Syllabus:
i) INTRODUCTION TO OCTAVE (OR APPROPRIATE PACKAGE): Octave as a calculator, Built-in Variables and Functions, Functions and Commands; Creating Matrices, Subscript Notation for Matrix Elements, Colon Notation, Deleting Elements from Vectors and Matrices, Mathematical Operations with Matrices, Reshaping Matrices, Strings, Working with Data from External Files, Plotting.Script. Scripts m-Files; Function m-Files; Input and Output Parameters; Relational Operators, if...else..., Case Selection with switch , forLoops, whileloops, breakCommand, return Command; Vectorization.
ii) Number representations, finite precision arithmetic, errors in computing. Convergence, iteration, Taylor series.
iii) SOLUTION OF A SINGLE NON-LINEAR EQUATION: Bisection method. Fixed point methods. Newtons method. Convergence to a root, rates of convergence.
iv) REVIEW OF APPLIED LINEAR ALGEBRA: Vectors and matrices. Basic operations, linear combinations, basis, range, rank, vector norms, matrix norms. Special matrices. Solving Systems of equations (Direct Methods): Linear systems. Solution of triangular systems. Gaussian elimination with pivoting. LU decomposition, multiple right-hand sides.
v) LEAST SQUARES FITTING OF DATA: Fitting a line to data. Generalized least squares. QR decomposition.
vi) INTERPOLATION: Polynomial interpolation by Lagrange polynomials. Alternate bases: Monomials, Newton, divided differences. Piecewise polynomial interpolation. Cubic Hermite polynomials and splines.
vii) NUMERICAL QUADRATURE: Newton - Cotes Methods: Trapezoid and Simpson quadrature. Gaussian quadrature. Adaptive quadrature.
viii) ORDINARY DIFFERENTIAL EQUATIONS: Eulers Method. Accuracy and Stability. Trapezoid method. Runge - Kutta method. Boundary value problems and finite differences.
Reference Texts:
(a) B. Kernighan and D. Ritchie: The C Programming Language.
(b) J. Nino and F. A. Hosch: An Introduction to Programming and Object Oriented Design using JAVA.
(c) G. Recketenwald: Numerical Methods with Matlab.
(d) Shilling and Harries: Applied Numerical methods for engineers using Matlab and C.
(e) S. D. Conte and C. De Boor: Elementary Numerical Analysis: An Algorithmic Approach.
(f) S. K. Bandopadhyay and K. N. Dey: Data Structures using C.
(g) J. Ullman and W. Jennifer: A first course in database systems.
- Teacher: Shreedhar Inamdar
Syllabus:
Determinant of n-th order and its elementary properties, expansion by a row or column, statement of Laplace expansion, determinant of a product, statement of Cauchy- Binet theorem, inverse through classical adjoint, Cramers rule, determinant of a partitioned matrix, Idempotent matrices. Norm and inner product on Rn and Cn, norm induced by an inner product, Orthonormal basis, Gram-Schmidt orthogonalization starting from any finite set of vectors, orthogonal complement, orthogonal projection into a subspace, orthogonal projector into the column space of A, orthogonal and unitary matrices. Characteristic roots, relation between characteristic polynomials of AB and BA when AB is square, Cayley-Hamilton theorem, idea of minimal polynomial, eigenvectors, algebraic and geometric multiplicities, characterization of diagonalizable matrices, spectral representation of Hermitian and real symmetric matrices, singular value decomposition. Quadratic form, category of a quadratic form, use in classification of conics, Lagranges reduction to diagonal form, rank and signature, Sylvesters law, determinant criteria for n.n.d. and p.d. quadratic forms, Hadamards inequality, extrema of a p. d. quadratic form, simultaneous diagonalization of two quadratic forms one of which is p.d., simultaneous orthogonal diagonalization of commuting real symmetric matrices, square-root method.
Note: Geometric meaning of various concepts like subspace and flat, linear independence, projection, determinant (as volume), inner product, norm, orthogonality, orthogonal projection, and eigenvector should be discussed. Only finite-dimensional vector spaces to be covered.
Reference Texts:
(a) C. R. Rao: Linear Statistical Inference and its Applications.
(b) A. Ramachandra Rao and P. Bhimasankaram: Linear Algebra.
(c) K. Hoffman and R. Kunze: Linear Algebra.
(d) F. E. Hohn: Elementary Matrix Algebra.
(e) P. R. Halmos: Finite Dimensional Vector Spaces.
(f) S. Axler: Linear Algebra Done Right!
(g) H. Helson: Linear Algebra. R Bapat: Linear Algebra and Linear Models.
(i) R. A. Horn and C. R. Johnson: Matrix Analysis.
(j) M. Artin: Algebra.
https://www.isibang.ac.in/~adean/infsys/database/Bmath/LAlg2.html
- Teacher: B V Rajarama Bhat
- Teacher: Renjith Thazhathethil
Syllabus:
Prerequisites: Probability 1 or 2 should cover statements of Law of Large Numbers, Strong Law of Large Numbers, Binomial Central Limit Theorem and Central Limit Theorem.
i) R- Basics: Installing R, Variables, Functions, Workspace, External packages and Data Sets.
ii) Introduction to exploratory Data analysis using R: Descriptive statistics; Graphical representation of data: Histogram, Stem-leaf diagram, Box-plot; Visualizing categorical data.
iii) Review of Basic Probability: Basic distributions, properties; simulating samples from standard distributions using R commands.
iv) Sampling distributions based on normal populations: t, 2 and F distributions.
v) Model fitting and model checking: Basics of estimation, method of moments, Basics of testing including goodness of fit tests, interval estimation; Distribution theory for transformations of random vectors;
vi) Nonparametric tests: Sign test, Signed rank test,Wilcoxon-Mann-Whitney test.
vii) Bivariate data: covariance, correlation and least squares.
viii) Resampling methods: Jackknife and Bootstrap.
Reference Texts:
(a) John Verzani: Using R for Introductory Statistics.
(b) James McClave and Terry Sincich: Statistics.
(c) Deborah Nolan and Terry Speed: Stat Labs.
(d) John A. Rice: Mathematical Statistics and Data Analysis.
- Teacher: Siva Athreya
- Teacher: Sangita Das