Applied mathematical methods

Applied mathematical methods

Applied Mathematical Methods covers the material vital for research in today's world and can be covered in a regular semester course. It is the consolidation of the efforts of teaching the compulsory first semester post-graduate applied mathematics course at the Department of Mechanical Enginee...

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Detalles Bibliográficos
Otros Autores: Daasagupta, Bhaskara Author (author)
Formato: Libro electrónico
Idioma:Inglés
Publicado: [Place of publication not identified] Pearson 2006
Edición:1st edition
Colección:Always learning.
Materias:
Mathematics
Physical Sciences & Mathematics
Mathematics - General
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009627786506719
Tabla de Contenidos:
  • Cover
  • Contents
  • List of Figures
  • List of Tables
  • Preface
  • Acknowledgements
  • Chapter 1: Preliminary Background
  • Theme of the Book
  • Course Contents
  • Sources for More Detailed Study
  • Directions for Using the Book
  • Expected Background
  • Preliminary Test
  • Prerequisite Problem Sets*
  • Problem set 1
  • Problem set 2
  • Problem set 3
  • Problem set 4
  • Problem set 5
  • Problem set 6
  • Chapter 2: Matrices and Linear Transformations
  • Matrices
  • Geometry and Algebra
  • Linear Transformations
  • Matrix Terminology
  • Exercises
  • Chapter 3: Operational Fundamentals of Linear Algebra
  • Range and Null Space: Rank and Nullity
  • Basis
  • Change of Basis
  • Elementary Transformations
  • Exercises
  • Chapter 4: Systems of Linear Equations
  • Nature of Solutions
  • Basic Idea of Solution Methodology
  • Homogeneous Systems
  • Pivoting
  • Partitioning and Block Operations
  • Exercises
  • Chapter 5: Gauss Elimination Family of Methods
  • Gauss-Jordan Elimination
  • Gaussian Elimination with Back-Substitution
  • LU Decomposition
  • Exercises
  • Chapter 6: Special Systems and Special Methods
  • Quadratic Forms, Symmetry and Positive Definiteness
  • Cholesky Decomposition
  • Sparse Systems*
  • Exercises
  • Chapter 7: Numerical Aspects in Linear Systems
  • Norms and Condition Numbers
  • Ill-conditioning and Sensitivity
  • Rectangular Systems
  • Singularity-Robust Solutions
  • Iterative Methods
  • Exercises
  • Chapter 8: Eigenvalues and Eigenvectors
  • Eigenvalue Problem
  • Generalized Eigenvalue Problem
  • Some Basic Theoretical Results
  • Eigenvalues of transpose
  • Diagonal and block diagonal matrices
  • Triangular and block triangular matrices
  • Shift theorem
  • Deflation
  • Eigenspace
  • Similarity transformation
  • Power Method
  • Exercises
  • Chapter 9: Diagonalization and Similarity Transformations
  • Diagonalizability.
  • Canonical Forms
  • Symmetric Matrices
  • Similarity Transformations
  • Exercises
  • Chapter 10: Jacobi and Givens Rotation Methods
  • Plane Rotations
  • Jacobi Rotation Method
  • Givens Rotation Method
  • Exercises
  • Chapter 11: Householder Transformation and Tridiagonal Matrices
  • Householder Reflection Transformation
  • Householder Method
  • Eigenvalues of Symmetric Tridiagonal Matrices
  • Exercises
  • Chapter 12: QR Decomposition Method
  • QR Decomposition
  • QR Iterations
  • Conceptual Basis of QR Method*
  • QR Algorithm with Shift*
  • Exercises
  • Chapter 13: Eigenvalue Problem of General Matrices
  • Introductory Remarks
  • Reduction to Hessenberg Form*
  • QR Algorithm on Hessenberg Matrices*
  • Inverse Iteration
  • Recommendation
  • Exercises
  • Chapter 14: Singular Value Decomposition
  • SVD Theorem and Construction
  • Properties of SVD
  • Pseudoinverse and Solution of Linear Systems
  • Optimality of Pseudoinverse Solution
  • SVD Algorithm
  • Exercises
  • Chapter 15: Vector Spaces: Fundamental Concepts*
  • Group
  • Field
  • Vector Space
  • Linear Transformation
  • Isomorphism
  • Inner Product Space
  • Function Space
  • Vector space of continuous functions
  • Linear dependence and independence
  • Inner product, norm and orthogonality
  • Linear transformations
  • Exercises
  • Chapter 16: Topics in Multivariate Calculus
  • Derivatives in Multi-Dimensional Spaces
  • Taylor's Series
  • Chain Rule and Change of Variables
  • Differentiation of implicit functions
  • Multiple integrals
  • Exact differentials
  • Differentiation under the integral sign
  • Numerical Differentiation
  • An Introduction to Tensors*
  • Exercises
  • Chapter 17: Vector Analysis: Curves and Surfaces
  • Recapitulation of Basic Notions
  • Curves in Space
  • Surfaces*
  • Exercises
  • Chapter 18: Scalar and Vector Fields
  • Differential Operations on Field Functions.
  • The del or nabla (∇) operator
  • Gradient
  • Divergence
  • Curl
  • Composite operations
  • Second order differential operators
  • Integral Operations on Field Functions
  • Green's theorem in the plane
  • Gauss's divergence theorem
  • Green's identities (theorem)
  • Stokes's theorem
  • Line integral
  • Surface and volume integrals
  • Closure
  • Exercises
  • Chapter 19: Polynomial Equations
  • Basic Principles
  • Analytical Solution
  • Cubic equations (Cardano)
  • Quartic equations (Ferrari)
  • General Polynomial Equations
  • Two Simultaneous Equations
  • Elimination Methods*
