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

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1.6.7 Taylor’s formulaWe derive Taylor’s formula by means of repeated integration by parts: Let a bea fixed point, x a real number and assume that the function f has at least nderivatives on the interval with endpoints a and a+x.Then we have the relationf(a + x) = f(a) +∫ a+xaf ′ (t) dt.Integrating by parts and remembering that the derivative with respect to t ofthe functionis 1 we gett − a − xf(a + x) = f(a) + [ a+xa f ′ (t)(t − a − x)] −Simplifying we findf(a + x) = f(a) + xf ′ (a) −A new integration by parts gives∫ a+xa∫ a+xa(t − a − x)f ′′ (t) dt(t − a − x)f ′′ (t) dt.[ ](t − a − x)f(a + x) = f(a) + xf ′ 2∫ a+x(a) −f ′′ (t) +2aUpon simplification we have∫ a+xf(a + x) = f(a) + xf ′ (a) + x22 f ′′ (a) +a(t − a − x) 2f (3) (t) dt.2(t − a − x) 2f (3) (t) dt.2This procedure is repeated several times and the formula in 13.1.10 emerges,upon using the mean-value theorem (13.2) on the last integral.1.7 Introduction exercisesIn Numerical Mathematics we take advantage of the results from mathematicalanalysis. For the present course it is necessary that the students recall whatthey have learnt in earlier courses in Calculus and Linear Algebra. They shouldbe able to use mathematical results from the literature and practise seeking suchresults in relevant books or in the internet. The exercises to follow illustratematerial which is deemed to be useful for the study of Numerical Mathematics.The problems may be solved analytically or by using a simple pocket calculator.15
1.7.1 Arithmetic seriesEvaluate the arithmetic sumCalculate A(100).Solution:A(n) = 1 + 3 + . . . + 2n − 1, n ≥ 1.A(n) = 1 +3 + . . . + 2n − 1, n ≥ 1.A(n) = 2n − 1 +2n − 3 + . . . + 1, n ≥ 1.1.7.2 Geometric seriesEvaluate the geometric seriesSolution:1.7.3 Limit valuesFind the following limit values:a)b)c)d)⇒ 2A(n) = n · 2n.A(n) = n 2 , A(100) = 10000.s = 2 + 0.4 + 0.08 + . . . + 2 · (0.2) n + . . .s = 2/(1 − 0.2) = 2.5limx→0sin xlimx→0 x .x 2e x − 1 − x .limx→+∞ x2 e −x .lim (1 +n→∞ 2x/n)n .e)2x 3 + 7x + 6limx→∞ x 3 .Solution:L’Hôpital’s rule in a) and b):a)sin xlimx→0 x= lim cos x= 1.x→0 116
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 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 30 and 31: Definition 2.2.1 We distinguish bet
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
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8.6 The fixed-point methodTo use th
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Chapter 9On approximation offunctio
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9.1.3 The square rootTo find √ a,
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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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