TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 1 / 38 QAC No. CC-09262020 COURSE DETAILS College Department Program College of Engineering Electrical Engineering Department Bachelor of Science in Electrical Engineering Course Title Code Credit Units Description Numerical Methods Lecture PEE 4-M 2 This course covers the concepts of numerical analysis and computer software tools in dealing with engineering problems. Numerical methods are extremely powerful problem-solving tools. They are capable of handling complex system of equations, non-linearities, and complicated geometries and that are often impossible to solve analytically. It includes techniques in finding the roots of an equation, solving systems of linear and non-linear equations, eigenvalue problems, polynomial approximation and interpolation, ordinary and partial differential equations. Academic Year Semester Schedule Class 2022-2023 1st Semester Monday 7am-10pm, Monday 10am-12pm, Wednesday 7am-9am, Wednesday 10am-12pm, Friday 7am-9am, Friday 3pm-5pm, BSEE, 3rd Year, A to F Prepared by Rank Date Reviewed by Designations Date Approved by Designation Date Melanie Traje Iradiel Instructor I October 5, 2022 NAME/S OF REVIEWER/S HERE Designation/s of the reviewer/s Date of review NAME OF APPROVING AUTHORITY Designation of the approving authority Date of preparation Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 2 / 38 QAC No. CC-09262020 GENERAL INSTRUCTIONS Note: Detail out here the requirements and general instructions for the student to be successful in the course. Also include contact details such as the person, email and phone number just in case the student will be needing help in the future. You will be graded according to your class standing and major examination results. Please see course requirement / grading system below as well as course policies and guidelines for your reference. For Lecture Subjects Class Standing 50% (Class Participation, Quizzes, Assignments, Seatwork… etc) Major Exams 50% (Prelim, Midterm, Finals) Attendance Attitude Teaching-Learning Activities (Quizzes, Seatwork, Assignment, Recitation, etc.) Major Exams Total Grade 1.00 1.25 1.50 1.75 2.00 2.25 2.50 1.75 3.00 5.00 Drp W* Withdrawn Transaction ID Signature Percentage Equivalent 99-100 96-98 93-95 90-92 87-89 84-86 81-83 78-80 75-77 74 and Below 5% 5% 40% 50% 100% Descriptive Rating Excellent Very Superior Superior High Average Average Low Average Satisfactory Fair Passed Failed Dropped TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 3 / 38 QAC No. CC-09262020 COURSE POLICIES/GUIDELINES The following are the general class polices and guidelines in the course: Homework/Assignments: Homework problems are assigned in the schedule. Homework is due to start of class, on the date shown in the schedule. Late homework will not be accepted. Any changes to the schedule will be announced in class. Problems are to be solved: one per page, front side only, and final answer clearly identified. Disorganized or incomplete work will not be graded, or it may earn reduced grade. You must show all of your work, and not just the final answer. Students should never copy from another source, nor allow their work to be copied. Exams: Exams dates are to be announce later. Any changes to the schedule will be announced in class. If a student must miss an exam for good reason (e.g., sickness, family emergency) they must contact the instructor prior to the exam date. There will be no make-up exams, except for documented sickness or family emergencies. Final Answer: When solving problems, students are encouraged to be neat, well-organized and logical. The correct final answer is important. The students are encouraged to check their work. Partial Credit & Review of Exams Grades: If a student believes they deserve more partial credit for an exam problem, they are encouraged to visit the instructor during the office hours and present a case for revising the grade. Students are asked to mark-up their exams using a pen, assuming a pencil was used in the exam. Do not add marks with pencil. Do not erase marks from the exam. Students should clearly show what was correct on the exam and explain the approach taken on the exam to help instructor fairly grade the exam. The instructor must read the work shown on the exam that was performed during the exam time period. It will not help, if