COMP 361/5611 Elementary Numerical Methods COMPLETE SLIDES WITH
SOLUTIONS- NEW UPDATE FALL 2025|2026 Concordia University
LECTURE NOTES
on
ELEMENTARY NUMERICAL METHODS
Eusebius Doedel
, TABLE OF CONTENTS
Vector and Matrix Norms 1
Banach Lemma 20
The Numerical Solution of Linear Systems 25
Gauss Elimination 25
Operation Count 29
Using the LU-decomposition for multiple right hand sides 34
Tridiagonal Systems 37
Inverses 40
Practical Considerations 47
Gauss Elimination with Pivoting 53
Error Analysis 56
The Numerical Solution of Nonlinear Equations 73
Some Methods for Scalar Nonlinear Equations 77
Bisection 78
Regula Falsi 80
Newton’s Method 83
The Chord Method 87
Newton’s Method for Systems of Nonlinear Equations 92
Residual Correction 99
Convergence Analysis for Scalar Equations 102
Convergence Analysis for Systems 145
,The Approximation of Functions 158
Function Norms 158
Lagrange Interpolation Polynomial 166
Lagrange Interpolation Theorem 176
Chebyshev Polynomials 185
Chebyshev Theorem 191
Taylor Polynomial 207
Taylor Theorem 211
Local Polynomial Interpolation 216
Numerical Differentiation 231
Best Approximation in the · 2 240
Best Approximation in R3 240
Best Approximation in General 247
Gram-Schmidt Orthogonalization 256
Best Approximation in Function Space 259
Numerical Integration 268
Trapezoidal Rule 270
Simpson’s Rule 273
Gauss Quadrature 287
Discrete Least Squares Approximation 296
Linear Least Squares 298
General Least Squares 306
, Smooth Interpolation by Piecewise Polynomials 326
Cubic Spline Interpolation 330
Numerical Methods for Initial Value Problems 341
Numerical Methods 347
Stability of Numerical Approximations 355
Stiff Differential Equations 365
Boundary Value Problems in ODE 384
A Nonlinear Boundary Value Problem 400
Diffusion Problems 404
Nonlinear Diffusion Equations 417
SOLUTIONS- NEW UPDATE FALL 2025|2026 Concordia University
LECTURE NOTES
on
ELEMENTARY NUMERICAL METHODS
Eusebius Doedel
, TABLE OF CONTENTS
Vector and Matrix Norms 1
Banach Lemma 20
The Numerical Solution of Linear Systems 25
Gauss Elimination 25
Operation Count 29
Using the LU-decomposition for multiple right hand sides 34
Tridiagonal Systems 37
Inverses 40
Practical Considerations 47
Gauss Elimination with Pivoting 53
Error Analysis 56
The Numerical Solution of Nonlinear Equations 73
Some Methods for Scalar Nonlinear Equations 77
Bisection 78
Regula Falsi 80
Newton’s Method 83
The Chord Method 87
Newton’s Method for Systems of Nonlinear Equations 92
Residual Correction 99
Convergence Analysis for Scalar Equations 102
Convergence Analysis for Systems 145
,The Approximation of Functions 158
Function Norms 158
Lagrange Interpolation Polynomial 166
Lagrange Interpolation Theorem 176
Chebyshev Polynomials 185
Chebyshev Theorem 191
Taylor Polynomial 207
Taylor Theorem 211
Local Polynomial Interpolation 216
Numerical Differentiation 231
Best Approximation in the · 2 240
Best Approximation in R3 240
Best Approximation in General 247
Gram-Schmidt Orthogonalization 256
Best Approximation in Function Space 259
Numerical Integration 268
Trapezoidal Rule 270
Simpson’s Rule 273
Gauss Quadrature 287
Discrete Least Squares Approximation 296
Linear Least Squares 298
General Least Squares 306
, Smooth Interpolation by Piecewise Polynomials 326
Cubic Spline Interpolation 330
Numerical Methods for Initial Value Problems 341
Numerical Methods 347
Stability of Numerical Approximations 355
Stiff Differential Equations 365
Boundary Value Problems in ODE 384
A Nonlinear Boundary Value Problem 400
Diffusion Problems 404
Nonlinear Diffusion Equations 417
