Topic

No. of hours

Lecture

Tutorial/Practical

Review of Required linear Algebra Concepts :-

Numerical Linear Algebra Problems and Their Importance.

Solving Numerical Linear Algebra Problems Using Obvious Approaches.

Vectors.

Orthogonality of Vectors and Subspaces.

Matrices.

Basic Concepts.

Range and Null Space.

Rank of a Matrix.

The Inverse of a Matrix.

Similar Matrices.

Orthogonality and Projections.

Projection of a Vector onto the Range and the Null Space of a Matrix.

 Some Special Matrices.

 Diagonal and Triangular Matrices.

Orthogonal Matrix.

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Permutation Matrix.

Hessenberg(Almost Triangular) Matrix.

Companion Matrix.

Nonderogatory Matrix.

Diagonally Dominant Matrix.

Positive Definite Matrix.

The Cayley-Hamilton Theorem.

Sinplm'Values.

Vectorand Matrix Norms.

VectorNonns.

MatrixNonns.

Convergence of a Matrix Sequence and Convergent Matrices.

Norms and Inverses.

NormInvariant Properties of Orthogonal Matrices.

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Floating-Point Numbers and Errors in Computations :-

Floating-Point Number Systems.

Rounding Errors.

Laws of Floating-Point Arithmetic.

A Additionofn Floating-Point Numbers.

Multiplication of n Floating-Point Numbers.

Inner Product Computation.

Error Bounds for Floating-Point Matrix Computations.

Roundoff Errors Due to Cancellation and Recursive Computations.

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Stability of Algorithms and Conditioning of Problems :-

Some Basic Algorithms.

Computing the Norm of a Vector.

Computing the Inner Product of Two Vectors.

Solution of an Upper Triangular System.

Computing the Inverse of an Upper Triangular Matrix.

Gaussian Elimination for Solving Ax = b.

Definitions and Concepts of Stability.

Conditioning of the Problem. Stability of the Algorithm. and Accuracy of the Solution.

The Wtlkinson Polynomial.

An Ill-Conditioned Linear System Problem.

Examples of ill-Conditioned Eigenvalue Problems.

Strong, Weak, and Mild Stability.

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Numerically Effective Algorithms and Mathematical Software :-

Definitions and Example.

Flop Count and Storage Considerations for Some Basic Algorithms.

Some Existing High-Quality Mathematical Software for Linear Algebra Problems.

UNPACK.

EISPACK.

LAPACK.

NETLIB.

NAG.

JMSL.

MATLAB.

MATLAB Codes and MATCOM.

The ACM Library.

ITPACK (Iterative Software Package).

The Software Package TEMPLATES (for the Solution of Linear Systems: Building Blocks for Iterative Methods).

SPARSKIT: A Basic Toolkit for Sparse Matrix Computations.

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Some Useful Transformations in Numerical linear Algebra and Their Applications :-

A Computational Methodology in numerical Linear Algebra.

Elementary Matrices and LU Factorization.

Gaussian Elimination without Pivoting.

Gaussian Elimination with Partial Pivoting.

Stability of Gaussian Elimination.

HooseholderTransfonnations and Applications to QR Factorization and Hessenberg Reduction.

Householder Matrices and QR Factorization.

Householder QR Factorization of a Nonsquare Matrix.

Householder Matrices and Reduction to Hessenberg Form.

Givens Matrices and Applications to QR Factorization and Hessenberg Reduction.

Givens Rotations and QR Factorization.

Uniqueness in QR Factorization.

Givens Rotations and Reduction to Hessenberg Form.

Uniqueness in Hessenberg Reduction.

Orthonormal Bases and Orthogonal Projections.

QR Factorization with Column Pivoting.

Modifying a QR Factorization.

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Numerical Solutions of Linear Systems.

Introduction.

Basic Results on the Existence, Uniqueness, and Invariance of Solutions.

Some Applications Giving Rise to Linear Systems Problems.

An Electric Circuit Problem.

Analysis of a Processing Plant Consisting of Interconnected Reactors.

Linear Systems Arising from Ordinary Differential Equations.

(Finite Difference Scheme)

Linear Systems Arising from Partial Differential Equations :-

A Case Study on Temperature Distribution.

Special Linear Systems Arising in Applications.

Linear Systems Arising from Finite Element Methods.

Approximation of a Function by a Polynomial: Hilbert System.

Direct Methods.

Solution of a Lower Triangular System.

Solution of the System Ax = bUsing Gaussian Elimination without Pivoting.

