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

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We next construct the general solution of the homogeneous equation.characteristic equation isr 2 + 4r + 3 = 0,which has the roots r 1 = −3 and r 2 = −1. Therefore the general solution of thedifferential equation becomeswithx(t) = Ae −3t + Be −t − 4/9 + t/3,ẋ(t) = −3Ae −3t − Be −t + 1/3.Using the initial conditions we obtain the following linear system of equationsfor determining the constants A and B:orgivingThe solution sought becomesx(0) = A + B − 4/9 = 1,ẋ(0) = −3A − B + 1/3 = 2,A + B = 13/9,−3A − B = 5/3,A = −14/9, B = 3.x(t) = − 14e−3t9+ 3e −t − 4 9 + t 3 .The21
Chapter 2On the representation ofnumbers in a computer2.1 IntroductionComputers may be used to process numbers in two different ways, namely asexact quantities which are processed by means of symbolic manipulations or asdata which are subject to numerical evaluation, often involving approximation.Software packages called computer algebras perform symbolic manipulations.Examples of such packets are: Maple, Mathematica, Derive and Macsyma.There is a principal limit to the performance of these processors, since onlya finite number of symbols may be stored in a given computer while it is possibleto devise mathematical expressions of any length, e.g.s =∞∑ln(cos 2 ( n√ n)) · e −n .n=1If we accept approximate values, then our calculations may be carried out bymeans of an algorithmic language like Matlab, Fortran77, Fortran90 or C. Bywriting a short program in any of these languages it would be easy to determinethe sum above to any desired accuracy, say 8 decimal places, by directsummation.2.2 Implementation of expressions on a computerSince a computer only may carry out a finite number of arithmetic operationsand a finite number of logical choices the most general expression which may beevaluated by means of a computer is a piecewise rational function. The value ofsuch an expression is affected by round-off errors, which may accumulate. Thisalso means that e.g. all standard functions must be approximated by piecewiserational functions. We now introduce22
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 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
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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