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Department of Mathematical Sciences
SYLLABUS
Semester II, 2003-2004 (032)
(M. Sarhan)
Course #:Math 471Title:Numerical Analysis IPrerequisite:Math 280; Math 321 or SE 301References:Forsyth and Moler: Computer Solution of Linear Algerbaic Systems. Prentice-Hall (1967).
Textbook Burden and Faires: Numerical Analysis, 7th ed. Brooks/Cole (2001).
Forsythe, Malcolm and Moler: Computer Methods for Mathematical Computations. Prentice-Hall (1977).Numerical Software:LINPACK, EISPACK and LAPACK.Attachments:List of Supplementary References (SR). Course Objectives.
Topic
No.No. of LecturesMaterial
I
1Course objectives and organizations. Main topics of Numerical Linear Algebra:
References: This syllabus; Secs. 1, 5 and 7 of I.
II
6Floating-Point Round-Off Analysis:
References: Sec. 20 of I; Sec. 1.2 of II; Sec. 2.1-2.3 of III.
Instability and Sensitivity: General comments and examples.
References: Sec. 1.3 of II; Secs. 2.4 and 2.5 of III.
III
6Review of selected topics from Linear Algebra and Matrix Theory:
Linear transformation and Matrices. Linear systems. Determinants. Vector and Matrix norms. Convergent sequences of vectors in EMBED Equation.DSMT4 . Convergent matrices.
References: Secs. 6.1, 6.3, 6.4 and 7.1 of II; Secs. 2, 3, 4 and 6 of I; SR(1).
IV
7Direct Methods for Solving Linear Systems: Gaussian Elimination. Forward elimination and Back substitution. LU Decomposition. Pivoting. Scaling. Subroutines DECOMP & SOLVE. Operations count. Computing determinants and inverses. LDU Decomposition. Crout and Doolittle Methods. Special classes of matrices. Choleskys Method. Iterative Improvement. Convergence of Iterative Improvement.
References: Secs. 9-14, 18, 22 and 23 of I; Secs. 6.1, 6.2, 6.5-6.7 and 7.4 of II; SR(2). (Sec. 22 of I is covered lightly.)
Syllabus Math 471
Page Two
Semester (032)
V
6Conditioning and Perturbation: Condition Number. Condition of a Linear System. Rounding error in Gaussian elimination. Nearly singular matrices. Inverses of perturbed matrices. Approximate inverses. Posteriori error estimates. Error analysis of perturbed systems.
References: Secs. 8, 15 and 21 of I; Secs. 7.4 and 7.6 of II; Secs. 3.1 and 3.2 of III; SR(3). (Details of proofs in Sec. 21 of I are omitted; however, exercises (21.43)-(21.46) are covered.)
VI
6Iterative Methods for Solving Linear Systems: The Jacobi Method. The Gauss-Seidel Method. Successive Over-Relaxation (SOR). Error analysis and Convergence theorems. Subroutines.
References: Secs. 7.3 and 7.6 of II.
VII
6The Linear Least-Square Problem: Least-squares and data fitting. Normal equations. Orthogonal polynomials. Normal equations in matrix form. The Singular Value Decomposition (SVD). Subroutine SVD.
References: Secs. 8.1 and 8.2 of II; Secs. 9.1, 9.2 and 9.5 of III; SR(4).
VIII
7Matrix Eigenvalue Problems: Variational principles. The Power Method. The Inverse Power Method. Deflation techniques. Shift of origin. Householders Method. The QR Algorithm. Subroutine SYMEIG.
References: Secs. 9.1-9.5 of II. (Secs. 9.3 and 9.4 are covered somewhat lightly.)
Software:
LINPACK is a package of FORTRAN subroutines that analyze and solve (by direct methods) various systems of linear equations and solve linear least squares problems.
EISPACK contains FROTRAN subroutines for computing eigenvalues and eigenvectors for a variety of different types of matrices.
LAPACK is a library of FOTRAN subroutines. It is a unified updated package of the subroutines in LINPACK and EISPACK.
IMSL and NAG are libraries of extensive numerical analysis subroutines.
These packages are high-quality Numerical Software: the subroutines are highly efficient, accurate and reliable. LINPACK, EISPACK and LAPACK are available in the public domain. IMSL and NAG are commercially available. See Sec. 1.4 of II for further details.
MATH 471
Numerical Analysis I
Semester II, 2003-04(032)
Supplementary References
(SR)
M. Sarhan
(1)For Topic III: Review of some topics from Linear Algebra and Matrix Theory Select any reference you prefer. The following reference is recommended:
G. Strang: Linear Algebra and Its Applications. Saunders HBJ(1988).
(2)
(a)
(b)
(c)Topic (IV)
M. Sarhan (lecture notes): Operations Count and Storage.
Tridiagonal Systems (p. 74):
H.B. Keller: Numerical Methods for Two-Point Boundary-Value Problems. Blaisdell (1968).
