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

More documents

Recommendations

Info

13.8.4 Inverse linear interpolation for f(x) = 0 . . . . . . . . . . 11213.8.5 Newton-Raphson’s method for f(x) = 0 . . . . . . . . . . 11213.8.6 Fix-point iteration for x = g(x) . . . . . . . . . . . . . . . 11213.9 Systems of nonlinear equations . . . . . . . . . . . . . . . . . . . 11313.9.1 Newton-Raphson’s method forF (x) = 0, x ∈ R n , F (x) ∈ R n . . . . . . . . . . . . . . . . 11313.9.2 Fix-point iteration for x = G(x), x ∈ R n , G(x) ∈ R n . . . 11313.10<strong>Numerical</strong> integration with the trapezoidal rule . . . . . . . . . . 11313.11Differential and difference equations with constant coefficients . . 11413.12Initial-value problems for a single ordinary differential equation . 11513.12.1 Piccard iteration . . . . . . . . . . . . . . . . . . . . . . . 11613.12.2 Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . 11613.12.3 The trapezoidal method . . . . . . . . . . . . . . . . . . . 11613.13Initial-value problems for systems of ordinary differential equations11614 Laboratory exercises and case studies 11714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11714.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11714.3 Three methods for solving a linear system of equations with atridiagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 11714.3.1 Similar problem . . . . . . . . . . . . . . . . . . . . . . . 11714.3.2 Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11714.4 A least squares problem . . . . . . . . . . . . . . . . . . . . . . . 11814.4.1 Similar problem . . . . . . . . . . . . . . . . . . . . . . . 11814.4.2 Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11814.5 Three methods for solving a nonlinear equation in one variable . 11814.5.1 Similar problem . . . . . . . . . . . . . . . . . . . . . . . 11814.5.2 Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11914.6 Analysing a table over the solution to an initial-value problem . . 11914.6.1 Similar problem . . . . . . . . . . . . . . . . . . . . . . . 11914.6.2 Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12014.7 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12114.8 Evaluation of equivalent sums . . . . . . . . . . . . . . . . . . . . 12114.8.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12114.8.2 Computational treatment . . . . . . . . . . . . . . . . . . 12114.8.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 12114.9 The stabilising effect of pivoting in Gauss elimination . . . . . . 12214.9.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12214.9.2 Computational results . . . . . . . . . . . . . . . . . . . . 12214.9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 12314.10Least squares fit . . . . . . . . . . . . . . . . . . . . . . . . . . . 12414.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 12414.10.2 Fitting a straight line to set of observations . . . . . . . . 12514.10.3 Derivation of the normal equations . . . . . . . . . . . . . 12814.11Interpolation with polynomials . . . . . . . . . . . . . . . . . . . 12814.11.1 Lagrange’s formula . . . . . . . . . . . . . . . . . . . . . . 1296
14.11.2 Estimate of the effect of round-off in given data on thecalculated value of P (x) . . . . . . . . . . . . . . . . . . . 12914.11.3 Estimation of truncation error . . . . . . . . . . . . . . . 13014.11.4 Newton’s interpolation formula . . . . . . . . . . . . . . . 13014.12<strong>Numerical</strong> integration problems using the trapezoidal method . . 13114.12.1 The trapezoidal rule on a periodic function over one period13114.12.2 The trapezoidal rule over the real line . . . . . . . . . . . 13114.12.3 The trapezoidal rule over the real line . . . . . . . . . . . 13214.12.4 Integral over a finite interval transformed to an integralover the real line . . . . . . . . . . . . . . . . . . . . . . . 13214.12.5 Integrand with integrable singularity . . . . . . . . . . . . 1337
Page 1: Lecture NotesCourse ÅMA 190Numeric
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 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/
Page 80 and 81:
whereF (h, w) = h∞∑n=−∞e iw
Page 82 and 83:
Proof PutThen we haveandM n = supr>
Page 84 and 85:
10.3 Numerical treatment of rapidly
Page 86 and 87:
n a n−1 s n1 1.00000 1.000002 −
Page 88 and 89:
phenomenon we needDefinition 10.4.1
Page 90 and 91:
a r and b r are given bya r = exp(
Page 92 and 93:
11.2 Constructing the trapezoidal r
Page 94 and 95:
In the case of (11.2) it is suffici
Page 96 and 97:
should give exact results for all l
Page 98 and 99:
h=0.2 :f(0) + f(1.6)T (0.2) = 0.2
Page 100 and 101:
h=0.5 :T (0.5) = 0.5 · f(0) + f(0.
Page 102 and 103:
A system of ordinary differential e
Page 104 and 105:
12.3.1 Euler’s methodx r+1 = x r
Page 106 and 107:
It is directly verified that the so
Page 108 and 109:
1.00 1.246834041.00 1.246818301.00
Page 110 and 111:
13.1.3 Stirling’s formula13.1.4 B
Page 112 and 113:
Power expansions for some elementar
Page 114 and 115:
TranspositionThenA ∈ R n×m , B =
Page 116 and 117:
13.5 Least squares solution of line
Page 118 and 119:
Let f be a function which may be di
Page 120 and 121:
13.9 Systems of nonlinear equations
Page 122 and 123:
may be writtenx(t) = x h (t) + x p
Page 124 and 125:
Chapter 14Laboratory exercises andc
Page 126 and 127:
Start with x 0 = 1b) Use the bisect
Page 128 and 129:
14.7 Case studies14.8 Evaluation of
Page 130 and 131:
n piv,sing nonpiv,sing piv,dbl nonp
Page 132 and 133:
We put c = 100 + u and solve the pr
Page 134 and 135:
and hence we may consider minimisin
Page 136 and 137:
x f(x)−1.0 0.3679−0.5 0.6065 .0
Page 138 and 139:
14.12 Numerical integration problem
Page 140 and 141:
14.12.5 Integrand with integrable s
show all

Lecture Notes Course Ã MA 190 Numerical Mathematics, First ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?

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