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, 7th 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.)

## 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.