Quiz Chapter 04.06: Gaussian Elimination

1. The goal of forward elimination steps in the Naïve Gauss elimination method is to reduce the coefficient matrix to a(n) _____ matrix.

2. Division by zero during forward elimination steps in Naïve Gaussian elimination of the set of equation \left[ A \right] \left[ X \right] = \left[ C \right] implies the coefficient matrix \left[ A \right] is

3. Using a computer with four significant digits with chopping, Naïve Gauss elimination solution to

• • • 0.0030x_{1} + 55.23x_{2} = 58.12
    • 6.239x_{1} - 7.123x_{2} = 47.23

is

4. Using a computer with four significant digits with chopping, Gauss elimination with partial pivoting solution to

• • • 0.0030x_{1} + 55.23x_{2} = 58.12
    • 6.239x_{1} - 7.123x_{2} = 47.23

is

5. At the end of forward elimination steps of Naïve Gauss Elimination method of the following equations

\begin{bmatrix} 4.2857 \times 10^{7}&-9.2307 \times 10^{5}&0&0 \\ 4.2857 \times 10^{7}&-5.4619 \times 10^{5}&-4.2857 \times 10^{7}&5.4619 \times 10^{5} \\ -6.5&-0.15384&6.5&0.15384 \\ 0&0&4.2857 \times 10^{7}&-3.6057 \times 10^{5} \\ \end{bmatrix} \begin{bmatrix} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \\ \end{bmatrix} = \begin{bmatrix}-7.887 \times 10^{3} \\ 0 \\ 0.007 \\ 0 \\ \end{bmatrix}

the resulting equations in the matrix form are given by

\begin{bmatrix} 4.2857 \times 10^{7}&-9.2307 \times 10^{5}&0&0 \\ 0&3.7688 \times 10^{5}&-4.2857 \times 10^{7}&5.4619 \times 10^{5} \\ 0&0&-26.9140&0.579684 \\ 0&0&0&5.62500 \times 10^{5} \\ \end{bmatrix} \begin{bmatrix} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \\ \end{bmatrix} = \begin{bmatrix}-7.887 \times 10^{3} \\ 7.887 \times 10^{3} \\1.19530 \times 10^{-2} \\ 1.90336 \times 10^{4} \\ \end{bmatrix}

The determinant of the original coefficient matrix is

6. The following data is given for the velocity of the rocket as a function of time. To find the velocity at t=21 s, you are asked to use a quadratic polynomial, v(t) = at^{2} + bt + c to approximate the velocity profile.

The correct set of equations will find a, b, and c are