Solution of Systems of Linear Equations:Introduction – Basic Ideas of Applied Linear Algebra – Systems of Linear Equations – Square Non-Singular Systems – Ill-Conditioned and Ill-Posed Systems;The Algebraic Eigenvalue Problem:The Algebraic Eigenvalue Problem – Canonical Forms, Symmetric Matrices – Methods of Plane Rotations – Householder Method, Tridiagonal Matrices – QR Decomposition, General Matrices;Selected Topics in Linear Algebra and Calculus:Singular Value Decomposition – Vector Space: Concepts – Multivariate Calculus – Vector Calculus in Geometry – Vector Calculus in Physics;An Introductory Outline of Optimization Techniques:Solution of Equations – Introdcution to Optimization – Multivariate Optimization – Constrained Optimization: Optimality Criteria – Constrained Optimization: Further Issues;Selected Topics in Numerical Analysis:Interpolation – Numerical Integration – Numerical Solution of ODE’s as IVP – Boundary Value Problems, Question of Stability in IVP Solution – Stiff Differential Equations, Existence and Uniqueness Theory:Ordinary Differential Equations – Theory of First Order ODE’s – Linear Second Order ODE’s – Methods of Linear ODE’s – ODE Systems – Stability of Dynamic Systems;Application of ODE’s in Approximation Theory:Series Solutions and Special Functions – Sturm-Liouville Theory – Approximation Theory and Fourier Series – Fourier Integral to Fourier Transform, Minimax Approximation;Overviews: PDE’s, Complex Analysis and Variational Calculus:Separation of Variables in PDE’s, Hyperbolic Equations – Parabolic and Elliptic Equations, Membrane Equation – Analytic Functions – Integration of Complex Functions – Singularities and Residues – Calculus of Variations

### Course Curriculum

 Introduction Details 52:23 Basic Ideas of Applied Linear Algebra Details 52:6 Systems of Linear Equations Details 51:43 Square Non-Singular Systems Details 59:50 Ill-Conditioned and Ill-Posed Systems Details 55:29 The Algebraic Eigenvalue Problem Details 56:31 Canonical Forms, Symmetric Matrices Details 55:13 Methods of Plane Rotations Details 56:16 Householder Method, Tridiagonal Matrices Details 1:16 QR Decomposition, General Matrices Details 57:37 Singular Value Decomposition Details 58:38 Vector Space: Concepts Details 54:51 Multivariate Calculus Details 56:17 Vector Calculus in Geometry Details 1:3:7 Vector Calculus in Physics Details 59:1 Solution of Equations Details 54:44 Introdcution to Optimization Details 54:39 Multivariate Optimization Details 55:12 Constrained Optimization: Optimality Criteria Details 55:43 Constrained Optimization: Further Issues Details 56:28 Interpolation Details 58:13 Numerical Integration Details 53:49 Numerical Solution of ODE’s as IVP Details 55:44 Boundary Value Problems, Question of Stability in IVP Solution Details 55:12 Stiff Differential Equations, Existence and Uniqueness Theory Details 58:29 Theory of First Order ODE’s Details 53:16 Linear Second Order ODE’s Details 1:1:4 Methods of Linear ODE’s Details 57:34 ODE Systems Details 53:6 Stability of Dynamic Systems Details 1:1:59 Series Solutions and Special Functions Details 55:53 Sturm-Liouville Theory Details 1:5 Approximation Theory and Fourier Series Details 55:1 Fourier Integral to Fourier Transform, Minimax Approximation Details 55:50 Separation of Variables in PDE’s, Hyperbolic Equations Details 54:19 Parabolic and Elliptic Equations, Membrane Equation Details 51:43 Analytic Functions Details 58:7 Integration of Complex Functions Details 0:56 Singularities and Residues Details 57:24 Calculus of Variations Details 55:37

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