the students says, “Now I Know how to solve the problem”. This is not an opportunity to earn extra credit by doing additional work. This is an opportunity to ensure the exam is graded fairly. Students are encouraged to present their best explanation of what they did during the exam, in order to earn the highest grade possible, yet they are expected to accept the instructor’s decision. Begin and end with a positive attitude. All reviews of an exam grade must be concluded no sooner than 24 hours after the exam is returned to the student, and no later than two weeks after the exam is returned to the class. Class Conduct: Students are expected to assist in maintaining a classroom environment that is conductive to learning for all students in the class. Please do not come late to class. Please do not leave early. If you must leave early, please inform the instructor before the class starts and sit in the back of the class to minimize the disruption. Please do not use cell phone, messaging, or games in class. If a laptop is open, it needs to be used for this class and never used to check email, play games, or search the internet. Calculators for Exams: Basic engineering/scientific calculators can be used on exams. Mini-laptop and calculators with communication capabilities; cannot be used on exams. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 4 / 38 QAC No. CC-09262020 Scholastic Dishonesty: The university expects each student to maintain a high standard of individual integrity. Scholastic dishonesty is a serious offense that includes, but it is not limited to, cheating on a test, plagiarism, or collusion. Withdrawing from a Course: Please make yourself aware of dates and policies about withdrawing from a course or withdrawing from the University. If you fall behind, don’t just give-up and quit attending. Contact the College of Engineering Advising Office and explore your options. If you drop a course, double-check to make sure it is done properly. Course Assessment There will be a course assessment in a form of a survey at the end of the course, before the final examination. Competencies/Skills After having the course, students are expected to: a) Solve system of linear equations using various numerical methods under direct and indirect methods. b) Understand the power and limitations of each methods discussed. Hardware Resources Software Resources Computer/cellphone/tablet, Wi-Fi, headset MS Teams, PowerPoint Presentation, e-books Academic Contact Person Admin Contact Person IT Contact Person Melanie Iradiel, melanie_iradiel@tup.edu.ph, 09458900047 Name, Email, Phone Number Name, Email, Phone Number Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 5 / 38 QAC No. CC-09262020 LESSON 3: SOLUTION OF SYSTEM OF LINEAR EQUATIONS (PART 2) 1. LEARNING OUTCOMES At the end of this lesson, the students are expected to: a. Solve linear systems using Direct methods such as LU Factorization methods using Row Operation Method, Doolittle’s Method and Crout’s method b. Understand the concept of iteration, convergence, and strictly diagonal matrices. c. Solve linear systems using Indirect methods such as Jacobi Iteration and Gauss-Seidel Iteration methods. d. Understand the power and the limitations of the various alternative numerical methods. 2. EXPECTED OUTPUTS At the end of this module, the students should complete or submit the following output: a. Assignment # 4 3. COMPLETION DEADLINE 1 week Note: Specify here how long this module should take in terms of number of hours/days. I. INTRODUCTION This lesson will introduce you to the remaining methods different methods of solving systems of linear equations. A system of linear equations (or linear system) is a collection of one or more linear equations involving the same set of variables. The methods to be used as discussed in this lesson under direct methods is LU Factorization methods using Row Operation Method, Doolittle’s Method and Crout’s method; and under Iterative methods are Jacobi Iteration and Gauss-Seidel Iteration methods. LU FACTORIZATION METHOD • • • • Another way of solving a system of equations is by using a factorization technique for matrices called LU factorization method. In other references it is called an LU Decomposition. LU Factorization is only applicable for square matrix. In an n × n system Ax = b, the matrix A can be factored as • where