Concept explainers
Investigate the meaning of numerical analysis and give its examples.
Explanation of Solution
In order to obtain the numerical solutions for the mathematical problems, numerical analysis is used. In the field of engineering and physics, numerical analysis plays a vital role.
Give the areas of study of numerical analysis as below.
- Value of functions
- Solving system of linear equations
- Differential equation
- Integral
- Eigen values and
vectors
Value of functions:
The function of numerical analysis includes square root, cube root, logarithmic etc.
Example:
Let us consider the value,
x=4
Square root:
The square root of x can be calculated as follows:
√x=√4=2
Cube root:
The cube root of x can be calculated as follows:
3√x=3√4=1.5874
Logarithmic function:
The logarithmic value of x can be calculated as follows:
log(x)=log(4)=0.602
Solving system of linear equations:
It is a collection of two or more linear equation involves same set of variables.
Consider the linear equation as follows:
x1+x2+x3=6 (1)
2x1+5x2+x3=15 (2)
−3x1+x2+5x3=14 (3)
Solve equation (1) and (2)
Multiply equation (1) by −2
Therefore equation (2) becomes,
−2x1−2x2−2x3=−12 (4)
Solve equation (2) and (4)
−2x1−2x2−2x3=−122x1+5x2+x3=15_ 3x2−x3=3 (5)
Multiply equation (1) by 3
Therefore equation (1) becomes,
3x1+3x2+3x3=18 (6)
Solve equation (6) and (3)
3x1+3x2+3x3=18−3x1+x2+5x3=14_ 4x2+8x3=32 x2+2x3=8 (7)
Solve equation (5) and (7)
Multiply equation (7) by −3
Therefore equation (7) becomes,
−3x2−6x3=−24 (8)
Solve equation (5) and (8)
3x2−x3=3−3x2−6x3−24_−7x3=−21 x3=3
Substitute 3 for x3 in equation (7)
x2+2(3)=8x2=8−6x2=2
Substitute 3 for x3, and 2 for x2 in equation (1)
x1+2+3=6x1=6−2−3=1
Therefore, the solution is x1=1, x2=2, x3=3.
Differential equation:
The differential equation is the equation which contains functions of derivatives represents their rate of change.
Let us consider the equation as follows:
y=x3+2x2+3x+4 (9)
Differentiate equation (9) with respect to x.
Therefore equation (9) becomes:
dydx=3x2+2(2)x+3(1)+0=3x2+4x+3
Thus, the rate of change of equation (9) is 3x2+4x+3.
Integral:
Integral calculus is the development of differential calculus. It is used to find the displacement, moment of inertia, area and volume in the mathematical concepts.
The indefinite integral can be represented by,
F(x)=∫f(x)dx (10)
Let us consider the function,
f(x)=x3+2x2+3x+4
Substitute x3+2x2+3x+4 for f(x) in equation (10)
F(x)=∫(x3+2x2+3x+4)dx=x44+2x33+3x22+4x
Eigen values and Eigen vectors:
The Eigen value is a non-zero vector that changes by a scalar factor when the linear transformation is applied.
Consider the Eigen value problem:
Ax=λx (11)
Here,
A is the matrix,
x is the vector,
λ is the Eigen values.
Consider the matrix A as follows:
A=[01−2−3]
The Eigen value of the matrix can be calculated by,
|A−λI|=0 . (12)
Here,
I is the identity matrix.
Substitute [1001] for I, and [01−2−3] for A in equation (12)
|[01−2−3]−λ[1001]|=0|[01−2−3]−[λ00λ]|=0|[−λ1−2−3−λ]|=0(−λ)(−3−λ)−(−2)(1)=0λ2+3λ+2=0 (13)
Solve the equation (13)
λ2+3λ+2=0λ2+2λ+λ+2=0λ(λ+2)+1(λ+2)=0(λ+1)(λ+2)=0 . (14)
Reduce equation (14) as follows,
λ+1=0 λ+2=0λ=−1 λ=−2
The Eigen values are,
λ1=−1,λ2=−2.
