MTH603 - Numerical Analysis

@zaasmi said in To apply Simpson's 1/3 rule, valid number of intervals are?: 7 8 5 3 Page 177 The Simpson’s 1/3 rule, we have used two sub-intervals of equal width. In order to get a composite formula, we shall divide the interval of integration [a, b] Into an even number The correct choice for applying Simpson’s 1/3 rule is 7 because it represents an odd number of points, which results in 6 intervals, and the number of intervals must be even for the rule to apply. So, in general: • 7 points → 6 intervals (valid for Simpson’s 1/3 rule), • 8 points → 7 intervals (invalid), • 5 points → 4 intervals (valid), • 3 points → 2 intervals (valid).

@zaasmi said in Newton's divided difference interpolation formula is used when the values of the independent variable are: Newton’s divided difference interpolation formula is used when the values of the independent variable are unequally spaced. Newton’s divided difference interpolation formula is used when the values of the independent variable are unequally spaced. This method is effective for cases where the data points (the independent variable values) do not have a uniform interval between them. It constructs an interpolating polynomial based on divided differences, making it more versatile compared to other methods like Newton’s forward or backward interpolation formulas, which are suited for equally spaced data points. unequally spaced

@zaasmi said in If there are (n+1) values of y corresponding to (n+1) values of x, then we can represent the function f(x) by a polynomial of degree: If there are (n+1) values of y corresponding to (n+1) values of x, then we can represent the function f(x) by a polynomial of degree If there are  values of  corresponding to  values of , the function  can be represented by a polynomial of degree . This is based on the concept of polynomial interpolation, specifically the Lagrange interpolation formula, where given  distinct points, a unique polynomial of degree  will pass through all those points.

@cyberian said in MTH603 Assignment 1 Solution and Discussion: @zaasmi said in MTH603 Assignment 1 Solution and Discussion: Assignment # 01 MTH603 (Summer 2024) Marks: 10 Due Date: 21.09 2024 DON’T MISS THESE: Important instructions before attempting the solution and submission of this assignment:  Lectures 23-30 are encompassed in Assignment 1.  Assignment 1 is due on 21 September 2024.  Properly Upload the solution of this assignment in MS Word format on LMS as per the previous practice. Question 1 [Marks 5] First construct the divided difference table and then find the interpolating polynomial of the following function 𝑦 = 𝑓(𝑥) by Newton’s Divided Difference Formula. 𝑥 0 𝜋 𝜋 2 𝑦 = 𝑓(𝑥) 0 1 0 Question 2 [Marks 5] Compute 𝑓′(1.5), from the following tabular data using the forward difference formula for derivative. 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 Summer 2024_MTH603_1.pdf Download

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion: Let A be an n ×n matrix. The number x is an eigenvalue of A if there exists a non-zero vector v such that _______. Av = xv Ax=xv Av + xv=0 Av = Ax1 Av = λv Av = λv Explanation: Eigenvalue and Eigenvector Definition: A number ( \lambda ) is an eigenvalue of an ( n \times n ) matrix ( A ) if there exists a non-zero vector ( v ) such that the equation ( Av = \lambda v ) holds true. Here, ( \lambda ) is the eigenvalue and ( v ) is the corresponding eigenvector. So, the correct option is: Av = λv

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@zaasmi said in mth603 final term solved papers by moaaz: @zaasmi said in mth603 final term solved papers by moaaz: @cyberian said in mth603 final term solved papers by moaaz: Check here mth603 final term solved papers by moaaz MTH603 Finals term.rar MTH603 Final term papers in one file.pdf MTH603 - Final Term Papers.pdf

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.