Download Assignment I 2022 (MCSC 202)

Assignment I
Subject: Numerical Methods (MCSC 202)
Group: GE/CE/MEI/CHE
1. rite the numbers of significant digits:
0.003207,
1.302076,
6.23000,
325.000,
2. Round off the following numbers:
3.237500 (3 decimal places), 3.234500 (3 decimal places),
3.23456 (3 decimal places), 3.1405 (Four significant figures)
40.00203,
10.203
1
4. The approximate values of the number 7 are given as 0.142, 0.140, and 0.143, which of these three
values is the best approximation? Why?
3. Compute: i) 1.3254 + 0.56 + 27.2879604 + 0.0375 ii) 4.6 0.128
iii)
0.995×1.53
1.592
iv) √2.01 −
√2 and find the absolute, relative and percentage errors.
4. Round of the number 47.5689 to four significant figures and then compute the absolute, relative and
percentage error.
5. Compute the maximum absolute and relative error:
a) 𝑢 =
5𝑥 3 𝑦2
when 𝑥 = 3.25, 𝑦 = 45.129 and 𝑧 = 0.577.
𝑧5
4𝑥 2 𝑦3
b) If 𝑢 =
𝑧4
and errors in 𝑥, 𝑦, 𝑧 be 0.001 when 𝑥 = 𝑦 = 𝑧 = 1.
6. If 𝑦 = 4𝑥 6 − 5𝑥, Find the percentage error in 𝑦 at 𝑥 = 1 if the error in 𝑥 = 0.04.
7. Find the relative error in computation of 𝑥 − 𝑦 for 𝑥 = 12.05 and 𝑦 = 8.02 having absolute error ∆𝑥 =
0.005, and ∆𝑦 = 0.001.
8. Prove that the relative error of a product of three non-zero numbers do not exceed the sum of the relative
errors of the given numbers.
9. Calculate a real root correct to four significant figures of the following equations:
a) Bisection method:
i) 𝑥 3 − 4𝑥 − 9 = 0, (2, 3)
ii)𝑥 3 + 𝑥 2 + 𝑥 + 7 = 0, (−3, −2)
b) False Position:
i) 𝑥𝑙𝑜𝑔10 𝑥 − 1.2 = 0, (2,3)
ii) 𝑥 3 − 4𝑥 − 9 = 0, (2,3)
c)Secant method:
i) 𝑒 𝑥 + 𝑥 2 − 𝑥 − 4 = 0, (0,4)
ii)𝑥 3 + 𝑥 2 + 𝑥 + 7 = 0, (−3, −2)
d) Newton-Raphson:
i) 𝑐𝑜𝑠𝑥 − 𝑥𝑒 𝑥 = 0, 𝑥0 = 1
ii)1.05 − 1.04𝑥 + 𝑙𝑛𝑥 = 0, 𝑥0 = −1
𝑥
e)Fixed point iteration: i) 𝑥𝑒 = 1, 𝑥0 = 1
ii) 𝑐𝑜𝑠𝑥 = 3𝑥 − 1, 𝑥0 = 2
10. Show that the equation 𝑓(𝑥 ) = 𝑥 3 − 𝑥 2 − 𝑥 + 1 = 0 has a double root at 𝑥 = 1, and find a double root
of the equation using generalized multiplicity Newton Raphson Method with initial guess 𝑥0 = 0.8 and
tolerance 𝜖 = 10−4
11. Solve the following systems of nonlinear equations using Newton-Raphson method accurate to 3
decimal places:
i) 𝑥 2 + 4𝑦 2 − 16 = 0, 𝑥𝑦 2 − 4 = 0, 𝑥0 = 𝑦0 = 1.5
ii) 𝑠𝑖𝑛𝑥 − 𝑦 = 0.9793, 𝑐𝑜𝑠𝑥 − 𝑥 = −0.6703 with 𝑥0 = 0.5, 𝑦0 = 1.5
12. Solve the following system of equations using iteration method correct to 3 significant figures
i) 𝑥 2 − 2𝑥 − 𝑦 + 0.5 = 0, 𝑥 2 + 4𝑦 2 − 4 = 0, 𝑥0 = 0, 𝑦0 = 1
ii) 𝑦 2 − 5𝑦 + 4 = 0,3𝑦𝑥 2 − 10𝑥 + 7 = 0, 𝑥0 = 1 , 𝑦0 = 1
13. Solve the following systems using LU decomposition method taking 𝑙𝑖𝑖 = 1
𝑥 + 2𝑦 + 3𝑧 = 14
2𝑥 + 20𝑦 + 26𝑧 = 120
3𝑥 + 26𝑦 + 70𝑧 = 265
14. Solve the following systems using LU decomposition method taking 𝑢𝑖𝑖 = 1
4𝑥 + 8𝑦 + 4𝑧 = 28
𝑥 + 5𝑦 + 4𝑧 − 3𝑤 = 13
𝑥 + 4𝑦 + 7𝑧 + 2𝑤 = 23
𝑥 + 3𝑦 − 2𝑤 = 4
15. Solve the following tidiagonal systems using Thomas Algorithm:
4𝑥 + 8𝑦 = 8
8𝑥 + 18𝑦 + 2𝑧 = 18
2𝑦 + 5𝑧 + 1.5𝑤 = 0.5
1.5𝑧 + 1.75𝑤 = −1.75
16. Solve the system of equations using Jacobi method accurate to 3 significant figures using the given
initial approximation:
4𝑥 + 𝑦 + 𝑧 + 𝑤 = −5
𝑥 + 8𝑦 + 2𝑧 + 3𝑤 = 23
, (0,0,0)
𝑥 + 2𝑦 − 5𝑧 = 9
−𝑥 + 2𝑧 + 4𝑤 = 4
17. State the condition of convergence of Gauss Seidel Iterative method. Apply this method to solve the
system correct to 3 decimal places:
28𝑥 + 4𝑦 − 𝑧 = 32
2𝑥 + 17𝑦 + 4𝑧 = 35
𝑥 + 3𝑦 + 10𝑧 = 24
The End