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

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Example 2.3.5 Consider a computer with working relative accuracy 1.0 · 10 −7Define the functions f and g according tof(x) = exp(x), g(x) =∑12r=0x r, −1 ≤ x ≤ 1.r!Then the exponential function f and the polynomial g are computationally equivalenton the interval [−1, +1].We next observe that there are only finitely many different computer representationsof a real number on a given computer with a pre-defined working accuracybut there are infinitely many real numbers. This means that there are infinitelymany reals having an identical computer representation. This conclusion maybe generalised to classes of computational problems as illustrated byExample 2.3.6 Let A be a matrix, b, c and x compatible vectors. Consider thetask of evaluatingv = b T x, when Ax = c.There are infinitely many different matrices and vectors A, b, c having identicalcomputer representations and they could conceivably define different values ofthe answer v. This is not desirable.25
Chapter 3Gaussian elimination forAx = b3.1 IntroductionThe Gauss elimination is based on systematic use of the followingLemma 3.1.1 Let x = (x 1 , . . . , x n ) T ∈ R n be a column vector, f 1 (x), f 2 (x)be functions having x as argument. Let finally c ≠ 0 be a constant. Then thefollowing two systems of equations have the same solution sets:f 1 (x) = 0, f 2 (x) = 0, (3.1)andf 1 (x) = 0, f 1 (x) + c · f 2 (x) = 0. (3.2)Proof: If x satisfies (3.1) then it immediately follows that x satisfies (3.2). Onthe other hand if f 1 (x) = 0, f 1 (x) + c · f 2 (x) = 0 we first see that f 1 (x) = 0giving that c · f 2 (x) = 0 and since c ≠ 0 we find that f 2 (x) = 0 as well, provingthe lemma.•We illustrate an application of the lemma with the followingExample 3.1.2 Solve the linear system3x + y = 1, (3.3)2x − y = 4 . (3.4)Using the lemma above we conclude that the system (3.3),(3.4) has the samesolution set as the system (3.5) below for all nonzero values of c:3x + y = 1, (2 + 3c)x + (c − 1)y = (4 + c). (3.5)26
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 30 and 31: Definition 2.2.1 We distinguish bet
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/
Page 80 and 81: 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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