17333xyyzxf+=∂∂,22329yxxzyf+=∂∂,23xyzzf=∂∂From eq. (8.4), we getzxyzyyxxzxxyyzzzfyyfxxfuΔ+Δ++Δ+=Δ∂∂+Δ∂∂+Δ∂∂≅Δ2223333)29()3(Given that2=x,1=y,1=z,005.0=Δx,001.0=Δy,005.0=Δzand therefore, we obtainzxyzyyxxzxxyyzuΔ+Δ++Δ+≤Δ2223333)29()3(085.0005.06001.020005.07=×+×+×=Hence the maximum absolute error inuis 0.085.The maximum relative error inuis given by()01062508085.0maxmax.uuEr=≈Δ=

1Chapter 2: Numerical Solutions of Algebraic and Transcendental equations1.IntroductionIn this chapter, we shall consider the problem of numerical computation of real root of a givenequation0)(=xf, which may be algebraic or trigonometric or transcendental. It will be assumedthat the function)(xfis continuously differentiable sufficient number of times.All the methods for numerical solution of equations will consist of two stages. The first stage,called location of root at which rough values of the root are obtained and the second stage whichconsists in improvement of rough value of each root to any desired degree of accuracy.In the second stage, a method of improvement of the rough value of a root will generate asequence of successive approximation or iterates}0{≥nxn, starting with initial rough value0xofthe rootαobtained in the first stage, such thatα→nxas∞→n.2.Basic concepts and definitions2.1Sequence of successive approximationsLet,}{nxbe a sequence of successive approximations for a desired rootαof the equation0)(=xf.The errornεat thenth iteration is defined bynnx-=αε(2.1.1)And we definednhby11++-=-=nnnnnxxhεε,(2.1.2)

2which may be considered as an approximation ofnε.The iteration process converges if and only if0→nεas∞→n.2.2 Order of convergenceDefinition:If an iterative method converges and two constants1≥pand0>Cexists such thatCpnnn=+∞→εε1lim(2.2.1)Thenpis called the order of convergence of the method andCis called asymptotic errorconstant.A sequence of iterates}0{≥nxnis said to converge with order of convergence1≥pto a rootαifpnnkεε≤+1,0≥n(2.2.2)for some0>k. If1=p, then the sequence of iterates}0{≥nxnis said to be linearly convergent. If2=p, then the iterative method is said to have quadratic convergence.3.Initial approximation3.1 Graphical methodIn this method, we plot the graph of the curve)(xfy=on the graph paper; the point at which thecurve crosses thex-axis, gives the root of the equation0)(=xf, any value in the neighbourhoodof this point may be taken as an initial approximation to the required root.

3Fig 1Graph ofxxexy-=cosSometimes the equation0)(=xfcan be written in the form)()(xhxg=where the graphs of)(xgy=and)(xhy=may be conveniently drawn. In that case the abscissae of the point ofintersection of the two graphs gives the required root of the0)(=xfand therefore any value inthe neighbourhood of this point can be taken as initial approximation to the required root. Fig.1shows the graph ofxxexy-=cosand Fig. 2 shows the graphs ofxy=andxexycos=. The abscissaeof the point of intersection of these two graphs give the required root of the0)(=xf.

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