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

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AbstractNumerical analysts study the algorithms of continuous mathematics, e.g. thesolution of linear and nonlinear systems of equations, numerical evaluation of integralsand the treatment of differential equations. Hence integer programmingdoes not belong to numerical analysis. The goal of the numerical treatment ofa mathematical problem is to determine a solution and to quantify the uncertaintyof the calculated results. A major topic of numerical mathematics is thestudy of the issues arising by the use of computers. The result of this work isoften represented in the form of packages of computer programs. Computersare used not only because the number of arithmetic operations is large, butalso because the computational schemes become large and complex. The userof the computer packages does not know all the details of the calculations tobe performed, but (s)he needs to know the main principles. The research anddevelopment in applied mathematics often could be described as a three-phaseprocess:• Analysis of the problem under study. The result of this investigation maybe represented in the form of theorems giving the mathematical propertiesof the solution sought.• Treatment of simple model problems, illustrating the theoretical resultsobtained.• Construction and documentation of computer programs solving the tasksabove.These lecture notes define the contents of Course ÅMA190 which gives 5 creditsin the European Credit Transfer System, ECTS. The author has taught thisand similar courses in Stavanger most academic years in the period 1986–2007.This is an introductory course and may even be looked upon as a user’s coursein the areas it introduces. It may not be sufficient for those aspiring to doprofessional work in the study and development of computational schemes. Thecourse gives an introduction to some basic concepts and methods in numericalmathematics. These topics include the study of error propagation, treatmentof linear systems of equations with a stable variant of the Gauss eliminationmethod, iterative schemes for linear and nonlinear systems of equations as wellas evaluating integrals by means of the trapezoidal and mid-point methods.The numerical treatment of ordinary differential equations by discretisation isillustrated by presenting the Euler and trapezoidal schemes. This knowledgewill often be useful for those using the results of computational mathematics.The written exam tests the ability to solve relatively simple tasks using thelecture notes and other mathematical literature. Hence this is not a test ofknowledge learnt by heart.
Chapter 1Introduction and somestandard definitions andresults1.1 Number systems in applied mathematicalanalysisThe following number systems are often encountered in applied mathematicalwork:• integers• rational numbers• real numbers• complex numbersWe note that each of the listed classes of numbers is a subset of the followingones. Integers and rational numbers may be processed by means of symbolicmanipulation, e.g. in Maple. Rational numbers are then represented in the formr = p/q,where p and q are integers (q ≠ 0). Complex numbers are written in the formz = x + iy,where x and y are reals. See also [11], page 4 – 9.2
Page 1: Lecture NotesCourse ÅMA 190Numeric
Page 6 and 7: 13.8.4 Inverse linear interpolation
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 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.
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7.2.2 MultiplicationPutwheres = x 1
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We now assume that f has a continuo
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functional values. Let s ∗ be an
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7.5.2 Example: additionLetWe then h
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Chapter 8Numerical treatment ofnonl
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8.3 Determination of approximate ro
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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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