Numerical Methods

Numerical Methods is a course subject in the IUP’s Civil Engineering. It is a 2-credit compulsory subject in the fourth semester. The subject covers the following topics.

  1. Introduction
    1. Mathematics in civil engineering
    2. Spreadsheet, computer programs
    3. Approximations and round-off errors
  2. Roots of equations
    1. Graphical methods
    2. The bisection methods
    3. The false-position methods
    4. Simple fixed-point iteration
    5. The Newton-Raphson method
    6. The secant method
    7. Multiple roots
  3. Linear algebraic equations
    1. The graphical method
    2. Cramer’s rule
    3. Elimination of unknowns
    4. Naive Gauss elimination
    5. Gauss-Jordan
    6. LU decomposition
    7. The matrix inverse
    8. Jacobi
    9. Gauss-Seidel
    10. Successive over-/under-relaxation
    11. Tri-diagonal matrix, Thomas algorithm
    12. Symmetric matrix, Cholesky decomposition
  4. Curve fitting
    1. Least-squares regression
    2. Interpolation
    3. Fourier approximation
  5. Numerical differentiation and integration
    1. The trapezoidal rule
    2. Simpson’s rule
    3. Gauss quadrature
  6. Ordinary differential equations (initial-value problems)
    1. Euler’s method
    2. Heun’s method
    3. The midpoint (improved polygon) method
    4. Runge-Kutta methods
    5. Stiffness
    6. Multistep methods
  7. Introduction to the finite difference approximation (boundary-value problems)
    1. Boundary-value problems
    2. Eigenvalue problems

Reference

Chapra, S.C., Canale, R.P., 2015, Numerical Methods for Engineers, 7th Ed., McGraw-Hill Book Co., New York.

Weekly Agenda

Week# Subject Description
1 Introduction
  • Course description
  • Mathematics in civil engineering
  • Spreadsheet and computer programs
  • Approximation and round-off errors
2 Roots of equations (1/2)
  • Graphical methods
  • The bisection methods
  • The false-position methods
  • Simple fixed-point iteration
  • The Newton-Raphson method
3 Roots of equations (2/2)
  • The secant method
  • Multiple roots
  • Exercise #1: roots of equations
4 Linear algebraic equations (1/3)
  • Gauss elimination
  • Gauss-Jordan
  • LU decomposition
5 Linear algebraic equations (2/3)
  • The matrix inverse
  • Jacobi
  • Gauss-Seidel
  • Successive over-/under-relaxation
6 Linear algebraic equations (3/3)
  • Tri-diagonal matrix, Thomas algorithm
  • Symmetric matrix, Cholesky decomposition
  • Exercise #2: linear algebraic equations
7 Regression
  • Polynomial regression
  • Multivariable regression
8 Midterm exam
9 Interpolation
  • Newton interpolation method
  • Lagrange interpolation method
  • Fourier approximation
10 Numerical differentiation and integration (1/2)
  • The trapezoidal rule
  • Simpson’s rule
11 Numerical differentiation and integration (2/2)
  • Gauss quadrature
  • Exercise #3: interpolation, numerical differentiation and integration
12 Initial-value problems (1/3)
  • Euler’s method
  • Heun’s method
  • The midpoint (improved polygon) method
13 Initial-value problems (2/3)
  • Runge-Kutta methods
14 Initial-value problems (3/3)
  • Stiffness
  • Multistep methods
  • Exercise #4: initial-value problems
15 Boundary-value problems
  • Introduction to FDA
  • Eigenvalue problems
16 Final exam

Lecture Notes

NM00_Course_Description
NM01_Math_and_Civil_Eng
NM02_Roots_of_Equations
NM03_Linear_Algebraic_Equations
NM04_Curve_Fitting