Taylor Polynomials restart;with(plots); with(student); with(numericalanalysis); with(student[numericalanalysis]);

Transcription

1 Talor Polnomials Talor's Theorem is used etensivel in Numerical Analsis and is the basis for the development of several important techniques. We begin b restarting and loading the plots package for some etra plotting commands. restart;with(plots); animate, animate3d, animatecurve, arrow, changecoords, compleplot, compleplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, densitplot, displa, dualaisplot, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d, inequal, interactive, interactiveparams, intersectplot, listcontplot, listcontplot3d, listdensitplot, listplot, listplot3d, loglogplot, logplot, matriplot, multiple, odeplot, pareto, plotcompare, pointplot, pointplot3d, polarplot, polgonplot, polgonplot3d, polhedra_supported, polhedraplot, rootlocus, semilogplot, setcolors, setoptions, setoptions3d, spacecurve, sparsematriplot, surfdata, tetplot, tetplot3d, tubeplot We net load the package called NumericalAnalsis in the Student package. It can be accessed in two was. The first wa is with(student); Calculus1, LinearAlgebra, MultivariateCalculus, NumericalAnalsis, Precalculus, SetColors, VectorCalculus with(numericalanalsis); AbsoluteError, AdamsBashforth, AdamsBashforthMoulton, AdamsMoulton, AdaptiveQuadrature, AddPoint, ApproimateEactUpperBound, ApproimateValue, BackSubstitution, BasisFunctions, Bisection, CubicSpline, DataPoints, Distance, DividedDifferenceTable, Draw, Euler, EulerTutor, EactValue, FalsePosition, FiedPointIteration, ForwardSubstitution, Function, InitialValueProblem, InitialValueProblemTutor, Interpolant, InterpolantRemainderTerm, IsConvergent, IsMatriShape, IterativeApproimate, IterativeFormula, IterativeFormulaTutor, LeadingPrincipalSubmatri, LinearSolve, LinearSstem, MatriConvergence, MatriDecomposition, MatriDecompositionTutor, ModifiedNewton, NevilleTable, Newton, NumberfSignificantDigits, PolnomialInterpolation, Quadrature, RatefConvergence, RelativeError, RemainderTerm, Roots, RungeKutta, Secant, SpectralRadius, Steffensen, Talor, TalorPolnomial, UpperBoundfRemainderTerm, VectorLimit Most of these new commands are appropriate to our class. Unless one is using several packages in Student simultaneousl, a more compact wa of opening our package is: with(student[numericalanalsis]); AbsoluteError, AdamsBashforth, AdamsBashforthMoulton, AdamsMoulton, AdaptiveQuadrature, AddPoint, ApproimateEactUpperBound, ApproimateValue, BackSubstitution, BasisFunctions, Bisection, CubicSpline, DataPoints, Distance, DividedDifferenceTable, Draw, Euler, EulerTutor, EactValue, FalsePosition, FiedPointIteration, ForwardSubstitution, Function, 1