Eigen vectors for the Eigen value can be calculated as follows:
Ax1=λ1x1(A−λ1).x1=0 (15)
Substitute [01−2−3] for A, and −1 for λ1 in equation (15)
([01−2−3]−[λ100λ1]).x1=0[−λ11−2−λ1−3]x1=0 (16)
Reduce the equation (16) as follows,
[−(−1)1−2−(−1)−3]x1=0[11−2−2]x1=0[11−2−2][x1,1x1,2]=0 (17)
Write the matrix form of equation (17) into linear equation as follows,
x1,1+x1,2=0 (18)
−2x1,1−2x1,2=0 (19)
Solving equation (18) and (19)
x1,1=−x1,2
Therefore, the Eigen vector of x1 can be written as
x1=[1−1]
Eigen vectors for the Eigen value can be calculated as follows:
Ax2=λ2x2(A−λ2).x2=0 (20)
Substitute [01−2−3] for A, and −1 for λ1 in equation (20)
([01−2−3]−[λ200λ2]).x2=0[−λ21−λ2−2−λ2−3]x2=0 (21)
Reduce the equation (16) as follows,
[−(−2)1−2−(−2)−3]x2=0[210−1]x2=0[21−2−1][x2,1x2,2]=0 (22)
Write the matrix form of equation (22) into linear equation as follows,
2x2,1+x2,2=0 (23)
−2x2,1−x2,2=0 (24)
Solving equation (23) and (24)
x2,2=−2x2,1
Therefore, the Eigen vector of x2 can be written as
x2=[1−2]
Conclusion:
Thus, the numerical analysis and its examples are explained.
Want to see more full solutions like this?
Chapter 18 Solutions
ENGINEERING FUNDAMENTALS
- Lab Assignment #2 Loads: UDL and Concentrated Name: TA 1. Use the provided beam models to solve for the equivalent concentrated load of each beam configuration. Draw the loading conditions showing the equivalent concentrated load(s). a) w = 30lbs/ft 6ft 6ft c) w = 50lbs/ft 12ft w = 70lbs/ft b) 4ft w = 20lbs/ft w = 40lbs/ft d) 9ft 2. Find the equivalent concentrated load(s) for the bags of cement stacked on the dock as shown here. Each bag weighs 100 lbs and is 12 inches long. Draw the loading conditions for each showing the equivalent concentrated load(s). 1 bag = 100lbs L= 12 ft L= 6ft L= 8ftarrow_forwardplease show the complete solution, step by step process, thanksarrow_forwardThe rectangular gate shown in figure rotates about an axis through N. If a=3.3 ft,b=1.3 ft, d=2 ft, and the width perpendicular to the plane of the figure is 3 ft, what torque(applied to the shaft through N) is required to hold the gate closed?arrow_forward
- An elevated tank feeds a simple pipe system as shown. There is a fire hydrant atpoint C. The minimum allowable pressure at point C is 22 psig for firefighting requirements.What are the maximum static head (in ft) as well as pressure (in psig) at point C (i.e. nodischarge in the system)? Do we meet the pressure requirement for firefighting? (Please donot worry about L or d in the figure below)arrow_forward12. For the beam loaded and supported as shown, determine the following using Point Load Analogous via Integration: a. the rotation at the left support. b. the deflection at midspan R1 1 . m 600 N/m 3 m + 2 m R2arrow_forward14. Find the reaction R and the moment at the wall for the propped beam shown below using Point Load Analogous via Integration: 16 kN/m 000 4.5m 4.5marrow_forward
- 13. Determine the moment at supports A and B of the fixed ended beam loaded as shown using Point Load Analogous via Integration: 10 kN/m 9 kN/m 3 m 3 m 12 kN/marrow_forwardHow does construction estimate inaccuracies lead to delays and complications that impact projects?arrow_forwardQ5: Given the following system: น -3 y= [4 -2] +3u Generate a model with states that are the sum and difference of the original states.arrow_forward
- 4. Draw a stress-strain curve (in tension and compression) for a reinforced concrete beam below. Label the important parts of the plot. Find the linear elastic approximation obtained using the transformed technique, and plot over the same strain ranges. 24" 4" 20" 16" f = 8,000 psi 8- #11 bars Grade 60 steel 4" (f, = 60 ksi and E₁ = 29000 ksi)arrow_forwardWhy is Historical Data important compared to other sourses of information when estimating construction projects?arrow_forwardNeed help, please show all work, steps, units and round to 3 significant figures. Thank you!!arrow_forward
- Engineering Fundamentals: An Introduction to Engi...Civil EngineeringISBN:9781305084766Author:Saeed MoaveniPublisher:Cengage LearningResidential Construction Academy: House Wiring (M...Civil EngineeringISBN:9781285852225Author:Gregory W FletcherPublisher:Cengage Learning