Solution of Ax = bUsing Pivoting Triangularization.

Solution of Ax = b without Explicit Factorization.

Solution of Ax = bUsing QR Factorization.

Solving a Linear System with Multiple Right-hand sides.

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Scaling.

LU Versus QR and a Table of comparisons.

Inverses and Determinants.

The Sherman-Morrison and Woodbury Fonnulas.

Avoiding Explicit Computation of the Inverses·.

Computing the Inverse of a Matrix.

Computing the Determinant of a Matrix.

Perturbation Analysis of the Linear System Problem.

Effect of Perturbation in the Right-Hand-Side Vector b.

Effect of Perturbation in Matrix A.

Effect of Perturbations in Both Matrix A and Vector b.

The Condition Number and Accuracy of Solution.

Some Well-known Ill-conditioned Matrices.

Effect of the Condition Number on Accuracy.

of the Computed Solution.

Size of the Condition Number Required for Ill-Conditioning.

The Condition Number and Nearness to Singularity.

Conditioning and Pivoting.

Conditioning and the Eigenvalue Problem.

Conditioning and Scaling.

Computing and Estimating the Condition Number.

Componentwise Perturbations and the Errors.

Iterative Refinement.

Iterative Methods.

The Jacobi Method.

The Gauss-Seidel Method.

Convergence of Iterative Methods.

The Successive Overrelaxation (SOR) Method.

The Conjugate Gradient Method.

The Arnoldi Process and GMRES.

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Least-Squares Solutions to linear Systems :-

Introduction.

A Simple Application Leading to an Over determined System.

Existence and Uniqueness.

Geometric Interpretation of the Least -Squares Problem.

Normal Equations and Polynomial Fitting.

Pseudoinverse and the Least-Squares Problem.

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Sensitivity of the Least-Squares Problem :-

Computational Methods for Over determined Least-Squares Problems.

The Normal Equations Method.

QR Factorization Methods for the Full-Rank Problem.

The QR Factorization Method for the Rank-Deficient Case.

Least-Squares Solution Using SVD.

Underdetermined Systems.

The Minimum-Norm Solution of the Full-Rank Underdetennined.

Problem Using Normal Equations.

TheQR Approach for the Full-Rank Underdetennined Problem.

Minimum-Norm Solution by MGS.

Iterative Refinement.

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Numerical Matrix Eigenvalue Problems.

Introduction.

Some Basic Results Involving Eigenvalues and Eigenvectors.

Eigenvalues and Eigenvectors.

The Schur Triangularization Theorem and Its Applications.

the Jordan Canonical Form.

The Eigenvalue Problems Arising in Practical Applications.

Stability Problems for Differential and Difference Equations.

Vibration Problem, Buckling Problem, and Simulating Transient.

Current of an Electrical Circuit.

An Example of the Eigenvalue Problem Arising in Statistics :-

Principal Components Analysis.

Localization of Eigenvalues.

The Gersgorin Disk Theorems.

Eigenvalue Bounds and Matrix Norms.

Inertia of a Matrix.

Computing Selected Eigenvalues and Eigenvectors.

Discussions on. the Importance of the Largest and Smallest Eigenvalues.

The Role of Dominant Eigenvalues and Eigenvectors in Dynamic Systems.

The Power Method, the Inverse Iteration, and the Rayleigh Quotient Iteration.

Computing the Subdominant Eigenvalues and Eigenvectors: Deflatio.

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Similarity Transformations and Eigenvalue :-

Difficulties with Using the Characteristic.

Eigenvalue Computations via the Jordan.

Eigenvalue Sensitivity.

The Bauer-Fike Theorem.

Sensitivity of the Individual Eigenvalues.

Eigenvector Sensitivity.

The Real Schur Form and QR Iterations.

The Basic QR Iteration.

The Hessenberg QR Iteration.

Convergence of the QR Iterations and the Shift of Origin.

The Single-Shift QR Iteration.

The Double-Shift QR Iteration.

Implicit QR Iteration.

Obtaining the Real Schur Form.

The Real Schur Form and Invariant Subspaces.

Computing the Eigenvectors.

The Hessenberg Inverse Iteration.

Calculating the Eigenvectors from the Real Schur Form.

The Symmetric Eigenvalue Problem.

The Sturm Sequence and the Bisection Method.

The Symmetric QR Iteration Method.

The Divide-and-Conquer Method.

The Lanczos Algorithm for Symmetric Matrices.

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