- Consider f âˆˆ C^4 [a, b], n even, h = (bâˆ’a)/n , x_j = a + jh, j = 0, 1, …, n:

(a) Determine the values of n and h required to approximate R Ï€ 0 cos (2x)dx to within 10âˆ’6 by the Composite Sympsonâ€™s rule.

(b) Write the approximation formula for n = 6 for the function given in part (a).

- Consider the matrix

A =[

1 2 âˆ’1

2 4 0

0 1 âˆ’1]

(a) Show that the matrix A is invertible. Is the matrix A symmetric ? Strictly

diagonally dominant ? Positive definite ? Justify your answers.

(b) Find a permutation matrix P such that P A can be factored into the product

LU, where L is lower triangular with 1â€™s on its diagonal, and U is upper triangular.

Find the matrices L and U.

(c) Using (b), solve the linear system

Ax =[

1

1

0]

3.True or False

(a) To solve a linear system of equations it is necessary to compute Aâˆ’1.

(b) It is not possible for a numerical integration formula to give an exact

answer.

(c) High order interpolation at equispaced points can be expected to give

large errors if there is noise present in the function being interpolated.

(d) In Gaussian quadrature one uses equispaced integration nodes and

determines the weights so that the formula is exact for as many monomials as

possible.

(e) For three distinct data points, an interpolating polynomial through

those points will always be a quadratic polynomial.

4. Multiple choice

#1.The following (x, f(x)) data is given

x 15 18 22

f(x) 48 74 50

The Newtonâ€™s divided difference second order polynomial for the above data is

given by

a_0 + a_1(x âˆ’ 15) + a_2(x âˆ’ 15)(x âˆ’ 18).

The value of a1 is most nearly

(A) -6.00

(B) 0.2857

(C) 8.666

(D) 48.00

#2.The polynomial that passes through the following (x, f(x)) data

x 18 22 24

f(x) ? 50 246

is given by 16.250x^2 âˆ’ 649.50x + 6474, 18 â‰¤ x â‰¤ 24

The corresponding polynomial using Newtonâ€™s divided difference polynomial is

given by a_0 + a_1(x âˆ’ 18) + a_2(x âˆ’ 18)(x âˆ’ 22).

The value of a_2 is:

(A) 0.5000

(B) 16.25

(C) 48.00

(D) not enough information

#3. The following table is given:

x 1.8 2.0 2.2 2.4 2.6

f(x) 6.0496 7.3890 9.0250 11.023 13.464

Using the forward finite difference formula with a step size of h = 0.2, the

derivative of the function f at x = 2 can be approximated by the value:

(A) 6.697

(B) 7.389

(C) 7.438

(D) 8.179

#4. The Composite Trapezoidal Rule for numerical integration is exact when integrating at most ……….. degree polynomials.

(A) first

(B) second

(C) third

(D) fourth

#5. The value of intergrate from 0.2 to xe^x dx by using the Composite Trapezoidal Rule with n = 3 subintervals is most nearly

(A) 11.672

(B) 11.807

(C) 12.811

(D) 14.633

#6. The value of intergrate form 0.2 to e^x dx by using Composite Simpsonâ€™s Rule using n = 4 subintervals is most nearly

(A) 7.8036

(B) 7.8062

(C) 7.8423

(D) 7.9655

#7 For a definite integral of any third degree polynomial, the two-point Gauss quadrature rule will give the same result as the

(A) Trapezoidal Rule

(B) Composite Tapezoidal Rule (two subintervals)

(C) Composite Trapezoidal Rule (three subintervals)

(D) Simpsonâ€™s Rule

#8. The value of intergrate from 0.2 to 2.2 x e^x dx by using the two-point Gauss quadrature rule is most nearly

(A) 11.672

(B) 11.807

(C) 12.811

(D) 14.633