Mathematical methods

Mathematical methods

Mathematics lays the basic foundation for engineering students to pursue their core subjects. Mathematical Methodscovers topics on matrices, linear systems of equations, eigen values, eigenvectors, quadratic forms, Fourier series, partial differential equations, Z-transforms, numerical methods of so...

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Detalles Bibliográficos
Otros Autores: Rukmangadachari, E. Author (author)
Formato: Libro electrónico
Idioma:Inglés
Publicado: [Place of publication not identified] Pearson 2009
Edición:1st edition
Materias:
Mathematics
Physical Sciences & Mathematics
Mathematics - General
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009628058006719
Tabla de Contenidos:
  • Cover
  • About the Author
  • Contents
  • Preface
  • Chapter 1: Matrices and Linear Systems of Equations
  • 1.1 Introduction
  • 1.1.1 Matrix: Definition
  • 1.1.2 Types of Matrices
  • 1.2 Algebra of Matrices
  • 1.3 Matrix Multiplication
  • 1.3.1 Properties
  • 1.4 Determinant of a Square Matrix
  • 1.4.1 Expansion of a Determinant of Third Order
  • 1.4.2 Expansion of the Determinant of a Matrix of any Order n
  • 1.4.3 Properties of Determinant of a Matrix A
  • 1.5 Related Matrices
  • 1.5.1 The Transpose of a Matrix: Properties
  • 1.5.2 Adjoint of a Square Matrix
  • 1.5.3 Invertible Matrix
  • 1.5.4 Submatrix of a Matrix
  • 1.6 Determinant-related Matrices
  • 1.6.1 Singular Matrix
  • 1.6.2 Nonsingular Matrix
  • 1.6.3 Properties of Invertible Matrices (Non Singular Matrices)
  • 1.7 Special Matrices
  • 1.7.1 Idempotent Matrix
  • 1.7.2 Nilpotent Matrix
  • 1.7.3 Involutory Matrix
  • 1.7.4 Periodic Matrix
  • Exercise 1.1
  • 1.8 Linear Systems of Equations
  • 1.8.1 Introduction
  • 1.9 Homogeneous (H) and Nonhomogeneous (NH) Systems of Equations
  • 1.9.1 Matrix Form of the Linear System
  • 1.10 Elementary Row and Column Operations (Transformations) for Matrices
  • 1.10.1 Equivalence of Matrices
  • 1.10.2 Vectors: Linear Dependence and Independence
  • 1.10.3 Rank of a Matrix: Definition 1
  • 1.10.4 Methods for Determining Linear Dependence (L.D.) and Linear Independence (L.I.) of Vectors
  • Exercise 1.2
  • 1.11 Inversion of a Nonsingular Matrix
  • 1.11.1 Method 1: Adjoint Method (or Determinants Method)
  • 1.11.2 Elementary Matrices
  • 1.11.3 Method 2: Gauss-Jordan6-7 Method of Finding the Inverse of a Matrix
  • Exercise 1.3
  • 1.12 Rank of a Matrix
  • 1.12.1 Rank of a Matrix: Definition 2
  • 1.13 Methods for Finding the Rank of a Matrix
  • 1.13.1 Method 1: Maximum Number of Linearly Independent Rows
  • 1.13.2 Method 2: Method of Minors (Enumeration Method).
