Syllabus
Course Topics
Calculations related to matrices
Vectors and drawing
Files and function definitions
Familiarity with some internal functions of the desired tool
An introduction to errors
Floating point system
The source of errors
Relative and absolute errors
Rounding, inherent and shear errors
Error propagation and process graph
Instability in numerical calculations
An introduction to finding the roots of a non-linear univariate function
Halving method
Displacement method
Chord method or decisive line
Newton-Raphson method
Iterative method or fixed point
Convergence rate of different methods
Necessary/sufficient conditions for the convergence of Newton-Raphson methods, string method and simple iteration method
Herner's method for calculating polynomial value
Generalized Newton-Raphson method for solving the system of nonlinear equations
Intuitive and mathematical proof of the mentioned methods
An introduction to interpolation, extrapolation and curve fitting
Different interpolation methods include Lagrange's method, Newton's divided difference method, Newton's forward, backward and central difference methods.
Proof of the mentioned methods and error analysis in them
Fitting to polynomials by least squares method
Fitting different curves with the help of linearization
Extrapolation
An introduction to numerical integration and differentiation
Various methods of numerical integration include rectangular method, midpoint method, trapezoidal method, Gauss-Legendre method, Simpson 1/3 method, Simpson 3/8 method, and Ramberg method.
Checking the error rate of the mentioned methods
Mathematical and intuitive proof for the mentioned methods
Numerical derivation using different methods including midpoint method, central difference method, three point method
Analyzing the error order of the mentioned methods and using the concept of Richardson's extrapolation to improve the numerical derivation results.
An introduction to differential equations
One-step methods include Taylor's method, Euler's method, modified Euler's method, 2nd-order Rang-Kutta methods (Hyun's method, midpoint and modified Euler), 3rd-order Rang-Kutta and 4th-order Rang-Kutta methods.
Multi-step methods such as the Adams-Moulton method
Error analysis of the mentioned methods and their comparison
Converting differential equations of higher degrees to linear differential equations
Converting single-step methods of solving linear differential equations into numerical methods that can be used to solve linear differential equations
An introduction to solving linear equations
An introduction to matrices
Direct methods for solving linear equations including inverse matrix method, Kramer method, Gaussian elimination method (progressive, backward and Gauss-Jordan), trigonometric analysis method (LU Choleski, Doolittle and Croat)
Iterative methods including Jacobi method and Gauss-Seidel method
Eigenvalues and eigenvectors, the power method to find an estimate of the dominant eigenvalue and its corresponding eigenvector, and Gerch Gorin's theorem
Vectors and drawing
Files and function definitions
Familiarity with some internal functions of the desired tool
An introduction to errors
Floating point system
The source of errors
Relative and absolute errors
Rounding, inherent and shear errors
Error propagation and process graph
Instability in numerical calculations
An introduction to finding the roots of a non-linear univariate function
Halving method
Displacement method
Chord method or decisive line
Newton-Raphson method
Iterative method or fixed point
Convergence rate of different methods
Necessary/sufficient conditions for the convergence of Newton-Raphson methods, string method and simple iteration method
Herner's method for calculating polynomial value
Generalized Newton-Raphson method for solving the system of nonlinear equations
Intuitive and mathematical proof of the mentioned methods
An introduction to interpolation, extrapolation and curve fitting
Different interpolation methods include Lagrange's method, Newton's divided difference method, Newton's forward, backward and central difference methods.
Proof of the mentioned methods and error analysis in them
Fitting to polynomials by least squares method
Fitting different curves with the help of linearization
Extrapolation
An introduction to numerical integration and differentiation
Various methods of numerical integration include rectangular method, midpoint method, trapezoidal method, Gauss-Legendre method, Simpson 1/3 method, Simpson 3/8 method, and Ramberg method.
Checking the error rate of the mentioned methods
Mathematical and intuitive proof for the mentioned methods
Numerical derivation using different methods including midpoint method, central difference method, three point method
Analyzing the error order of the mentioned methods and using the concept of Richardson's extrapolation to improve the numerical derivation results.
An introduction to differential equations
One-step methods include Taylor's method, Euler's method, modified Euler's method, 2nd-order Rang-Kutta methods (Hyun's method, midpoint and modified Euler), 3rd-order Rang-Kutta and 4th-order Rang-Kutta methods.
Multi-step methods such as the Adams-Moulton method
Error analysis of the mentioned methods and their comparison
Converting differential equations of higher degrees to linear differential equations
Converting single-step methods of solving linear differential equations into numerical methods that can be used to solve linear differential equations
An introduction to solving linear equations
An introduction to matrices
Direct methods for solving linear equations including inverse matrix method, Kramer method, Gaussian elimination method (progressive, backward and Gauss-Jordan), trigonometric analysis method (LU Choleski, Doolittle and Croat)
Iterative methods including Jacobi method and Gauss-Seidel method
Eigenvalues and eigenvectors, the power method to find an estimate of the dominant eigenvalue and its corresponding eigenvector, and Gerch Gorin's theorem
Slides
We will post slides on the course website after each lecture.
Grading Policy
- Assignments: 50% (20% Homework Assignments + 30% Programming Exercises)
- Midterm Exam: 25%
- Final Exam: 25%