Andrew Aggarwala 9/16/99

HW #1

1) Involves running and explaining the previously written program epsi.f. The output is given below.

0.500000 0.707107

0.250000 0.500000

0.125000 0.353553

6.25000E-02 0.250000

3.12500E-02 0.176777

1.56250E-02 0.125000

7.81250E-03 8.83883E-02

3.90625E-03 6.25000E-02

1.95312E-03 4.41942E-02

9.76562E-04 3.12500E-02

4.88281E-04 2.20971E-02

2.44141E-04 1.56250E-02

1.22070E-04 1.10485E-02

6.10352E-05 7.81250E-03

3.05176E-05 5.52427E-03

1.52588E-05 3.90625E-03

7.62939E-06 2.76214E-03

3.81470E-06 1.95312E-03

1.90735E-06 1.38107E-03

9.53674E-07 9.76562E-04

4.76837E-07 6.90534E-04

2.38419E-07 4.88281E-04

1.19209E-07 3.45267E-04

5.96046E-08 2.44141E-04

The purpose of epsi.f is to determine the "machine epsilon" (eps), which is defined as the smallest floating point number such that (1 + eps > 1). After being initialized with a value of 1.0, eps is used in a recursive loop where it's value is halved each iteration. The loop will end only when (eps + 1) is greater then 1. The fact that the loop ever ends proves that round off error occurs within the computer, and that the value of

(1+epsmax) where epsmax <= 1.00E-08 is equal to 1 in the computers calculations.

2) Work out the Taylor Polynomial for cos(x) using Maple and explain.

> fun:=cos(x); [define cos(x)]

fun := cos(x)

> tylrs:=taylor(fun,x=a); [using the taylor function, develop the taylor series expansion of cos(x) in the general form]

tylrs := cos(a) - sin(a) (x - a) - 1/2 cos(a) (x - a)^2 + 1/6 sin(a) (x - a)^3

+ 1/24 cos(a) (x - a)^4 - 1/120 sin(a) (x - a) ^5 + O((x - a) ^6)

> tylrp:=convert(tylrs,polynom); [converts the taylor series expansion into a taylor polynomial by dropping the Remainder term]

tylrp := cos(a) - sin(a) (x - a) - 1/2 cos(a) (x - a) ^2 + 1/6 sin(a) (x - a)^3

+ 1/24 cos(a) (x - a)^4 - 1/120 sin(a) (x - a)^5

> tylrs3:=taylor(fun,x=0,4);tylrs9:=taylor(fun,x=0,10); [form the taylor series of thrid and ninth order respectively]

tylrs3 := 1 - 1/2 x^2 + O(x^4 )

tylrs9 := 1 - 1/2 x^2 + 1/24 x^4 - 1/720 x^6 + 1/40320 x^8 + O(x^10 )

> tylrp3:=convert(tylrs3,polynom);tylrp9:=convert(tylrs9,polynom); [convert the above taylor series into Taylor Polynomials]

tylrp3 := 1 - 1/2 x^2

tylrp9 := 1 - 1/2 x^2 + 1/24 x^4 - 1/720 x^6 + 1/40320 x^8

> y0:=evalf(subs(x=0.1,fun));y3:=evalf(s\

> ubs(x=0.1,tylrp3));y9:=evalf(subs(x=0.1,tylrp9)); [Evaluate the actual function cos(x), the 3^rd and 9^th order Taylor Polynomials at x=0.1]

y0 := .9950041653

y3 := .9950000000

y9 := .9950041653

> error3:=abs(y3-y0);error9:=abs(y9-y0); [Computes the error term between the 3^rd order approximation with the actual function, and the 9^th order approx. with the actual function. As expected the higher order approximation gives the exact solutionàexact to the machine epsilon of 1.00E-08]

error3 := .41653 10E-5

error9 := 0