MTH603 - Numerical Analysis

Question # 3 of 10 ( Start time: 09:05:56 PM ) Total Marks: 1 To take the derivative of f(x) = 2x in the interval [-3,3], which of the following partition of subintervals will be suitable? Select correct option: Equally spaced Unequally spaced Union of equally spaced and unequally spaced intervals. Any arbitrary partition will work

@asad-saab Fall 2023 MTH603 Assignment # 1 Section In charge: Husna Muzaffar Total Marks 20 Instructions To solve this assignment you need to have a good grip on lectures 1-15. The course is segmented into four sections, each of which is supervised by a different faculty member. Information regarding the section in charge can be found in the course information section on the LMS. A distinct assignment file has been given to each section, resulting in a total of four separate assignment files. The relevant assignment file can be downloaded from the announcement section of the course. It is important to note that students can only view the announcements relevant to their respective sections. You will prepare the solution of assignment on Word file and upload at the assignment interface on LMS as per usual practice. Plagiarism in the submitted assignment will lead to a zero grade. Additionally, any student who submits a solution file that is not applicable to their section will also get a zero grade. 𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧# 𝟏: Marks 10 Solve the system of equations by using Crout’s method. 2𝑥 + 5𝑦 + 3𝑧 = 16 𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧# 𝟐: Marks 10 3𝑥 + 𝑦 + 2𝑧 = 11 −3𝑥 + 7𝑦 + 8𝑧 = 10 Solve the following system of equations by using Jacobi′s iterative method for the first three iterations by taking initial starting of solution vector as (0,0,0). 8𝑥 − 2𝑦 − 2𝑧 = 3 −2𝑥 + 6𝑦 + 𝑧 = 9 −2𝑥+𝑦+7𝑧= 6 [center][image: wNTUZAq.png][/center]

@zaasmi said in MTH603 Quiz 2 Solution and Discussion: [image: X4n0EMk.png] [image: OwwuQuC.png]

@zaasmi said in MTH603 Assignment 2 Solution and Discussion: @zaasmi said in MTH603 Assignment 2 Solution and Discussion: Assignment NO. 2 MTH603 (Spring 2021) Maximum Marks: 20 Due Date: July 30, 2021 DON’T MISS THESE: Important instructions before attempting the solution of this assignment: • To solve this assignment, you should have good command over 23 - 30 lectures. Try to get the concepts, consolidate your concepts and ideas from these questions which you learn in the 23-30 lectures. • Upload assignments properly through LMS, No Assignment will be accepted through email. • Write your ID on the top of your solution file. Don’t use colourful back grounds in your solution files. Use Math Type or Equation Editor Etc. for mathematical symbols. You should remember thatif we found the solution files of some students are same then we will reward zero marks to all those students. Try to make solution by yourself and protect your work from other students, otherwise you and the student who send same solution file as you will be given zero mark. Also remember that you are supposed to submit your assignment in Word format any other like scan images etc. will not be accepted and we will give zero mark corresponding to these assignments. Question 1: Find the first and second derivative of function f(x) at x=1.5 if: x 1.5 2.0 2.5 3.0 3.5 4.0 f(x) 3.375 7.000 13.625 24.000 38.875 59.000 MARKS 10 Question 2: Using Newton’s forward interpolation formula, find the value of function f(1.6) if: x 1 1.4 1.8 2.2 f(x) 3.49 4.82 5.96 6.5 MARKS 10 https://www.youtube.com/watch?v=BtdgWZ0wy4Q MTH603 Assignment 2 Solution Spring 2021-converted.docx MTH603 Assignment 2 Solution Spring 2021.pdf

@zaasmi said in MTH603 Grand Quiz Solution and Discussion: A series 16+8+4+2+1 is replaced by the series 16+8+4+2, then it is called Each number in the sequence is half the value of the number receding it. So the common difference in the series is dividing by two. 16÷2=8 8÷2=4 4÷2=2 2÷2=1 1÷2=½ The answer is ½ or 0.5 When you keep dividing by two, you will notice an interesting pattern: the denominator continues to increase by two, while the numerator value remains the same. That’s fascinating because in natural, whole numbers the numbers in the series would decrease by two. 1/4 , 1/8 , 1/16 etc.

@zainab-ayub said in MTH603 Quiz 2 Solution and Discussion: Mth603 ka koi student hai tu plz yeh question bta dy kis trha solve ho ga Given the following data x:1 2 5 y:1 4 10 Value of 1st order divided difference f[2 , 5] is [image: JgeUzkI.png]

@zareen said in MTH603 Assignment 1 Solution and Discussion: Question #2: Solve the system of linear equations with the help of Gaussian elimination method. 2x + y + z = 9;3x −2y + 4z = 9;x +y-2z = 3 System of Linear Equations entered : [1] 2x + y + z = 9 [2] 3x - 2y + 4z = 9 [3] x + y - 2z = 3 Solve by Substitution : // Solve equation [3] for the variable y [3] y = -x + 2z + 3 // Plug this in for variable y in equation [1] [1] 2x + (-x +2z+3) + z = 9 [1] x + 3z = 6 // Plug this in for variable y in equation [2] [2] 3x - 2•(-x +2z+3) + 4z = 9 [2] 5x = 15 // Solve equation [2] for the variable x [2] 5x = 15 [2] x = 3 // Plug this in for variable x in equation [1] [1] (3) + 3z = 6 [1] 3z = 3 // Solve equation [1] for the variable z [1] 3z = 3 [1] z = 1 // By now we know this much : x = 3 y = -x+2z+3 z = 1 // Use the x and z values to solve for y y = -(3)+2(1)+3 = 2 Solution : {x,y,z} = {3,2,1}

Solution idea

https://youtu.be/tQRwrupC2Ao Spring 2020_MTH603_1_SOL.docx

Assignment No: 01 Question #1: Find the root of the equation, Perform three iteration of the equation, ln (x −1) + sinx =0 by using Newton Raphson method. Ans: Let f(x) = ln(x+1) + sinx = 0 and f(x) = 1/(x-1) + cosx F (1.5) = ln(0.5) + (1.5) = - 0.0667 F(2) = ln(1) + sin(2) = 0.035 Since f (1.5) f (2) < 0 so roots lies in interval [1.5, 2] Let x0 = 1.75 . x0 can be taken in the interval any real number [ 1.5 , 2 ], we let mid point of this interval . As we know Newton Raphson method is Xn+1 = xn – f ( xn ) / f(xn) First iteration X1 = x0 –f(x0) / f(x0) = 1.75 - f(1.75) / f(1.75) = 1.75 – (-0.2571 / 2.3329) = 1.8602 Second iteration: X2 = x1 - f(x) / f(x) = 1.8602 –[ f(1.8602) / f(1.8602)] = 1.8602 - ( -0.1181 / 2.1620 ) = 1.9148 Third iteration: X3 = x2- f(x2) / f(x2) = 1.9148 –f(1.9148) / f(1.9148) = 1.9148 – [-0.0556/2.0926] = 1.9414 Question #2: Solve the system of linear equations with the help of Gaussian elimination method. x + y + z = 6;2x − y + z = 3;x + z = 4 ANS: In Gaussian elimination method we convert the augmented matrix into reduce Echelon form therefore, Augmented matrix is R2- 2R1 , R3 – R1 -1R2 , -1R3 R23 R3-3R2 X + Y+ Z = 6 ;………………….(1) Y = 2, Z = 3 Put into eq (1), we get X = 1 ,