1.7.2. Applied Area.

We will present two applications. The first involves the unknown trajectory of a mass (artillery shell). The second application is somewhat related and requires a polynomial approximation to a function that is difficult to compute. Later we will illustrate some of the numerical difficulties.

Application to Trajectories. A mass is observed several times. Based upon these observations and basic laws of motion, we want to determine the path of the mass. This will allow us compute where the mass came from and where it will land.

Figure: Trajectory

The simplest model of motion is in 2D with the only force being gravitation in the vertical direction. In this case the path can be described in terms of the initial position (x(0),y(0)) = (x₀,y₀) and initial velocity (u(0),v(0) = (u₀,v₀) as

₀

y(t) = v₀t + y₀- 16t².

These four parameters are the unknowns, and they will require four equations which can be gotten from two observations of the position. Let (x₁,y₁) and (x₂,y₂) be the observed positions at times t₁ and t₂ so that

₀

₁

₀

₁

v₀t₁ + y₀- 16t₁²= y₁,

u₀t₂ + x₀= x₂and

v₀t₂ + y₀- 16t₂² = y_2.

_{The first and third equations decouple and have the matrix form}

If the mass is moving fast, the observations will have to be close together which means the t₁ and t₂ will be close. Thus the lines in the (x₀,u₀) space will be almost parallel. Another way of viewing this is to do the first row operation

If t₂- t₁is small, then significant roundoff errors can occur in the calculation of u₀.

Application to Function Approximation. Many functions are very expensive to evaluate, but often segments of the graphs look like graphs of polynomials. In order to reduce the function evaluation times, these functions can be approximated by a variety of polynomials. One type is the Lagrange interpolation polynomials. The coefficients in an n^th order polynomial are determined by setting the polynomial at distinct points equal to the function evaluated at these points. For example, if the functions looks like a quadratic polynomial near the points x₁, x₂ and x_3,then we can find the three coefficients of

P(x) = a₀ + a₁x + a₂x²

by solving the system associated with P(x_i) = f(x_i) for i = 1,2,3. The matrix form is

If two of the interpolation points are close, then the computed solution may have significant error.

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