Numerical Methods Problems

Some numerical methods problems from B S Grewal...

Regula - falsi

1) Find a real root of the equation x³-2x-5=0 by the method of false position correct to three decimal places.

2) Find the root of the equation cos x=xe^x using regular-falsi method correct to four decimal places.

3) Find the real root of the equation x log ₁₀ x = 1.2 by regula-falsi method correct upto four decimal places.

4) Use the method of false position, to find the fourth root of 32 correct upto three decimal places.

5) Using regula-falsi method, find the real root of the following equations correct to three decimal places:

i) xe^x=2                 ii) cos x=3x-1      iii) xe^x=sinx    iv) x tanx=-1            v) 2x-logx=7       vi) 3x+sinx=e^x

6) Find the fourth root of 12 correct to three decimal places by interpolation method(regula falsi)

7) Locate the root of f(x) = x¹⁰ -1=0 between 0 and 1.3 using method of false position.

Newton – Raphson

1) Find the positive root of x⁴-x=10 correct to three decimal places using newton-raphson method.

2) Find by newton’s method, the real root of the equation 3x=cosx+1 correct to four decimal places.

3) Using Newton’s iterative method, find the real root of x log 10 x = 1.2 correct to five decimal places.

4) Find by newton-raphson method, a root of the following equations correct upto 3 decimal places.

i) x³-3x+1=0     ii) x³-2x-5=0      iii) x³-5x+3=0       iv) 3x³-9x²+8=0

5) Using newton’s iterative method, find a root of the following equations correct to 4 decimal places:

i) x⁴+x³-7x²-x+5=0 which lies between 2 and 3.

ii) x⁵ – 5 x² +3=0

6) Find the negative root of equation x³ -21x +3500=0 correct to 2 decimal places by newton’s method.

7) Using Newton-Raphson method, find a root of the following equations correct to 3 decimal places:

i) x²+4sinx=0

ii) x sinx+cosx=0

iii) e^x=x³+cos25x which is near 4.5

iv) x log ₁₀ x=12.34 start with x₀=4.5

v) cosx = x e^x

vi) 10x+x-4=0

8) use newton’s method to find the smallest root of the equation e^x sinx=1 to four decimal places.

9) Develop an algorithm using N.R method, to find the fourth root of a positive number N and hence find .(fourth root of 32).

10) evaluate the following(correct to 3 decimal places) by using newton-raphson method.

i) 1/18                   ii) 1/ (square root of 15)             iii) (28)^-1/4

11) State and explain the newton-raphson method to find the roots of a differentiable equation.

Gauss Elimination Method:

1) Apply the gauss elimination method to solve the equations x+4y-z=-5; x+y-6z=-12; 3x-y-z=4

2)Solve 10x-7y+3z+5u=6; -6x+8y-z-4u=5; 3x+y+4y+11u=2; 5x-9y-2z+4u=7 by gauss elimination.

3) Using gauss elimination method, solve the equations x+2y+3z-u=10, 2x+3y-3z-u=1, 2x-y+2z+3u=7, 3x+2y-4z+3u=2.

4) Solve the following equations by gauss elimination method:

i) x + y + z=9; 2x - 3y + 4z = 13; 3x + 4y + 5z = 40

ii) 2x + 2y + z = 12; 3x + 2y + 2z = 8; 5x + 10y -8z =10

iii) 2x – y + 3z = 9; x + y + z = 6; x – y + z=2

iv) 2x₁ + 4x₂ + x₃ = 3; 3x₁ + 2x₂ – 2x₃= -2; x₁ – x₂ + x₃ = 6

v) 5x₁ + x₂ + x₃ + x₄ = 4; x₁ + 7x₂ + x₃ + x₄ = 12; x₁ + x₂ + 6x₃ + x₄ = -5; x₁ + x₂ + x₃ + 4x₄= -6

Gauss-jordan method:

1) Apply Gauss-jordan method to solve the equations

 x + y + z=9; 2x - 3y + 4z = 13; 3x + 4y + 5z = 40

2) Solve the equations 10x-7y+3z+5u=6;

                                                -6x+8y-z-4u=5;

                                                3x+y+4z+11u=2;

                                                5x-9y-2z+4u=7

3) solve the following equations by Gauss-Jordan Method:

i) 2x+5y+7z=52; 2x+y-z=0; x+y+z=9

ii) 2x – 3y +z=-1; x +4y +5z=25; 3x-4y+z=2

iii) x + y+ z=9; 2x+y-z=0; 2x+5y+7z=52

iv) x + 3y + 3z=16; x + 4y + 3z=18; x + 3y + 4z=19

v) 2x₁+x₂+5x₃+x₄=5; x₁+x₂-3x₃+4x₄=-1

    3x₁+6x₂-2x₃+x₄=8; 2x₁+2x₂+2x₃-3x₄=2

LU decomposition:

1) Solve the following equations by factorization method:

i) 2x+3y+z=9; x+2y+3z=6; 3x+y+2z=8

ii) 10x+y+z=12; 2x+10y+z=13; 2x+2y+10z=14

iii) 10x+y+2z=13; 3x+10y+z=14; 2x+3y+10z=15

iv) 2x₁-x₂+x₃=-1; 2x₂-x₃+x₄=1; x₁+2x₃-x₄=-1; x₁+x₂+2x₄=3

Iterative methods:

Jacobi’s Iteration:

1) Solve by Jacobi’s iteration method, the equations

20x+y-2z=17;

3x+20y-z=-18;

2x-3y+20z=25

2) Solve by jacobi’s iteration method, the equations 10x + y – z=11.19

                                                                                                       x+10y+z=28.08;

                                                                                                      -x+y+10z=35.61

3) Solve the equations:

10x₁-2x₂-x₃-x₄=3;

-2x₁+10x₂-x₃-x₄=15;

-x₁-x₂+10x₃-2x₄=27;

-x₁-x₂-2x₃+10x₄=-9

4) Solve by jacobi’s method, the equations: 5x-y+z=10;

                                                                                     2x+4y=12;

                                                                                     x+y+5z=-1 starting with the solution{2,3,0}

5) Solve by jacobi’s method the equations:

13x + 5y-3z+u=18;

2x+12y+z-4u=13;

x-4y+10z+u=29;

2x+y-3z+9u=31;

Gauss-Seidel Method:

1) Apply Gauss-Seidel iteration method to solve the equations 20x+y-2z=17

3x+20y-z=-18; 2x-3y+20z=25

2) Solve the equations 27x+6y-z=85;

                                                x+y+54z=110;

                                                6x+15y+2z=72;

3) Apply Gauss-Seidel iteration method to solve the equations:

10x₁-2x₂-x₃-x₄=3;

-2x₁+10x₂-x₃-x₄=15;

-x₁-x₂+10x₃+2x₄=27;

-x₁-x₂-2x₃+10x₄=-9

4) Solve the following equations by Gauss-seidel method:

i) 2x+y+6z=9; 8x+3y+2z=13; x+5y+z=7

ii) 28x+4y-z=32; x+3y+10z=24; 2x+17y+4z=35;

iii) 10x+y+z=12; 2x+10y+z=13; 2x+2y+10z=14;

iv) 7x₁+52x₂+13x₃=104; 83x₁+11x₂-4x₃=95; 3x₁+8x₂+29x₃=71

v) 3x₁-0.1x₂-0.2x₃=7.85; 0.1x₁+7x₂-0.3x₃=-19.3; 0.3x₁-0.2x₂+10x₃=71.4