Numerical Analysis Questions and Answers – Iteration Method

This set of Numerical Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Iteration Method”.

1. What will be the root of the equation f(x) = 3x4 – 2x2 using iteration method?
a) 0
b) 1
c) 2
d) 3
View Answer

Answer: a
Explanation: Given,
Let f(x) = 3x4-2x2
Here 3x4-2x2 = 0
3x4 = 2x2
x4 = ((2x2)/3)
x = \(\sqrt[4]{\frac{(2x^2)}{3}}\)
\(\phi(x) = \sqrt[4]{\frac{(2x^2)}{3}}\)

Here

x 0 1
f(x) 0 1

Here f(0) = 0

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Root of the equation 3x4-2x2 is 0.

2. What will be the root of the equation f(x) = tan(x) using iteration method?
a) -0.0573
b) -0.0572
c) 2.4587
d) -0.0571
View Answer

Answer: d
Explanation: Given,
Considering f(x) = tan(x) as \(\phi(x)\) = tan(x)
We can also directly input \(\phi(x)\) value like x = value format, e.g. x = cos(x)/exp(x)
x = tan(x)
\(\phi(x)\) = tan(x)
x1 = \(\phi(x0) = \phi(2)\) = -2.185
x2 = \(\phi(x1) = \phi(-2.185)\) = 1.4179
x3 = \(\phi(x2) = \phi(1.4179)\) = 6.4906
x4 = \(\phi(x3) = \phi(6.4906)\) = 0.2104
x5 = \(\phi(x4) = \phi(0.2104)\) = 0.2136
x6 = \(\phi(x5) = \phi(0.2136)\) = 0.2169
x7 = \(\phi(x6) = \phi(0.2169)\) = 0.2203
x8 = \(\phi(x7) = \phi(0.2203)\) = 0.224
x9 = \(\phi(x8) = \phi(0.224)\) = 0.2278
x10 = \(\phi(x9) = \phi(0.2278)\) = 0.2318
x11 = \(\phi(x10) = \phi(0.2318)\) = 0.2361

We can calculate in this mode

Approximate root of the equation tan(x) using iteration method is -0.0571.

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3. What will be the root of the equation f(x) = 4cos(x)-x using iteration method, with the method of
f(x) = cos(x)-x*ex?
a) 0.1282
b) It is divergent
c) 0.1284
d) -0.8723
View Answer

Answer: b
Explanation: Given,
Considering f(x) = 4cos(x)-x as \(\phi(x)\) = 4cos(x)-x
We can also directly input \(\phi(x)\) value like x = value format, e.g. x = cos(x)/exp(x)
x = 4cos(x)-x
\(\phi(x) = 4cos(x)-x\)
x1 = \(\phi(x0) = \phi(2)\) = -3.6646
x2 = \(\phi(x1) = \phi(-3.6646)\) = 0.1993
x3 = \(\phi(x2) = \phi(0.1993)\) = 3.7216
x4 = \(\phi(x3) = \phi(3.7216)\) = -7.0675
x5 = \(\phi(x4) = \phi(-7.0675)\) = 9.899
x6 = \(\phi(x5) = \phi(9.899)\) = -13.4576
x7 = \(\phi(x6) = \phi(-13.4576)\) = 15.9715
x8 = \(\phi(x7) = \phi(15.9715)\) = -19.8334
x9 = \(\phi(x8) = \phi(-19.8334)\) = 22.0487
x10 = \(\phi(9) = \phi(22.0487)\) = -26.0421
x11 = \(\phi(x10) = \phi(-26.0421)\) = 28.4991
x12 = \(\phi(x11) = \phi(28.4991)\) = -32.3985
x13 = \(\phi(x12) = \phi(-32.3985)\) = 34.6181
x14 = \(\phi(x13) = \phi(34.6181)\) = -38.6107
x15 = \(\phi(x14) = \phi(-38.6107)\) = 41.0606
x16 = \(\phi(x15) = \phi(41.0606)\) = -44.9643
x17 = \(\phi(x16) = \phi(-44.9643)\) = 47.1858
x18 = \(\phi(x17) = \phi(47.1858)\) = -51.1781
x19 = \(\phi(x18) = \phi(-51.1781)\) = 53.6248
x20 = \(\phi(x19) = \phi(53.6248)\) = -57.5304
x21 = \(\phi(x20) = \phi(-57.5304)\) = 59.7528
x22 = \(\phi(x21) = \phi(59.7528)\) = -63.745
x23 = \(\phi(x22) = \phi(-63.745)\) = 66.19
x24 = \(\phi(x23) = \phi(66.19)\) = -70.0966

We can calculate this mode

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The process clue us away from the solution, so it is divergent.