L is the lower triangular matrix and U is the upper triangular matrix An LU decomposition is not unique. There can be more than one such LU decomposition for a matrix. A = LU Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 6 / 38 QAC No. CC-09262020 There are three ways to find the L and U matrices: 1. Row operation method 2. Doolitle’s method 3. Crout’s method Each method has its own advantages and disadvantages. It is up to the user to choose what will be the most appropriate and efficient to use for a given problem. A. ROW OPERATION METHOD • When given a square matrix A we want to find L (a lower triangular matrix) and U (an upper triangular matrix) such that A = LU • We get U by reducing the matrix A to an upper triangular matrix form using the row operations without interchanging rows. If you swap rows, then an LU decomposition will not exist. π = [0 0 0 • πΏ=[ Transaction ID Signature ] 0 For L start with Identity matrix but leaving the elements below than main diagonal blank 1 • 0 0 0 1 0 0 1 0 0] 0 1 We obtain the value of blank elements for L by using the opposites of the multiples used in row operations method in U. For example, π π + ππ π , just get the opposite value of k which is −k TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 7 / 38 QAC No. CC-09262020 Consider the previous system of linear equations −3π₯ + 2π¦ − π§ = −1 6π₯ − 6π¦ + 7π§ = −7 3π₯ − 4π¦ + 4π§ = −6 Write it in matrix form Ax=B: Where A is the coefficient matrix FINDING THE UPPER TRIANGULAR MATRIX (U) To get U, transform Matrix A to upper triangular form. First, the goal is to make a21, and a31 become zero. - k = -2 -k = -1 Take note that to get the missing value of elements in L, just get the value −π: π π + ππ π , Then the matrix is now Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 8 / 38 QAC No. CC-09262020 The next goal is to make a32 to become zero. Then the matrix π is now FINDING THE LOWER TRIANGULAR MATRIX (L) Start with the identity matrix leaving the elements below the main diagonal blank. 1 [ 0 1 0 0] 1 Plug-in the missing value of elements in L by just getting the value −π: π π + ππ π, obtained from the row operations conducted in finding U 1 L = [−π −π 0 1 π 0 0] 1 Now, our A = L U Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN −3 [π π 2 −6 −π −1 1 7 ] = [−π 4 −π 0 1 π −3 0 0] [ π π 1 2 −2 π Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 9 / 38 QAC No. CC-09262020 −1 5] −2 Steps to solve a system using an LU factorization: 1. Set up the equation π΄π₯ = B 2. Find an LU factorization for A. This will yield to the equations: π΄π₯ = B where A = LU LUπ₯ = B let Uπ₯ = C (equation 1) LC = B (Equation 2) 3. Solve for the value of C using Equation 2 4. Solve Equation 1: Uπ₯ = C to find x. • Solving Equation 2: LC = B 1 [−π −π 0 1 π −1 0 π1 0] [π2 ] = [−7] −6 1 π3 (1)π1 + (0)π2 + (0)π3 −1 [(−2)π1 + (1)π2 + (0)π3 ] = [−7] (−1)π1 + (1)π2 + (1)π3 −6 Simplifying the matrix equation, we will get ππ = −π −2π1 + π2 = −7 πΈππ 3. −π1 + π2 + π3 = −6 Eqn. 4 To find π2 , substitute π1 to Eq. 3 −2(−1) + π2 = −7 ππ = −7 − 2 = −π To find π3 , use Eq. 4 −(−1) + (−9) + π3 = −6 π3 = −6 + 8 = π Therefore Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 10 / 38 QAC No. CC-09262020 −1 πΆ = [−9] 2 • Solve Equation 1: Uπ₯ = C to find x. −3 [π π 2 −2 π −1 π₯ −1 5 ] [π¦] = [−9] −2 π§ 2 −3π₯ + 2π¦ − π§ −1 [ −2π¦ + 5π§ ] = [−9] 2 −2π§ −3π₯ + 2π¦ − π§ = −1 −2π¦ + 5π§ = −9 −2π§ = 2 Solving for z: −2π§ = 2 π = −π Solving for y: −2π¦ + 5π§ = −9 −2π¦ + 5(−1) = −9 2π¦ = 4 π=π Solving for x: −3π₯ + 2π¦ − π§ = −1 −3π₯ + 2(2) − (−1) = −1 3π₯ = 6 π=π We got the same answer. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 11 / 38 QAC No. CC-09262020 B. DOOLITTLE’S METHOD Thus far though, we have found an πΏπ factorization of a matrix by first applying the row operation method to π΄ to get π, and then examining the multipliers in the row operations process to determine the entries below the main diagonal of πΏ. We will now look at another method for finding an πΏπ factorization of matrix without going through the process of row operation method. Doolittle’s Method takes an π × π matrix π΄ and assume that an πΏπ factorization exists. πΏ is a lower triangular matrix with ones on the main diagonal and π is an upper triangular matrix, We then match the entries of π΄ with the products or necessary entries from πΏ and π. Then