  • Sylvester's dialytic method
  • Bezout's method
  • Advanced Techniques*
  • Exercises
  • Chapter 20: Solution of Nonlinear Equations and Systems
  • Methods for Nonlinear Equations
  • Bracketing and bisection
  • Fixed point iteration
  • Newton-Raphson method
  • Secant method and method of false position
  • Quadratic interpolation method
  • Van Wijngaarden-Dekker Brent method
  • Systems of Nonlinear Equations
  • Newton's method for systems of equations
  • Broyden's secant method
  • Closure
  • Exercises
  • Chapter 21: Optimization: Introduction
  • The Methodology of Optimization
  • Single-Variable Optimization
  • Optimality criteria
  • Iterative methods
  • Conceptual Background of Multivariate Optimization
  • Optimality criteria
  • Convexity
  • Trust region and line search strategies
  • Global and local convergence
  • Exercises
  • Chapter 22: Multivariate Optimization
  • Direct Methods
  • Cyclic coordinate search
  • Rosenbrock's method
  • Hooke-Jeeves pattern search
  • Box's complex method
  • Nelder and Mead's simplex search
  • Remarks
  • Steepest Descent (Cauchy) Method
  • Newton's Method
  • Modified Newton's method
  • Hybrid (Levenberg-Marquardt) Method
  • Least Square Problems
  • Exercises
  • Chapter 23: Methods of Nonlinear Optimization*
  • Conjugate Direction Methods.
  • Conjugate gradient method
  • Extension to general (non-quadratic) functions
  • Powell's conjugate direction method
  • Quasi-Newton Methods
  • Closure
  • Exercises
  • Chapter 24: Constrained Optimization
  • Constraints
  • Optimality Criteria
  • Sensitivity
  • Duality*
  • Structure of Methods: An Overview*
  • Penalty methods
  • Primal methods
  • Dual methods
  • Lagrange methods
  • Exercises
  • Chapter 25: Linear and Quadratic Programming Problems*
  • Linear Programming
  • The simplex method
  • General perspective
  • Quadratic Programming
  • Active set method
  • Linear complementary problem
  • A trust region method
  • Duality
  • Exercises
  • Chapter 26: Interpolation and Approximation
  • Polynomial Interpolation
  • Lagrange interpolation
  • Newton interpolation
  • Limitations of single-polynomial interpolation
  • Hermite interpolation
  • Piecewise Polynomial Interpolation
  • Piecewise cubic interpolation
  • Spline interpolation
  • Interpolation of Multivariate Functions
  • Piecewise bilinear interpolation
  • Piecewise bicubic interpolation
  • A Note on Approximation of Functions
  • Modelling of Curves and Surfaces*
  • Exercises
  • Chapter 27: Basic Methods of Numerical Integration
  • Newton-Cotes Integration Formulae
  • Mid-point rule
  • Trapezoidal rule
  • Simpson's rules
  • Richardson Extrapolation and Romberg Integration
  • Further Issues
  • Exercises
  • Chapter 28: Advanced Topics in Numerical Integration*
  • Gaussian Quadrature
  • Gauss-Legendre quadrature
  • Weight functions in Gaussian quadrature
  • Multiple Integrals
  • Double integral on rectangular domain
  • Monte Carlo integration
  • Exercises
  • Chapter 29: Numerical Solution of Ordinary Differential Equations
  • Single-Step Methods
  • Euler's method
  • Improved Euler's method or Heun's method
  • Runge-Kutta methods
  • Practical Implementation of Single-Step Methods.
  • Runge-Kutta method with adaptive step size
  • Extrapolation based methods
  • Systems of ODE's
  • Multi-Step Methods*
  • Exercises
  • Chapter 30: ODE Solutions: Advanced Issues
  • Stability Analysis
  • Implicit Methods
  • Stiff Differential Equations
  • Boundary Value Problems
  • Shooting method
  • Finite difference (relaxation) method
  • Finite element method
  • Exercises
  • Chapter 31: Existence and Uniqueness Theory
  • Well-Posedness of Initial Value Problems
  • Existence of a solution
  • Uniqueness of a solution
  • Continuous dependence on initial condition
  • Uniqueness Theorems
  • Extension to ODE Systems
  • Closure
  • Exercises
  • Chapter 32: First Order Ordinary Differential Equations
  • Formation of Differential Equations and Their Solutions
  • Separation of Variables
  • ODE's with Rational Slope Functions
  • Some Special ODE's
  • Clairaut's equation
  • Second order ODE's with the function not appearing explicitly
  • Second order ODE's with independent variable not appearing explicitly
  • Exact Differential Equations and Reduction to the Exact Form
  • First Order Linear (Leibnitz) ODE and Associated Forms
  • Orthogonal Trajectories
  • Modelling and Simulation
  • Exercises
  • Chapter 33: Second Order Linear Homogeneous ODE's
  • Introduction
  • Homogeneous Equations with Constant Coefficients
  • Euler-Cauchy Equation
  • Theory of the Homogeneous Equations
  • Basis for Solutions
  • Exercises
  • Chapter 34: Second Order Linear Non-Homogeneous ODE's
  • Linear ODE's and Their Solutions
  • Method of Undetermined Coefficients
  • Method of Variation of Parameters
  • Closure
  • Exercises
  • Chapter 35: Higher Order Linear ODE's
  • Theory of Linear ODE's
  • Homogeneous Equations with Constant Coefficients
  • Non-Homogeneous Equations
  • Euler-Cauchy Equation of Higher Order
  • Exercises
  • Chapter 36: Laplace Transforms
  • Introduction.
  • Basic Properties and Results.

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