4. What will be the root of the equation f(x) = x4+2x3+x2+x-1 using iteration method?
a) 0.4848
b) 0.4835
c) 0.4838
d) -0.4835
View Answer

Answer: c
Explanation: Given,
Let f(x) = x4+2x3+x2+x-1
Here x4+2x3+x2+x-1 = 0
x4+2x3+x2+x = 1
x(x3+2x2+x+1) = 1
x= 1/(x3+2x2+x+1)
\(\phi(x)\) = 1/(x3+2x2+x+1)

Here

x 0 1
f(x) -1 4

Here f(0) = -1<0 and f(1) = 4>0
Root lies between 0 and 1.
x0 = 0+12 = 0.5
x1 = \(\phi(x0) = \phi(0.5)\) = 0.4706
x2 = \(\phi(x1) = \phi(0.4706)\) = 0.4956
x3 = \(\phi(x2) = \phi(0.4956)\) = 0.4742
x4 = \(\phi(x3) = \phi(0.4742)\) = 0.4924
x5 = \(\phi(x4) = \phi(0.4924)\) = 0.4769
x6 = \(\phi(x5) = \phi(0.4769)\) = 0.4901
x7 = \(\phi(x6) = \phi(0.4901)\) = 0.4789
x8 = \(\phi(x7) = \phi(0.4789)\) = 0.4885
x9 = \(\phi(x8) = \phi(0.4885)\) = 0.4803
x10 = \(\phi(x9) = \phi(0.4803)\) = 0.4872
x11 = \(\phi(x10) = \phi(0.4872)\) = 0.4813
x12 = \(\phi(x11) = \phi(0.4813)\) = 0.4864
x13 = \(\phi(x12) = \phi(0.4864)\) = 0.482
x14 = \(\phi(x13) = \phi(0.482)\) = 0.4857
x15 = \(\phi(x14) = \phi(0.4857)\) = 0.4826
x16 = \(\phi(x15) = \phi(0.4826)\) = 0.4853
x17 = \(\phi(x16) = \phi(0.4853)\) = 0.483
x18 = \(\phi(x17) = \phi(0.483)\) = 0.4849
x19 = \(\phi(x18) = \phi(0.4849)\) = 0.4833
x20 = \(\phi(x19) = \phi(0.4833)\) = 0.4847
x21 = \(\phi(x20) = \phi(0.4847)\) = 0.4835
x22 = \(\phi(x21) = \phi(0.4835)\) = 0.4845
x23 = \(\phi(x22) = \phi(0.4845)\) = 0.4836
x24 = \(\phi(x23) = \phi(0.4836)\) = 0.4844
We can analyze in this custom

Approximate root of the equation x4+2x3+x2+x-1 using iteration method is 0.4838.

5. What will be the root of the equation f(x) = 2x-x-1.5 using iteration method?
a) 0.3602
b) 0.3607
c) -0.3602
d) -0.3607
View Answer

Answer: d
Explanation: Given,
Considering f(x) = 2x-x-1.5 as \(\phi(x)\) = -x+2x-1.5
We can also directly input \(\phi(x)\) value like x = value format, e.g. x= cos(x)/exp(x).
x = -x+2x-1.5
\(\phi(x)\) = -x+2x-1.5
x1 = \(\phi(x0) = \phi(2)\) = 0.5
x2 = \(\phi(x1) = \phi(0.5)\) = -0.5858
x3 = \(\phi(x2) = \phi(-0.5858)\) = -0.2479
x4 = \(\phi(x3) = \phi(-0.2479)\) = -0.41
x5 = \(\phi(x4) = \phi(-0.41)\) = -0.3374
x6 = \(\phi(x5) = \phi(-0.3374)\) = -0.3711
x7 = \(\phi(x6) = \phi(-0.3711)\) = -0.3557
x8 = \(\phi(x7) = \phi(-0.3557)\) = -0.3628
x9 = \(\phi(x8) = \phi(-0.3628)\) = -0.3595
x10 = \(\phi(x9) = \phi(-0.3595)\) = -0.361
x11 = \(\phi(x10) = \phi(-0.361)\) = -0.3604
x12 = \(\phi(x11) = \phi(-0.3604)\) = -0.3607

Approximate root of the equation 2x-x-1.5 using iteration method is -0.3607.