  • 1.13.3 Method 3: Reduction to Normal or Canonical Form by Elementary Transformations
  • 1.13.4 Rank of a Product of Matrices
  • 1.13.5 Method 4: Reduction of an m × n Matrix to a Normal Form by Finding Nonsingular Matrices P and Q Such That PAQ = N
  • 1.13.6 Method 5: Reduction of Matrix A to Echelon Form
  • Exercise 1.4
  • 1.14 Existence and Uniqueness of Solutions of a System of Linear Equations
  • 1.14.1 The System of NH Equations: Consistency and Inconsistency
  • 1.14.2 Existence of a Unique Solution for NH System of n Equations in n Unknowns
  • 1.14.3 Existence of a Solution for NH System of m Equations in n Unknowns
  • 1.14.4 Solutions of NH and H Systems of Equations
  • 1.15 Methods of Solution of NH and H Equations
  • 1.15.1 Method 1: Method of Determinants (Cramer's Rule)
  • 1.15.2 Method 2: Method of Matrix Inversion (or Adjoint Method)
  • 1.15.3 Method 3: Gauss's Elimination Method
  • 1.15.4 Method 4: Gauss-Jordan Elimination Method
  • 1.15.5 Method 5: LU Decomposition (Triangular Decomposition) Method
  • 1.15.6 Tridiagonal System: Solution by LU Decomposition Method
  • 1.16 Homogeneous System of Equations (H)
  • Exercise 1.5
  • Chapter 2: Eigenvalues and Eigenvectors
  • 2.1 Introduction
  • 2.1.1 Matrix Polynomial
  • 2.2 Linear Transformation
  • 2.3 Characteristic Value Problem
  • 2.3.1 Characteristic Equation of Matrix A
  • 2.3.2 Spectrum of A
  • 2.3.3 Procedure for Finding Eigenvalues and Eigenvectors
  • Exercise 2.1
  • 2.4 Properties of Eigenvalues and Eigenvectors
  • 2.4.1 Characteristic Polynomial Pn(l)
  • 2.5 Cayley-Hamilton Theorem
  • 2.5.1 Inverse of a Matrix by Cayley-Hamilton Theorem
  • Exercise 2.2
  • 2.6 Reduction of a Square Matrix to Diagonal Form
  • 2.6.1 Diagonalisation-Powers of a Square Matrix A
  • 2.6.2 Modal Matrix and Spectral Matrix of a Square Matrix A
  • 2.6.3 Similarity of Matrices.
  • 2.6.4 Diagonalisation-Conditions for Diagonalisability of a Matrix A
  • 2.6.5 Orthogonalisation of a Symmetric Matrix
  • 2.7 Powers of a Square Matrix A - Finding of Modal Matrix P and Inverse Matrix A−1
  • 2.7.1 Solution over the Complex Field-Eigenvectors of a Real Matrix Over Complex Field
  • Exercise 2.3
  • Chapter 3: Real and Complex Matrices
  • 3.1 Introduction
  • 3.2 Orthogonal/Orthonormal System of Vectors
  • 3.2.1 Norm of a Vector
  • 3.2.2 Orthonormal System of Vectors
  • 3.3 Real Matrices
  • 3.3.1 Symmetric Matrix
  • 3.3.2 Skew-Symmetric Matrix
  • 3.3.3 Properties of Symmetric and Skew-Symmetric Matrices
  • 3.3.4 Orthogonal Matrix
  • 3.3.5 Properties of Orthogonal Matrix
  • Exercise 3.1
  • 3.4 Complex Matrices
  • 3.4.1 Conjugate of a Matrix
  • 3.4.2 Properties of Conjugate Matrices
  • 3.4.3 Transposed Conjugate (Tranjugate) of a Matrix
  • 3.4.4 Properties of Tranjugate Matrices
  • 3.4.5 Hermitian1 Matrix
  • 3.4.6 Skew-Hermitian Matrix
  • 3.4.7 Unitary Matrix
  • 3.5 Properties of Hermitian, Skew-Hermitian and Unitary Matrices
  • Exercise 3.2
  • Chapter 4: Quadratic Forms
  • 4.1 Introduction
  • 4.2 Quadratic Forms
  • 4.2.1 Quadratic Form: Definition
  • 4.3 Canonical Form (or) Sum of the Squares Form
  • 4.3.1 Index and Signature of a Real Quadratic Form
  • 4.4 Nature of Real Quadratic Forms
  • 4.4.1 Positive Definite
  • 4.4.2 Negative Definite
  • 4.4.3 Positive Semi-Definite
  • 4.4.4 Negative Semi-Definite
  • 4.4.5 Indefinite
  • 4.5 Reduction of a Quadratic Form to Canonical Form
  • 4.6 Sylvestor's Law of Inertia
  • 4.7 Methods of Reduction of a Quadratic Form to a Canonical Form
  • 4.7.1 Diagonalisation (by Simultaneous Application of Row and Column Transformations)
  • 4.7.2 Orthogonalisation
  • 4.7.3 Lagrange's Method of Reduction (Completing Squares)
  • Exercise 4.1
  • Chapter 5: Solution of Algebraic and Transcendental Equations.
