Quantitative Methods Answers

a) Find the interval of unit length that contains the smallest positive root of the
equation f(x) = x^3 −5x^2 +1 = 0. Starting with this interval, find an interval of
length 0.05 or less that contains the root, by Bisection method.
b) Taking the endpoints of the last interval obtained in part a) above as the initial
approximations, perform two iterations of the secant method to approximate the
root.
c) Determine the maximum error in quadratic interpolation at equispaced points.

Set up the iteration method in matrix form for the following system of linear
equations:
3x −t = 1
4x −y + z = 4
x +2z = 1
−x −y−t = 2
Further, determine whether the method converges or not.

Civil and transportation engineers must often estimate the future traffic flow on roads and bridges to plan for maintenance or possible future expansion. The following data gives the number of vehicles (in millions) crossing a bridge each year for 10 years. Fit a cubic polynomial to the data and use the fit to estimate the flow in year 2000.
Year 1990 1991 1992 1993 1994 Vehicle flow (millions) 2.1 3.4 4.5 5.3 6.2
Year 1990 1991 1992 1993 1994 Vehicle flow (millions) 6.6 6.8 7 7.4 7.8

Determine the value of the integral
I =
Z 1
0
x
p
1−x
2 dx
by composite trapezoidal rule with 3 and 5 ordinates. Improve the result by using
extrapolation technique.

The interpolating polynomial of degree \\(\\leq n\\)with the nodes\\(x_{0}, x_{1},\\cdots, x_{n}\\) can be written as

9.One of these describes the Lagrange’s interpolating polynomial P(x)
a.\\(p(x) = L_{1}(x)f_{0} + L_{2}(x)f_{1} + L_{3}(x)f_{2} + L_{4}(x)f_{3}\\)
b.\\(P(x) = L_{0}(x)f_{0} + L_{1}(x)f_{1} + L_{2}(x)f_{2} + L_{3}(x)f_{3}\\)
c.\\(P(x) = L_{0}(x)f_{3} + L_{1}(x)f_{2} + L_{2}(x)f_{1} + L_{3}(x)f_{0}\\)
d.\\(P(x) = L_{0}(x)f_{3} + L_{1}(x)f_{2} + L_{2}(x)f_{1} + L_{3}(x)f_{0}\\)

10.One of these is a method of solving system of linear equations:
a.Square Method
b.Inverse method
c.Lagrange Method
d.Transformation Method

7.Iterative methods of the solutions of systems of equations are ________________
a.finite
b.sequential
c.infinite
d.non-sequential

8.Stirling\'s formula for interpolation is given by
a.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2}d^2f_0\\)
b.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2!}d^2f_0\\)
c.\\(P_n (x) = f (x_0) + \\frac{s - 1}{2}[df_1+ d_\\frac{-1}{2}] + \\frac{s^2}{2!}d^2f_0\\)
d.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2!}df_0\\)

5.The technique of determining an approximate value of f(x) for a non-tabular value of x which lies in the internal [a, b] is referred to as
a.interpolation
b.extrapolation
c.exterpolation
d.intrapolation

6.If all the elements above the main diagonal of a square matrix vanish, the matrix is known as
a.side triangular
b.half triangular
c.lower triangular
d.upper triangular

3.The zeroeth divided difference of the function f, with respect to \\( x_{i} \\) denoted by \\( f[x_{i}]\\) can be written as
a.\\(f[x] = f(xi)\\)
b.\\(f[x_{0}] = f(x_{i})\\)
c.\\(f[x] = f(x_{i})\\)
d.\\( f[x_{i}] = f(x_{i})\\)

4.When the number of rows is equal to the number of columns in a matrix, it is called _________ matrix
a.square
b.equal
c.similar
d.singular

1.The interpolating polynomial of degree \\(\\leq n\\)with the nodes\\(x_{0}, x_{1},\\cdots, x_{n}\\) can be written as
a.\\( f[x_{i}] = f(x_{i})\\)
b.\\( f[x_{i}] + f(x^i)=1\\)
c.\\( f[x_{i}] - f(x_{i}=0)\\)
d.\\( f[x_{i}] +f(x_{i}=0)\\)

2. A matrix in which all the non-diagonal elements vanish is called ___________ matrix.
a.diagonal
b.vanishing
c.non-diagonal
d.non-vanishing