solving the system π΄π₯ = π΅ becomes a matter of simply applying substitution and backward substitution. • • • • Doolittle’s Method is best explained with an example. Suppose that π΄ is a 3 × 3 matrix and that an πΏπ factorization exists. • If we multiply πΏ and π and equate to the first row of π΄ we immediately get that: π11 = (1) (π’11)+ (0)(0) + (0)(0) π11 = π’11 π12 = (1) (π’12) + (0)(π’22) + (0)(0) π12 = π’12 π13 = (1) (π’13) + (0)(π’23) + (0)(π’33) π13 = π’13 • We then multiply πΏ and π for the second row of π΄ and we have that: π21 = (π21)(π’11) + (1)(0) + (0)(0) π21 = π21 π’11 Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 12 / 38 QAC No. CC-09262020 π22 = (π21)(π’12) + (1)(π’22) + (0)(0) π22 = π21π’12 + π’22 π23 = (π21)(π’13) + (1)(π’23) + (0)(π’33) π23 = π21π’13 + π’23 • • From these equations, we can solve for π21, π’22, and π’23. Lastly, we then multiply πΏ and π for the third row of A π31 = (π31)(π’11) + (π32)(0) + (1)(0) π31 = π31π’11 π32 = (π31)(π’12) + (π32)(π’22) + (1)(0) π32 = π31π’12 + π32π’22 π33 = (π31)(π’13) + (π32)(π’23) + (1)(π’33) π33 = π31π’13 + π32π’23 + π’33 • For a less general example, suppose that we want to find an πΏπ factorization to the matrix: • We first assume that an πΏπ factorization exists. Then we have that: We immediately have that: π11 = π’11 −3 = π’11 • Transaction ID Signature π12 = π’12 2 = π’12 π13 = π’13 −1 = π’13 Moving onto the second row of π΄, we will first get π21. Then we can substitute π21 to the next equation to get π’22 and π’23 TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN π21 = π21π’11 6 = π21(−3) π21 = −2 • • REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 13 / 38 QAC No. CC-09262020 π23 = π21π’13 + π’23 7 = (−2)(−1) + π’23 π’23 = 5 Moving onto the third row of π΄, we will first get π31. Then we can substitute π31 to the next equation to get π32. Plugging both into the third equation and we will have π’33. π31 = π31π’11 3 = π31(−3) π31 = −1 • π22 = π21π’12 + π’22 −6 = (−2)(2) + π’22 π’22 = −2 Index No. π32 = π31π’12 + π32π’22 −4 = (−1)(2) + π32(−2) π32 = 1 π33 = π31π’13 + π32π’23 + π’33 4 = (−1)(−1) + (1)(5) + π’33 π’33 = −2 So, we have all entries for both πΏ and π and so: Then we go through the same process to solve for x. First, let us now solve for C using Equation 1: LC = B We will multiply L with C to get the equations. Simplifying the matrix equation, we will get π1 = −1 −2π1 + π2 = −7 −π1 + π2 + π3 = −6 Eqn. 3 Eqn. 4 Now, we will do forward substitution. Substitute π1 to Eq. 3 Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 14 / 38 QAC No. CC-09262020 −2(−1) + π2 = −7 π2 = −7 − 2 π2 = −9 Then, substitute π1 and π2 to Eq. 4 −(−1) + (−9) + π3 = −6 π3 = −6 + 8 π3 = 2 This is now the matrix πΆ • Recalling Equation 1: Uπ₯ = C to find x. (π)π₯ = C We will multiply U with x to get the equations. Simplifying the matrix equation, we will get Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 15 / 38 QAC No. CC-09262020 Now, we will do backward substitution. Substitute π§ to Eq. 6 Next, substitute π¦ and π§ to Eq. 5 This is now the matrix x Finally, we simplify the augmented matrix. We can write the equations We get the same answer using LU factorization by the Doolittle’s method. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 16 / 38 QAC No. CC-09262020 C. CROUT’S METHOD • • • • Another way to find the lower and upper triangular matrices that is almost the same as the Doolittle’s method. Just like the Doolittle’s method, the Crout’s Method takes an π × π matrix π΄ and assume that an πΏπ factorization exists. Only that the diagonal one’s is on the upper triangular matrix not on the lower. We then match the entries of π΄ with the products or necessary entries from πΏ and U. Suppose that π΄ is a 3 × 3 matrix and that an πΏπ factorization exists. • For the first column of A, let’s multiply πΏ and π and we immediately get that: • π11 = (π11)(1) + (0)(0) + (0)(0) π11 = π’11 π21 = (π21)(1) + (1)(0) + (0)(0) π21 = π21 π31 = (π31)(1) + (π32)(0) + (π33)(0) π31 = π31 • For the second column of A, let’s multiply πΏ and π and we get that: π12 = (π11)(π’12) + (0)(1) + (0)(0) π12 = π11 π’12 Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 17 / 38 QAC No. CC-09262020 π22 = (π21)(π’12) + (π22)(1) + (0)(0) π22 = π21π’12 + π22 π32 = (π31)(π’12) + (π32)(1) + (π33)(0) π32 = π31π’12 + π32 • From these equations, we can solve