Z. Nashed (lecture notes):
Numerical Inversion of Upper Triangular Matrices.
(3)
(a)
(b)
(c)Topic (V)
M. Sarhan (lecture notes):
Rounding Error in Gaussian Elimination
Estimating the Condition Number of a Matrix.
Z. Nashed (lecture notes):
Error Analysis of Perturbed Systems of Linear Equations.
Convergent Matrices and A Posteriori Error Estimates
(Secs 1.1.1 and 2.1.3):
E. Isaacson and H.B. Keller: Analysis of Numerical Methods. John Wiley (1966).
(4)Topic (VII)
Orthogonal Polynomials and Least-Squares
(Parts of Secs. 6.3 and 6.4):
S.D. Conte and C. de Boor: Elementary Numerical Analysis: An Algorithmic Approach, 3rd ed. McGraw-Hill (1980).
Remarks:
Lecture notes (taken by the students) supplement the assigned materials in the textbook and the references.
Assignment sheets contain remarks and supplementary problems as appropriate.
COURSE OBJECTIVES
MATH 471
Numerical Analysis I
(3-0-3)
This course is addressed to senior and graduate students of Sciences and Engineering. Its objectives are to introduce the students to some of the more important topics of Numerical Linear Algebra including aspects of the theory by which its algorithms may be analyzed and to high-quality Numerical Software of Linear Algebra.
The core topics are Direct Methods for Solving Linear Systems, Conditioning and Perturbation, Iterative Techniques for Solving Linear Systems, the Linear Least-Squares Problem and the Matrix Eigenvalue Problem. The treatment of these topics is balanced with respect to theory and computations and it is firm and rigorous. Hence, it calls for a good coverage of some elements of Floating-Point Round-Off Analysis, a study of rudiments of Matrix Theory and a brief review of selected topics from Linear Algebra as bases.
The objectives are consistent with the course description (Undergraduate Bulletin, 2001-03) and they are certainly within the reach of a good student a student who had done good on the Prerequisites (Math 280 and Math 321 or SE 301), who has good recollection of the materials covered there and who is willing to do good and serious work.
Forward Outlook: Math 471 (Numerical Analysis I) and Math 472 (Numerical Analysis II) form a total package that contains standard topics of numerical analysis. Thus, for further study in this field, the next recommended course is Math 472.
(A copy of the syllabus is attached.)
M. Sarhan
Course Instructor
MATH 471
Semester II, 2003-04 (032)
Submitted: January 2004
MATH 471
Numerical Analysis I
Semester II, 2003-04(032)
Prelude
To the Student
Math 471 is a rigorous course in Numerical Linear Algebra. The student is assumed to have had introductory courses in linear algebra and numerical analysis such as Math 280 and Math 321 or SE 301. He should be able to do programming in FORTRAN.
A copy of the course syllabus is attached to this prelude. It clearly indicates what is to be covered, the references and the rate of coverage. A copy of the list of Supplementary References (SR) and a copy of the course objectives are also attached.
Each lecture is an integral part of the course. Some lectures contain supplementary material. Thus, good notes of lectures must be taken.
There certainly will be reading and studying assignments and homework and programming assignments. A deadline will be specified for each assignment and it must be normally met by all students. Reading and studying assignments may contain material that is not covered in class; such assignments should be taken very seriously. All assignments that are required for submission must be done strictly in clear, clean and neat form. A certain format will be explained in class.
Students should be ready and able to do some simple programming in FORTRAN by the end of the second week of classes. Proficiency in programming is expected at a later stage.
With respect to grading, the general outline of the course is as follows: There will be a Midterm exam (25%), Prefinal (15%) and Final (10%) exams. Each exam will be based largely on all new material covered in the lectures preceding it. All exams will be both computational and theoretical in nature. Homework and programming assignments will count as 50% of the total grade.
It is likely that Help Sessions will be scheduled. Such sessions are meant to remedy deficient background and to assist in learning new materials.
Homepages and e-mail will certainly facilitate contacts. Details are given below. Effective use of all recourses is certainly encouraged.
I wish you a productive and a rewarding experience and I look forward to having a pleasant semester with all of you.
Course Homepage: http://faculty.kfupm.edu.sa/math/msarhan/math471.html
Instructor Homepage: http://faculty.kfupm.edu.sa/math/msarhan
Instructor e-mail: msarhan@kfupm.edu.sa
Dr. Mahmoud Sarhan
Instructor
Math 471(032)
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