is suffici
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should give exact results for all l
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h=0.2 :f(0) + f(1.6)T (0.2) = 0.2
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h=0.5 :T (0.5) = 0.5 · f(0) + f(0.
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A system of ordinary differential e
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12.3.1 Euler’s methodx r+1 = x r
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It is directly verified that the so
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1.00 1.246834041.00 1.246818301.00
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13.1.3 Stirling’s formula13.1.4 B
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Power expansions for some elementar
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TranspositionThenA ∈ R n×m , B =
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13.5 Least squares solution of line
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Let f be a function which may be di
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13.9 Systems of nonlinear equations
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may be writtenx(t) = x h (t) + x p
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Chapter 14Laboratory exercises andc
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Start with x 0 = 1b) Use the bisect
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14.7 Case studies14.8 Evaluation of
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n piv,sing nonpiv,sing piv,dbl nonp
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We put c = 100 + u and solve the pr
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and hence we may consider minimisin
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x f(x)−1.0 0.3679−0.5 0.6065 .0
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14.12 Numerical integration problem
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14.12.5 Integrand with integrable s
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