for π’12, π22, and π32. • Lastly, we then multiply πΏ and π for the third column of π΄ and we have that: π13 = (π11)(π’13) + (0)(π’23) + (0)(1) π13 = π11π’13 π23 = (π21)(π’13) + (π22)(π’23) + (0)(1) π23 = π21π’13 + π22π’23 π33 = (π31)(π’13) + (π32)(π’23) + (π33)(1) π33 = π31π’13 + π32π’23 + π33 • For a less general example, suppose that we want to find an πΏπ factorization to the matrix: We first assume that an πΏπ factorization exists. Then we have that: We immediately have that: π11 = π11 −3 = π 11 π21 = π21 6 = π21 π31 = π31 3 = π31 Moving onto the second column of π΄, we will first get π’12. Then we can substitute π’12 to the next equations to get π22 and π32. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 18 / 38 QAC No. CC-09262020 Moving onto the third column of π΄, we will first get π31. Then we can substitute π31 to the next equation to get π32. Plugging both into the third equation and we will have π’33. So, we have all entries for both πΏ and π and so: • Then we go through the same process to solve for x. First, let us now solve for C using Equation 1: LC = B We will multiply L with C to get the equations. Simplifying the matrix equation, we will get Now, we will do forward substitution. Substitute π1 to Eq. 7 Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 19 / 38 QAC No. CC-09262020 Then, substitute π1 and π2 to Eq. 8 This is now the matrix πΆ Recalling now equation no. 1, let us solve for x We will multiply U with x to get the equations Simplifying the matrix equation, we will get Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 20 / 38 QAC No. CC-09262020 Now, we will do backward substitution. Substitute π§ to Eq. 10 Next, substitute π¦ and π§ to Eq. 9 This is now the matrix x Finally, we simplify the augmented matrix. We can write the equations: π=π π=π π = −π We get the same answer using LU factorization by the Crout’s method. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 21 / 38 QAC No. CC-09262020 ITERATIVE METHODS Another way of solving a system of equations is by using Iterative methods. The word Iteration is a repetition of a mathematical or computational procedure applied to the result of a previous application, typically as a means of obtaining successively closer approximations to the solution of a problem. For the method of iteration, we approximate solutions quickly and with low percentage of error. As iteration techniques, the idea is to find a procedure for computing several “rounds” of approximations, each better than the last. In this section you will look at two iterative methods – the Jacobi Iteration and Gauss-Seidel Iteration both for approximating the solution of a system of n linear equations in n variables RECALL (NATURE OF ITERATIVE METHODS) Input: Initial estimate of the solution Process (Method) Output: closer approximate of the solution Is output up to desired accuracy? Print output as approximated desired solution to the problem STOP Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 22 / 38 QAC No. CC-09262020 JACOBI ITERATION METHOD It is named after Carl Gustav Jacob Jacobi (1804–1851). The Jacobi method is used for determining the solutions of a system of linear equations approximately. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This method makes two assumptions: (1) that the system given by a square system of n linear equation has a unique solution. (2) that the coefficient matrix A has no zeros on its main diagonal. If any of the diagonal entries are zero, then rows or columns must be interchanged to obtain a coefficient matrix that has non-zero entries on the main diagonal. Writing it in matrix form π΄π₯ = B Then A can be decomposed into a diagonal component D, a strictly lower triangular part L and strictly upper triangular part U. Note that it is a strictly triangular matrix if all the entries on the main diagonal of a (upper or lower) triangular matrix are all zeros (0). π΄=π·+πΏ+π Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 23 / 38 QAC No. CC-09262020 The solution is then obtained iteratively via Matrix form: Equation form: where π₯[π] – is the kth approximation or iteration of x π₯[π+1] – is the next or the k+1 iteration of x USING THE EQUATION FORM To begin the Jacobi method, solve the first equation for π₯1, the second equation for π₯2 , the third equation for π₯3 and so on. Then make an initial approximation