King
Fahd University of Petroleum and Minerals
Department
of Mathematical Sciences
SYLLABUS
Semester II, 2003-2004 (032)
(M. Sarhan)
Course #: |
Math
471 |
Title: |
Numerical
Analysis I |
Prerequisite: |
Math
280; Math 321 or SE 301 |
References: |
I)
Forsyth and Moler:
Computer Solution of Linear Algerbaic Systems.
Prentice-Hall (1967). II) Textbook – Burden and Faires: Numerical Analysis, 7^{th} ed. Brooks/Cole (2001).III)
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 Lectures |
Material |
I |
1 |
Course objectives and
organizations. Main topics of Numerical Linear Algebra: References: This syllabus; Secs. 1, 5
and 7 of I. |
II |
6 |
Floating-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 |
6 |
Review 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
_{}. 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 |
7 |
Direct 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. Cholesky’s 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 |
6 |
Conditioning 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 |
6 |
Iterative 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 |
6 |
The 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 |
7 |
Matrix Eigenvalue Problems: Variational principles. The Power Method. The Inverse
Power Method. Deflation techniques. Shift of origin. Householder’s Method.
The QR Algorithm. Subroutine SYMEIG. References: Secs. 9.1-9.5 of II. (Secs. 9.3 and 9.4 are covered somewhat lightly.) |
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.