6. What will be the root of the equation f(x) = x3-x-1 using iteration method, using the method
x = \(\phi(x)\) = cos(x)/ex?
a) -0.3595
b) 0.3595
c) It is divergent
d) -0.3604
View Answer

Answer: c
Explanation: Given,
x = x3-x-1
\(\phi(x)\) = x3-x-1
x1 = \(\phi(x0) = \phi(2) \) = 5
x2 = \(\phi(x1) = \phi(5) \) = 119
x3 = \(\phi(x2) = \phi(119) \) = 1685039

The method lead us away from the solution, so it is divergent.

n x0 x1 = \(\phi(x0)\) Update Difference
x1-x0
2 2 5 x0 = x1 3
3 5 119 x0 = x1 114
4 119 1685039 x0 = x1 1684920

7. What will be the root of the equation f(x) = sin(2)(x) + 1/tan(x) using iteration method?
a) 1.8568
b) 1.9564
c) -1.9568
d) It is divergent
View Answer

Answer: d
Explanation: Given,
Considering f(x) = sin2(x) + 1/tan(x) as \(\phi(x)\) = sin2(x) + 1/tan(x)
We can also directly input \(\phi(x)\) value like x = value format, e.g. x = cos(x)/exp(x)

x = sin2(x) + 1/tan(x)
\(\phi(x)\) = sin2(x) + 1/tan(x)
x1 = \(\phi(x0) = \phi(2)\) = 2.9257
x2 = \(\phi(x1) = \phi(2.9257)\) = 3.9018
x3 = \(\phi(x2) = \phi(3.9018)\) = 4.8678
x4 = \(\phi(x3) = \phi(4.8678)\) = 5.7945
x5 = \(\phi(x4) = \phi(5.7945)\) = 6.6688
x6 = \(\phi(x5) = \phi(6.6688)\) = 7.4859
x7 = \(\phi(x6) = \phi(7.4859)\) = 8.2449
x8 = \(\phi(x7) = \phi(8.2449)\) = 8.9472
x9 = \(\phi(x8) = \phi(8.9472)\) = 9.5951
x10 = \(\phi(x9) = \phi(9.5951)\) = 10.1918
x11 = \(\phi(x10) = \phi(10.1918)\) = 10.7403
x12 = \(\phi(x11) = \phi(10.7403)\) = 11.2441
x13 = \(\phi(x12) = \phi(11.2441)\) = 11.7063
x14 = \(\phi(x13) = \phi(11.7063)\) = 12.1301
x15 = \(\phi(x14) = \phi(12.1301)\) = 12.5184
x16 = \(\phi(x15) = \phi(12.5184)\) = 12.8741
x17 = \(\phi(x16) = \phi(12.8741)\) = 13.1996
x18 = \(\phi(x17) = \phi(13.1996)\) = 13.4976
x19 = \(\phi(x18) = \phi(13.4976)\) = 13.7702
x20 = \(\phi(x19) = \phi(13.7702)\) = 14.0195
x21 = \(\phi(x20) = \phi(14.0195)\) = 14.2475
x22 = \(\phi(x21) = \phi(14.2475)\) = 14.4559
x23 = \(\phi(x22) = \phi(14.4559)\) = 14.6464
x24 = \(\phi(x23) = \phi(14.6464)\) = 14.8206

The method lead us away from the solution, so it is divergent.

8. What will be the root of the equation f(x) = sin3(x) + x3 – exp(2x) using iteration method?
a) It is divergent
b) 0.4811
c) -0.4814
d) -0.4825
View Answer

Answer: a
Explanation: Given,
Considering f(x) = sin3(x) + x3 – exp(2x) as \(\phi(x)\) = x3 + sin3(x) – e(2x)
We can also directly input \(\phi(x)\) value like x = value format, e.g. x = cos(x)/exp(x)

x = x3 + sin3(x) – e(2x)
\(\phi(x)\) = x3 + sin3(x) – e(2x)
x1 = \(\phi(x0) = \phi(2)\) = -46.3159
x2 = \(\phi(x1) = \phi(-46.3159)\) = -99361.7367

The method chief us away from the solution, so it is divergent.

n x0 x1 = \(\phi(x0)\) Update Difference
x1-x0
2 2 -46.3159 x0 = x1 48.3159
3 -46.3159 -99361.7367 x0 = x1 99315.4208

9. Iteration method is used for the purpose of approximating the solutions.
a) False
b) True
View Answer

Answer: b
Explanation: Iteration method is used for the calculation of approximate solution of problems which are connected with the differential equation. As the name suggests the solution can be found by the iteration procedure for a closer accuracy.

10. When should one stop the computation of iteration in the iteration method?
a) Until |f(xi)-f(xi-1)| \(\approx\) 0
b) Until |f(xi)-f(xi-1)| < 0
c) Until |f(xi)-f(xi-1)| > 0
d) Until |f(xi)-f(xi)| < 0
View Answer

Answer: a
Explanation: Firstly, we need to write the equation x = \(\phi(x)\). Then, find points a and b such that a < b and f(a).f(b) < 0. Later, if f(a) is more closer to 0 than f(b), then x0 = a else x0 = b.
Lastly, x1 = \(\phi(x0)\)
x2 = \(\phi(x1)\)
x3 = \(\phi(x2)\)

Repeat until |f(xi)-f(xi-1)| \(\approx\) 0.

Sanfoundry Global Education & Learning Series – Numerical Analysis.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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