of the solution (π₯1, π₯2,π₯3, … … . , π₯π) and substitute these values of π₯π into the right-hand side of the rewritten equations to obtain the first approximation. As new values are generated, they are not immediately used but rather are retained for the next iteration. By repeated iterations, you will perform a sequence of approximations that often converges to the actual solution. EXAMPLE Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 24 / 38 QAC No. CC-09262020 Use the Jacobi method to approximate the solution of the following system of linear equations. Continue the iterations until two successive approximations are identical when rounded to three significant digits. SOLUTION Step 1: Write the system in the form Step 2: Make an initial approximation of the solution since you do not know the actual solution. Initial Approximation: π₯1 = 0 π₯2 = 0 π₯ 3 = 0 Step 3: Substitute the values of the initial approximation to the equations to get your first iteration values. Iteration 1 Repeat the process by substituting the current values obtained to the equations. Continue the iterations until two successive approximations are identical when rounded to three significant digits. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 25 / 38 QAC No. CC-09262020 Step 4: Create a table to track the iterations. Because the last two columns in Table are identical, you can conclude that to three significant digits the solution is ππ = π. πππ ππ = π. πππ ππ = −π. ππ3 Below is the solution for iterations 2 to 7 Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 26 / 38 QAC No. CC-09262020 For the system of linear equations given in Example, the Jacobi method is said to converge. That is, repeated iterations succeed in producing an approximation that is correct to three significant digits. As is generally true for iterative methods, greater accuracy would require more iterations. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 27 / 38 QAC No. CC-09262020 GAUSS-SEIDEL ITERATION METHOD It is the modification of the Jacobi method, named after Carl Friedrich Gauss (1777–1855) and Philipp L. Seidel (1821–1896). This modification is no more difficult to use than the Jacobi method, and it often requires fewer iterations to produce the same degree of accuracy. With the Jacobi method, the values of π₯π obtained in the nth approximation remain unchanged until the entire (n+1) approximation has been calculated. With the Gauss-Seidel method, on the other hand, as each new x value is computed it is immediately used in the next equation to determine another x value. That is, once you have determined π₯1 from the first equation, its value is then used in the second equation to obtain the new π₯2. Similarly, the new π₯1 and π₯2 are used in the third equation to obtain the new π₯3 and so on EXAMPLE Use the Gauss-Seidel iteration method to approximate the solution to the system of equations given in Example in Jacobi iteration method. Continue the iterations until two successive approximations are identical when rounded to three significant digits. SOLUTION Step 1: Write the system in the form Step 2: Make an initial approximation of the solution since you do not know the actual solution. Initial Approximation: π₯1 = 0 π₯2 = 0 π₯3 = 0 Step 3: Substitute the values of the initial approximation to the first equation. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 28 / 38 QAC No. CC-09262020 Iteration 1 Now that you have a new value for π₯1 , use it now to compute the new value for π₯2 Similarly, use new values of π₯1 πππ π₯2 to compute a new value for π₯3 Step 4: Create a table to track the iterations. Note that after only six iterations of the Gauss-Seidel method, you achieved the same accuracy as was obtained with seven iterations of the Jacobi method ππ = π. πππ Transaction ID Signature ππ = π. πππ ππ = −π. ππ3 TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 29 / 38 QAC No. CC-09262020 Below is the solution for iterations 2 to 6 Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 30 / 38 QAC No. CC-09262020 Note: Neither of the iterative methods presented in this section always converges. There are times that applying the Jacobi method or the Gauss-Seidel method to a system of linear equations obtain a divergent sequence of approximations. In such cases, it is said that the method diverges. EXAMPLE OF DIVERGENCE