Quantitative Methods Answers

A new design of nitrogen charged accumulator, for use in a hydraulic circuit, is being factory tested for certification purposes.(http://www.accumulators.co.uk/). The following data has been collected by a technician:
o Gas volume: V = 0.8 m3
o Gas pressure: p = 160 kPa
o Compression index: n = 1.4
o Pressure/volume relationship: pV ⁿ = C
o Work done =
i. Evaluate C.
ii. Calculate the work done in compressing the nitrogen from 0.8 m3 to 0.6 m3 using:
o integral calculus
o Simpson’s rule
o the mid-ordinate rule.

A new design of nitrogen charged accumulator, for use in a hydraulic circuit, is being factory tested for certification purposes.(http://www.accumulators.co.uk/). The following data has been collected by a technician:
o Gas volume: V = 0.8 m3
o Gas pressure: p = 160 kPa
o Compression index: n = 1.4
o Pressure/volume relationship: pV ⁿ = C
o Work done =
i. Evaluate C.
ii. Calculate the work done in compressing the nitrogen from 0.8 m3 to 0.6 m3 using:
o integral calculus
o Simpson’s rule
o the mid-ordinate rule.

iii. Comment on the accuracy of the two numerical methods.

A new design of nitrogen charged accumulator, for use in a hydraulic circuit, is being factory tested for certification purposes.(http://www.accumulators.co.uk/). The following data has been collected by a technician:
o Gas volume: V = 0.8 m3
o Gas pressure: p = 160 kPa
o Compression index: n = 1.4
o Pressure/volume relationship: pV ⁿ = C
o Work done =
i. Evaluate C.
ii. Calculate the work done in compressing the nitrogen from 0.8 m3 to 0.6 m3 using:
o integral calculus
o Simpson’s rule

The following table of values of f(x) is given:
x →0.25 0.30 0.35
f(x)→1.25 1.90 1.68

Compute f'(x) at x=0.30 using an O(h²) formula.

6 0 0.5| x' =1
0.5 2 - 1 | x'' = - 1
3 1 0 | x''' =2

Solve the system by LU decomposition method

Find the Gershgorin bounds on the eigenvalues of the matrix
6 2 1
1 - 5 0
2 1 4

And also draw a rough sketch of the bounds

Determine a unique polynomial f(x) of degree ≤3 such that f(x_0)=1,f'(x_0)=-1,f(x_1)=0 and f'(x_1)=-1,where x_0+h=x_1.

Perform three iterations of the inverse power method to obtain the smallest eigenvalue in magnitude of the matrix [{2,5},{3,6}].
Take appropriate initial approximation to the eigenvector.

Set up a Gauss Seidel iteration scheme for the system of equations given below
[{6,0,1/2},{1/2,2,-1},{3,1,0}] [{x_1},{x_2},{x_3}]=[{1},{-1},{2}].
How many iteration are required to reach the accuracy of 5×10^(-4) ?

Find by Newton's method the root of the equation
xlog_(10 )x=4
near x=6 correct to 3 decimal places of digits.