Homework III

Consider data points (0,0), (1,1), (2,4), (3,3).

Set up the linear system of equations using piecewise cubic polynomial interpolation (cubic spline) with the knots placed at the midpoints of the data intervals.

Determine the coefficients of the linear system.

Write the interpolating piecewise polynomial for each data interval.

On I_i=[x_i,x_i+1], assume the cubic has the following form:

P_i(x) = a_i(x_i+1-x)³/ h_i + a_i+1(x-x_i)³/ h_i + b_i(x_i+1-x)+c_i(x-x_i)

Define the cubic polynomials for each interval as:

§ P₁(x), x Є [z₁,z₂]

§ P₂(x), x Є [z₂,z₃]

§ P₃(x), x Є [z₃,z₄]

§ P₄(x), x Є [z₄,z₅]

Please show all your work clearly.