Apply the Jacobi method to the system π₯1 − 5π₯2 = −4 7π₯1 − π₯2 = 6 using the initial approximation (π₯1, π₯2) = (0,0) and find out that the solution diverges. SOLUTION Step 1: Write the system in the form π₯1 = −4 + 5π₯2 π₯2 = −6 + 7π₯1 Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 31 / 38 QAC No. CC-09262020 Step 2: Substitute the values of the initial approximation to the equations to get your first iteration values. Step 3: Create a table to track the iterations. The repeated iterations produce a sequence of approximations that diverges. For this particular system of linear equations, you can determine that the actual solution is π₯1 = 1 and π₯2 = 1 but you can see from table of iterations that the approximations given by the Jacobi method become progressively worse instead of better. The problem of divergence in the given example is not resolved by using the Gauss-Seidel method rather than the Jacobi method. In fact, for this particular system the Gauss-Seidel method diverges more rapidly. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 32 / 38 QAC No. CC-09262020 Table of Iterations Using Gauss-Seidel Method With the given initial approximation (π₯1, π₯2) = (0,0) neither the Jacobi nor Gauss-Seidel method converges to the solution of linear equations given. But a special type of coefficient matrix A, called a strictly diagonally dominant matrix, for which it is guaranteed that both methods will converge. Definition of Strictly Diagonally Dominant Matrix An n x n matrix A is strictly diagonally dominant if the absolute value of each entry on the main diagonal is greater than the sum of the absolute values of the other entries in the same row. That is, EXAMPLE: Strictly Diagonally Dominant Matrices Which of the following systems of linear equations has a strictly diagonally dominant coefficient matrix? Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 33 / 38 QAC No. CC-09262020 (a) The coefficient matrix is strictly diagonally dominant because (b) The coefficient matrix is NOT strictly diagonally dominant because the entries in the second and third rows do not conform to the definition. But interchanging the second and third rows in the original system of linear equations, however, produces a strictly diagonally dominant matrix as shown below. 4π₯1 + 2π₯2 − π₯3 = −1 3π₯1 − 5π₯2 + π₯3 = 3 π₯1 + 2π₯3 = −4 Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 34 / 38 QAC No. CC-09262020 Convergence of the Jacobi and Gauss-Seidel Methods For Jacobi Method If coefficient matrix A is strictly diagonally dominant, then the system of linear equations given by π΄π₯ = π has a unique solution to which the Jacobi method will converge for any initial approximation. For Gauss-Seidel Method The convergence properties of the Gauss–Seidel method are dependent on the matrix A. The procedure is known to converge if either: Matrix A is strictly diagonally dominant or Matrix A is symmetric positive-definite. Note: The Jacobi and Gauss–Seidel method sometimes converges even if these conditions are not satisfied. Remember: A symmetric matrix is a square matrix that is equal to its transpose. It is a positive definite matrix if all the determinants of each sub-diagonal n x n matrices are all greater than zero (0). Symmetric Positive Definite Matrix Example: Check if the symmetric matrices A and b are positive definite. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 35 / 38 QAC No. CC-09262020 Interchanging Rows to Obtain Convergence Using the previous example that diverge, π₯1 − 5π₯2 = −4 7π₯1 − π₯2 = 6 Interchange the rows of the system to make coefficient matrix strictly diagonally dominant 7π₯1 − π₯2 = 6 π₯1 − 5π₯2 = −4 Then apply the Gauss- Seidel method to approximate the solution to four significant digits. Using the initial approximation (π₯1, π₯2) = (0,0) you can obtain the sequence of approximations shown in the table below. The solution is π₯1 = 1 and π₯2 = 1. