Contents
Lesson 1 - Error and Related Concepts (Ch 1) ......................................................................................... 3
Error .................................................................................................................................................... 3
Lesson 2 – Nonlinear Equations (Ch 2 & 10.2) ........................................................................................ 3
Bisection Method: ............................................................................................................................... 3
Fixed Point Method: ........................................................................................................................... 4
Newton’s Method: .............................................................................................................................. 5
Secant Method:................................................................................................................................... 6
Method of False Position (Regula Falsi Method): ............................................................................... 7
Methods for Polynomial Functions: .................................................................................................... 7
Horner’s method: ............................................................................................................................ 7
Muller’s method: ............................................................................................................................ 8
Newton’s Method for Multi-variable Functions: ............................................................................ 8
Lesson 3 – Linear Systems: Direct Methods (Ch 6) ................................................................................. 9
Gaussian Elimination........................................................................................................................... 9
Gaussian elimination with Pivoting..................................................................................................... 9
Scaled Partial Pivoting: .................................................................................................................... 9
Matrix Factorization Methods and Special Matrices: ..................................................................... 9
Special Matrix Properties: ............................................................................................................. 10
Error and Residual of an Approximate Solution: .............................................................................. 10
Condition of a Linear System: ........................................................................................................... 10
Lesson 4 – Linear Systems: Iterative (Indirect) Methods (Ch 7) ........................................................... 11
Notation: ........................................................................................................................................... 11
Jacobi Method: ................................................................................................................................. 11
Gauss Seidel Method: ....................................................................................................................... 13
Relaxation Methods: ......................................................................................................................... 14
Lesson 5 – Polynomial Interpolation and Approximation (Ch 3 & 8.1) ................................................ 15
Lagrange Interpolating Polynomial: .................................................................................................. 15
Neville’s Interpolating Polynomial: ................................................................................................... 16
Newton’s Difference Interpolating Polynomial: ............................................................................... 17
Error Estimation: ............................................................................................................................... 18
Osculating Polynomials: .................................................................................................................... 18
Hermite Polynomials: .................................................................................................................... 18
Cubic Splines: ................................................................................................................................ 18
, Bezier Curves:................................................................................................................................ 20
Least Squares Approximation: ...................................................................................................... 20
Lesson 6 – Numerical Differentiation and Integration (Ch 4) ............................................................... 24
Numerical Differentiation ................................................................................................................. 24
Numerical Integration ....................................................................................................................... 25
Trapezoidal Rule: .............................................................................................................................. 25
Simpson’s Rule: ................................................................................................................................. 26
Newton-Cotes Formulas: .................................................................................................................. 26
Quadrature Methods: ....................................................................................................................... 27
, COS2633
Lesson 1 - Error and Related Concepts (Ch 1)
Error → Loss of accuracy due to rounding decimals
Types of errors:
• Experimental error → Due to inaccurate data (bad input)
• Truncation error → Due to replacing exact method with approximation
• Round-off error → Caused by physical limit to the number of digits that can be
retained
• Machine error → Due to data representation and manipulation
Lesson 2 – Nonlinear Equations (Ch 2 & 10.2)
Root problem form (RP) → f(x) = 0
Fixed-point problem form (FP) → x = g(x)
Bisection Method:
• Uses RP form
• Requires that chosen interval (a, b) contains the root
• Root is calculated by halving current interval
• Interval containing the root is confirmed by Intermediate Value Theorem ( f(a) . f(b) < 0 )
• Convergence:
o Ensures convergence to root cuz it is bracted by endpoints
• Termination:
o Number of iterations n required to attain a tolerance of e is found by solving
Lesson 1 - Error and Related Concepts (Ch 1) ......................................................................................... 3
Error .................................................................................................................................................... 3
Lesson 2 – Nonlinear Equations (Ch 2 & 10.2) ........................................................................................ 3
Bisection Method: ............................................................................................................................... 3
Fixed Point Method: ........................................................................................................................... 4
Newton’s Method: .............................................................................................................................. 5
Secant Method:................................................................................................................................... 6
Method of False Position (Regula Falsi Method): ............................................................................... 7
Methods for Polynomial Functions: .................................................................................................... 7
Horner’s method: ............................................................................................................................ 7
Muller’s method: ............................................................................................................................ 8
Newton’s Method for Multi-variable Functions: ............................................................................ 8
Lesson 3 – Linear Systems: Direct Methods (Ch 6) ................................................................................. 9
Gaussian Elimination........................................................................................................................... 9
Gaussian elimination with Pivoting..................................................................................................... 9
Scaled Partial Pivoting: .................................................................................................................... 9
Matrix Factorization Methods and Special Matrices: ..................................................................... 9
Special Matrix Properties: ............................................................................................................. 10
Error and Residual of an Approximate Solution: .............................................................................. 10
Condition of a Linear System: ........................................................................................................... 10
Lesson 4 – Linear Systems: Iterative (Indirect) Methods (Ch 7) ........................................................... 11
Notation: ........................................................................................................................................... 11
Jacobi Method: ................................................................................................................................. 11
Gauss Seidel Method: ....................................................................................................................... 13
Relaxation Methods: ......................................................................................................................... 14
Lesson 5 – Polynomial Interpolation and Approximation (Ch 3 & 8.1) ................................................ 15
Lagrange Interpolating Polynomial: .................................................................................................. 15
Neville’s Interpolating Polynomial: ................................................................................................... 16
Newton’s Difference Interpolating Polynomial: ............................................................................... 17
Error Estimation: ............................................................................................................................... 18
Osculating Polynomials: .................................................................................................................... 18
Hermite Polynomials: .................................................................................................................... 18
Cubic Splines: ................................................................................................................................ 18
, Bezier Curves:................................................................................................................................ 20
Least Squares Approximation: ...................................................................................................... 20
Lesson 6 – Numerical Differentiation and Integration (Ch 4) ............................................................... 24
Numerical Differentiation ................................................................................................................. 24
Numerical Integration ....................................................................................................................... 25
Trapezoidal Rule: .............................................................................................................................. 25
Simpson’s Rule: ................................................................................................................................. 26
Newton-Cotes Formulas: .................................................................................................................. 26
Quadrature Methods: ....................................................................................................................... 27
, COS2633
Lesson 1 - Error and Related Concepts (Ch 1)
Error → Loss of accuracy due to rounding decimals
Types of errors:
• Experimental error → Due to inaccurate data (bad input)
• Truncation error → Due to replacing exact method with approximation
• Round-off error → Caused by physical limit to the number of digits that can be
retained
• Machine error → Due to data representation and manipulation
Lesson 2 – Nonlinear Equations (Ch 2 & 10.2)
Root problem form (RP) → f(x) = 0
Fixed-point problem form (FP) → x = g(x)
Bisection Method:
• Uses RP form
• Requires that chosen interval (a, b) contains the root
• Root is calculated by halving current interval
• Interval containing the root is confirmed by Intermediate Value Theorem ( f(a) . f(b) < 0 )
• Convergence:
o Ensures convergence to root cuz it is bracted by endpoints
• Termination:
o Number of iterations n required to attain a tolerance of e is found by solving
