Reference no: EM132641028

Write a computer program for each problem.

Question 1. Use the lagrange interpolation polynomial that pisses through the following data points:

to estimate the ue of y at x = -2.5.

2. For the following data set

(a) construct two types of cubic spline approximation (with the efficient implementation) using the following boundary conditions:
i. f"(x_o) = f"(x_n) = 0
ii = f"(x_o) = f"(x₁) and f"(x_n) = f"(x_n-1)
(b) plot the two cubic splines in the same figure over the interval x  ∈ (0,31 for comparison.

(c) use the above cubic splines to evaluate the y value at x = 2.5.

3. Using the data in the following table to fit a quadratic polynomial to the data:

namely, find a quadratic polynomial

y = a_o + a₁x + a₂x²

where a₀, a₁ and a₂ are the three coefficients to be determined using the least square approach.

(a) Approach I: form an overdetermined system Ax = b
y_t = a_o + a₁x₁ + a₂x_t² for 0 ≤  i ≤ 5
then form a square system by multiplying the overdetermined system by A^T, the transpose of A, i.e.
A^TAx = A^Tb
which can be solved using LAPACK's LU decomposition routines.

(b) Approach 2: use the following LAPACK routine

void dgels_(char *TRANS ,int *M, int *N, int *NRHS, double *A, int *LDA, double *b, int *LDB, double *WORK, int *LWORK, int *INFO):
to directly solve the overdetermined system.

(c) Approach 3: follow the standard least square procedure to minimize the sum of error square

minimize ε = ∑⁵_t=0(a₀ + a₁x₁ + a₂x_i² - y_t)²

which can be achieved by setting the partial derivative of ε with respect to a₀, a₁ and a₂ equal to zero:
∂ε/∂a₀ = 0

∂ε/∂a₁ = 0

∂ε/∂a₂ = 0

which forms a 3 x 3 linear system to be solved.