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 36 / 38 QAC No. CC-09262020 II. ACTIVITY Note: In this section, discuss the required hands-on exercises at self-pacing, collaborative, experimental or simulation approaches. This section should answer the enabling or demonstrative domains of learning outcomes as facilitated though synchronous, asynchronous and offline modalities. Include the link/s of the actual worksheets using Google Suite or online Microsoft Office applications or similar platforms or multimedia. ASSIGNMENT # 4 1. Find the solution of the following system of linear equations using: (a) LU factorization using row operation method (b) LU factorization using Doolittle’s method (c) LU factorization using Crout’s method π₯ + π¦ + π§ + π€ = 13 2π₯ + 3π¦ − π€ = −1 −3π₯ + 4π¦ + π§ + 2π€ = 10 π₯ + 2π¦ − π§ + π€ = 1 2. Determine whether the matrix is strictly diagonally dominant. 3. Apply the Jacobi and Gauss-Seidel method to the following system of linear equations, using the initial approximation (π₯1, π₯2, … . , π₯π ) = (0,0, … … ,0). Continue performing iterations until two successive approximations are identical when rounded to three significant digits. 4. The coefficient matrix of the system of linear equations is NOT strictly diagonally dominant. Show that the Jacobi and Gauss-Seidel methods converge using an initial approximation of (π₯1, π₯2, … . , π₯π ) = (0,0, … … ,0). Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 37 / 38 QAC No. CC-09262020 5. Show that the Gauss-Seidel method diverges for the given system using the initial approximation (π₯1, π₯2, … . , π₯π ) = (0,0, … … ,0). III. EVALUATION Note: Discuss here the mode of evaluation to be conducted, which are not limited to objective types of quizzes, long exams, or essays, etc. Include the link/s of the actual worksheets using Google Suite or online Microsoft Office applications or similar platforms or multimedia. 1. SYNCHRONOUS: Recitation and seatwork 2. ASYNCHRONOUS: Assignment IV. IMPROVEMENT PLANS Note: Specify here also any means of improving student competencies through assignment, research, coaching, mentoring, remedial classes, or similar activities using the 3 modalities of learning. 1. SYNCHRONOUS: Solving problem exercises and Q&A 2. ASYNCHRONOUS: Assignment V. REFERENCES Note: Specify here the list of references used in the design and development of the course learning materials. Use APA Version 7 and arrange them alphabetically. [1] "Elementary Linear," 2019. [Online]. Available: https://college.cengage.com/mathematics/larson/elementary_linear/5e/students/ch08- 10/chap_10_2.pdf. [Accessed 10 October 2020]. [2] M. Pedrick, "Index of /~mpedrick/teaching/supplemental," Department of Mathematics University of California, Santa Barbara, 5 January 2020. [Online]. Available: https://web.math.ucsb.edu/~mpedrick/teaching/supplemental/m4a_supp_2.pdf. [Accessed 15 October 2020]. [3] Wolfram Research, Inc, "https://mathworld.wolfram.com/," 23 October 2020. [Online]. Available: https://mathworld.wolfram.com/JacobiMethod.html. [Accessed 20 October 2020]. [4] M. O. Author, "Applying The Jacobi Iteration Method," [Online]. Available: http://mathonline.wikidot.com/applying-the-jacobi-iteration-method. [Accessed 14 October 2020]. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version. TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES Ayala Blvd., Ermita, Manila, 1000, Philippines Tel No. +632-301-3001 local 102 | Fax No. +632-521-4063 Email: vpaa@tup.edu.ph | Website: www.tup.edu.ph VAA-ERD STANDARD MODULE DESIGN Index No. REF-ERD-INT-6.1-SMD Issue No. 01 Revision No. 00 Date 09262020 Page 38 / 38 QAC No. CC-09262020 [5] Wikipedia, "https://en.wikipedia.org/," 16 October 2020. [Online]. Available: https://en.wikipedia.org/wiki/Jacobi_method#:~:text=In%20numerical%20linear%20algebra%2C %20the,then%20iterated%20until%20it%20converges.. [Accessed 15 October 2020]. [6] ScienceDirect, "Iterative Convergence," 2020. [Online]. Available: https://www.sciencedirect.com/topics/engineering/iterativeconvergence#:~:text=Iterative%20convergence %20relates%20to%20the,step%20in%20an%20u nsteady%20problem.. [Accessed 15 October 2020]. [7] S. C. C. a. R. P. Canale, "CHAPTER 11 Special Matrices and Gauss-Seidel," in Numerical Methods for Engineers, New York City, McGraw-Hill Education, 2015, pp. 300-311. Transaction ID Signature TUPM-COE-SMD-MTI-10052022-0700AM No part of this document shall be copied or reproduced without the prior written permission of the President through the Vice President for Academic Affairs and Director for Educational Resource Development Services. The bearer of this document shall verify with the Quality Assurance Office if such document is officially registered with the Quality Management System (QMS) or if it is the latest version.
