1. Introduction
• Motivational Examples (full text search, computation of string/membrane deflection using mesh methods, signal and image analysis)
• Fundamentals of Linear Algebra (vector space, basis, linear representation, matrix, scalar multiplication, orthogonality, norm)
• Correctness, Stability, Types of Errors
• Numerical Approximation on a Computer
• Insight into the Analysis of Computational Demands and Complexity
• Storage Formats for Dense and Sparse Matrices (CSR, CSC, …)
2. Direct Solvers for Systems of Linear Equations
• Summary of Systems Types and Their Solvability
• Gaussian Elimination
• Inverse Matrix
• LU Decomposition
• Cholesky and LDLT Decomposition
• Stabilization with Partial and Complete Pivoting
3. Orthogonal and Spectral Problems
• Gram-Schmidt process, Its Versions (classical, modified, iterative)
• Householder Transformation, Givens Transformation
• QR Decomposition
• Eigenvalue and Spectral Decomposition
• Eigenvalue Estimations
• Dominant Eigenvalue Computation (power method, Lanczos method, spectral shift and reduction)
• Calculation of Spectral Decomposition using QR algorithm
• Singular Value Decomposition (SVD)
• Generalized Inversion
4. Iterative Solvers for Systems of Linear Equations
• Linear Methods (Jacobi, Gauss-Seidel, and Successive Over-relaxation (SOR) methods)
• Gradient Methods (method of steepest descent, Krylov methods)
• Preconditioning
5. Numerical Methods for Solving Non-linear Equations
• Root Separation
• Bisection Method
• Simple Iteration Method
• Newton’s Method
6. Interpolation and Approximation problems
• Polynomial Interpolation
• Lagrange Polynomial Interpolation
• Newton Polynomial Interpolation
• Linear and Cubic Spline
• Method of Least Squares
• Orthogonal Systems of Functions
7. Numerical Differentiation and Integration
• Motivational Examples (full text search, computation of string/membrane deflection using mesh methods, signal and image analysis)
• Fundamentals of Linear Algebra (vector space, basis, linear representation, matrix, scalar multiplication, orthogonality, norm)
• Correctness, Stability, Types of Errors
• Numerical Approximation on a Computer
• Insight into the Analysis of Computational Demands and Complexity
• Storage Formats for Dense and Sparse Matrices (CSR, CSC, …)
2. Direct Solvers for Systems of Linear Equations
• Summary of Systems Types and Their Solvability
• Gaussian Elimination
• Inverse Matrix
• LU Decomposition
• Cholesky and LDLT Decomposition
• Stabilization with Partial and Complete Pivoting
3. Orthogonal and Spectral Problems
• Gram-Schmidt process, Its Versions (classical, modified, iterative)
• Householder Transformation, Givens Transformation
• QR Decomposition
• Eigenvalue and Spectral Decomposition
• Eigenvalue Estimations
• Dominant Eigenvalue Computation (power method, Lanczos method, spectral shift and reduction)
• Calculation of Spectral Decomposition using QR algorithm
• Singular Value Decomposition (SVD)
• Generalized Inversion
4. Iterative Solvers for Systems of Linear Equations
• Linear Methods (Jacobi, Gauss-Seidel, and Successive Over-relaxation (SOR) methods)
• Gradient Methods (method of steepest descent, Krylov methods)
• Preconditioning
5. Numerical Methods for Solving Non-linear Equations
• Root Separation
• Bisection Method
• Simple Iteration Method
• Newton’s Method
6. Interpolation and Approximation problems
• Polynomial Interpolation
• Lagrange Polynomial Interpolation
• Newton Polynomial Interpolation
• Linear and Cubic Spline
• Method of Least Squares
• Orthogonal Systems of Functions
7. Numerical Differentiation and Integration