2 InitialValueProblem, InitialValueProblemTutor, Interpolant, InterpolantRemainderTerm, IsConvergent, IsMatriShape, IterativeApproimate, IterativeFormula, IterativeFormulaTutor, LeadingPrincipalSubmatri, LinearSolve, LinearSstem, MatriConvergence, MatriDecomposition, MatriDecompositionTutor, ModifiedNewton, NevilleTable, Newton, NumberfSignificantDigits, PolnomialInterpolation, Quadrature, RatefConvergence, RelativeError, RemainderTerm, Roots, RungeKutta, Secant, SpectralRadius, Steffensen, Talor, TalorPolnomial, UpperBoundfRemainderTerm, VectorLimit To help understand Talor's Theorem, we look at several Talor polnomials P n ( ) for the function f ( ) = 1 about = 1. We begin b entering the function as an epression. restart;with(plots):with(student[numericalanalsis]): f:=1/; f := 1 We use the command TalorPolnomial to form the Talor polnomials for the function. The first argument is the function epression, the second is the, the third is the list of the orders we want. t:=talorpolnomial(f, = 1,order=[1,2,3,4,,6,3]); t := 2 K, 2 K C K 1 2, 2 K C K 1 2 K K 1 3, 2 K C K 1 2 K K 1 3 C K 1 4, 2 K C K 1 2 K K 1 3 C K 1 4 K K 1, 2 K C K 1 2 K K 1 3 C K 1 4 K K 1 C K 1 6, 2 K C K 1 2 K K 1 3 C K 1 4 K K 1 C K 1 6 K K 1 7 C K 1 8 K K 1 9 C K 1 1 K K 1 11 C K 1 12 K K 1 13 C K 1 14 K K 1 1 C K 1 16 K K 1 17 C K 1 18 K K 1 19 C K 1 2 K K 1 21 C K 1 22 K K 1 23 C K 1 24 K K 1 2 C K 1 26 K K 1 27 C K 1 28 K K 1 29 C K 1 3 Note that, besides a list, order= can be followed b a single number or a range such as t is now a Maple list of 7 polnomials. We use a for loop to epand the polnomials. for i from 1 to 6 do P[i]:=epand(t[i]) end do; P[3]:=epand(t[7]); P 1 := 2 K P 2 := 3 K 3 C 2 P 3 := 4 K 6 C 4 2 K 3 P 4 := K 1 C 1 2 K 3 C 4 P := 6 K 1 C 2 2 K 1 3 C 6 4 K P 6 := 7 K 21 K 3 3 C 21 4 K 7 C 6 C 3 2 P 3 := 31 K 46 K C K C K C

3 C K C K C K C K C K C K K C C K C K C K C K C 3 Since TalorPolnomial can onl take positive integers for order, we define P = 1. P[]:=1; P := 1 Now let's plot P (in blue) and f (in red) together. plot:=plot(f,=..4,=-..1,color=red): plot1:=plot(p[],=..4,=-..1,color=blue): displa(plot,plot1); 1 K Note that the function and the degree Talor polnomial have the same value at = 1. However, as ou var from = 1, the constant polnomial is hardl a good approimation to the function. Let's see what happens as we move up a degree to a first order Talor polnomial. plot1:=plot(p[1],=..4,=-..1,color=blue): displa(plot,plot1); This is the linear approimation to f at = 1. Here both the function and the degree one Talor 3

4 polnomial have the same value and the same derivative at = 1, so if we sta close enough to = 1, we can use the polnomial, which is alwas eas to evaluate, to approimate the function. Let's advance to a second order Talor polnomial. 1 K plot1:=plot(p[2],=..4,=-..1,color=blue): displa(plot,plot1); 4

5 1 K Now the graphs also have the same second derivative at = 1 (both are concave up), allowing us to move further from = 1 than before and still get good approimations. Using diff (for taking the derivative of an epression) and eval, let's check that the first and second derivatives for the Talor polnomial and function are reall both the same. For the first derivatives: eval(diff(f,),=1); K1 So the first derivatives are reall the same. Note the use of the $ operator to get the second derivative. For the second derivatives: eval(diff(f,$2),=1); 2 eval(diff(p[2],),=1); eval(diff(p[2],$2),=1); K1 same as the degree of the Talor polnomial, giving ever better approimations further and further from = 1. For the third degree: plot1:=plot(p[3],=..4,=-..1,color=blue): 2 The are also the same. It continues that the number of derivatives matching at the chosen value is the

6 displa(plot,plot1); 1 K Here the Talor polnomial looks to be an ecellent approimation from.6 to 1.6. For the fourth degree: plot1:=plot(p[4],=..4,=-..1,color=blue): displa(plot,plot1); 6

7 1 K This looks good from. to 1.7. For the fifth degree: plot1:=plot(p[],=..4,=-..1,color=blue): displa(plot,plot1); 7

8 1 K Now we get good approimations from.4 to 1.7. For degree si: plot1:=plot(p[6],=..4,=-..1,color=blue): displa(plot,plot1); 8

9 1 K This looks good from.4 to 1.8. Now let's jump to degree 3: plot1:=plot(p[3],=..4,=-..1,color=blue): displa(plot,plot1); 9

10 1 K This now looks good from.2 to 1.9. Let's check the amount of error in the Talor polnomial approimation for = actual:=evalf(subs(=18/1,f)); actual :=.6 appro3:=evalf(subs(=18/1,p[3])); appro3 := err:=abs(appro3-actual); err :=.19 We are off b about 1. Prett good. 2 1

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