Curriculum and Syllabi - Indian Institute of Technology Bhubaneswar

oots; Diophantine equations: linear Diophantine equations, Pythagorean triples, Fermat's lasttheorem, Tell's, Bachet's and Catalan's equations, sums of squares; Diophantine approximations:continued fractions, convergent, approximation theorems; quadratic fields: primes and uniquefactorization.Texts:1. Koshy, Elementary Number Theory with Applications, Academic Press.2. D.M. Burton, Elementary Number Theory, 5th Ed. McGraw Hill.References:3. Kenneth H. Rosen, Elementary Number Theory (and it’s applications), Fifth Edition, PearsonAddison- Wesley.4. I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to the Theory of Numbers, Wiley,1991.5. K. Chandrasekaran, An Introduction to Analytic Number Theory, Springer, 1968.6. G.H. Hardy and E.M. Wright, An introduction to the Theory of Numbers, 5th Ed. Oxford UniversityPress.(ii) Advanced Matrix Theory (MA5007):3-0-0: 3 Credits Prerequisite: Linear AlgebraEigenvalues, eigenvectors and similarity, Unitary equivalence and normal matrices, Schur’s theorem,Spectral theorems for normal and Hermitian matrices; Jordan canonical form, Application of Jordancanonical form, Minimal polynomial, Companion matrices, Functions of matrices; Variationalcharacterizations of eigenvalues of Hermitian matrices, Rayleigh-Ritz theorem, Courant-Fischertheorem, Weyl theorem, Cauchy interlacing theorem, Inertia and congruence, Sylvester's law ofinertia; Matrix norms, Location and perturbation of eigenvalues Gerschgorin disk theorem; Positivesemidefiniteness, Singular value decomposition, Polar decomposition, Schur and Kronecker products;Positive and nonnegative matrices, Irreducible nonnegative matrices.Texts:1. R. A. Horn and C. R. Johnson, Matrix Analysis, CUP, 1985.References:2. P. Lancaster and M. Tismenetsky, The Theory of Matrices, second ed., Academic Press, 1985.3. F. R. Gantmacher, The Theory of Matrices, Vol-I, Chelsea, 1959.(iii) Numerical Linear Algebra (MA5008):3-0-0: 3 Credits Prerequisite: Linear AlgebraFundamentals. Linear systems, LU decompositions, Gaussian elimination with partial pivoting,Banded systems, Positive definite systems, Cholesky decomposition. Vector and matrix norms,Perturbation theory of linear systems, Condition numbers, Estimating condition numbers, IEEEfloating point arithmetic, Analysis of round off errors. Gram-Schmidt orthonormal process,Orthogonal matrices, Householder transformation, Givens rotations, QR factorization, Roundoff erroranalysis of orthogonal matrices, Stability of QR factorization. Solution of linear least squaresproblems, Normal equations, Singular Value Decomposition(SVD), Polar decomposition, Moore-Penrose inverse, Rank deficient least squares problems, Sensitivity analysis of least-squares problems.Review of eigenvalues and canonical forms of matrices, Sensitivity of eigenvalues and eigenvectors,Reduction to Hessenberg and tridiagonal forms, Power and inverse power methods, Rayleigh quotientiteration, Explicit and implicit QR algorithms for symmetric and non-symmetric matrices,Implementation of implicit QR algorithm. Computing the SVD, Sensitivity analysis of singular values