  • 5.1 Introduction to Numerical Methods
  • 5.2 Errors and their Computation
  • 5.2.1 Exact and Approximate Numbers
  • 5.2.2 Significant Digits
  • 5.2.3 Loss of Significant Digits
  • 5.2.4 Rounding off
  • 5.2.5 Rules for Rounding off
  • 5.2.6 Absolute, Relative and Percentage Errors
  • 5.3 Formulas for Errors
  • 5.3.1 Relative Error
  • 5.3.2 Error Bound for ã
  • 5.3.3 Error Propagation
  • 5.3.4 Error in Rounding
  • 5.3.5 Programming Errors
  • 5.3.6 Errors of Numerical Results
  • 5.4 Mathematical Pre-requisites
  • 5.5 Solution of Algebraic and Transcendental Equations
  • 5.5.1 Introduction
  • 5.5.2 Zero or Root of a Function
  • 5.6 Direct Methods of Solution
  • 5.6.1 Descartes' 4 Rule of Signs
  • 5.7 Numerical Methods of Solution of Equations of the Form f (x) = 0
  • 5.7.1 Fixed Point Iteration (Successive Approximation) Method
  • 5.7.2 Bolzano's5 (Bisection or Interval-Halving) Method
  • 5.7.3 Newton-Raphson6 Method
  • 5.7.4 Secants Method (or Chords Method)
  • 5.7.5 Method of False Position (Regula Falsi)
  • Exercise 5.1
  • Chapter 6: Interpolation
  • 6.1 Introduction
  • 6.1.1 Formula for Errors in Polynomial Interpolation
  • 6.2 Interpolation with Equal Intervals
  • 6.2.1 Finite Differences
  • 6.2.2 Forward (Advancing) Difference Operator D
  • 6.2.3 Properties Satisfi ed by D
  • 6.2.4 Backward Difference Operator ∇
  • 6.3 Symbolic Relations and Separation of Symbols
  • 6.3.1 Factorial Function
  • 6.3.2 The Enlargement or Displacement or Shift Operator E
  • 6.3.3 The Relations Between D, -, E and 1
  • Exercise 6.1
  • 6.4 Interpolation
  • 6.4.1 The Differential Operator D: Relation Between D, -, E and D
  • 6.5 Interpolation Formulas for Equal Intervals
  • 6.5.1 Newton1-Gregory2 Forward Interpolation Formula
  • 6.5.2 Newton-Gregory Backward Interpolation Formula
  • Exercise 6.2
  • 6.6 Interpolation with Unequal Intervals
  • 6.6.1 Divided Differences.
  • 6.6.2 Divided Differences:Notation
  • 6.7 Properties Satisfied by D'
  • 6.7.1 Linearity Property
  • 6.7.2 Symmetrical Property
  • 6.7.3 Vanishing of (n + 1) Divided Differences
  • 6.7.4 Special Case: Equally spaced Divided Differences
  • 6.8 Divided Difference Interpolation Formula
  • 6.8.1 Newton's Divided Difference Formula
  • 6.8.2 Sheppard's3 Zig-Zag Rule
  • 6.8.3 Lagrange's4 Formula for Unequal Intervals
  • 6.9 Inverse Interpolation Using Lagrange's Interpolation Formula
  • 6.10 Central Difference Formulas
  • 6.10.1 Gauss's5 Interpolation Formulae
  • 6.10.2 Stirling's6 Formula
  • 6.10.3 Bessel7 Formula
  • Exercise 6.3
  • Chapter 7: Curve Fitting
  • 7.1 Introduction
  • 7.1.1 Curve Fitting: Method of Least Squares
  • 7.1.2 Some Standard Approximating Curves
  • 7.2 Curve Fitting by the Method of Least Squares
  • 7.2.1 Least Squares Straight Line Fit or Linear Regression
  • 7.2.2 Least Squares Parabolic (Quadratic) Curve
  • 7.2.3 Nonlinear Curves
  • 7.3.2 Transcendental Curves
  • 7.3 Curvilinear (or Nonlinear) Regression
  • 7.3.1 Polynomial Regression
  • 7.3.2 Transcendental Curves
  • 7.4 Curve fitting by a Sum of Exponentials
  • 7.5.1 Linear Weighted Least Squares Approximation
  • 7.5.2 Nonlinear Weighted Least Squares Approximation
  • 7.5 Weighted Least Squares Approximation
  • 7.5.1 Linear Weighted Least Squares Approximation
  • 7.5.2 Nonlinear Weighted Least Squares Approximation
  • Exercise 7.1
  • Chapter 8: Numerical Differentiation and Integration
  • 8.1 Introduction
  • 8.1.1 Numerical Differentiation
  • 8.1.2 Numerical Differentiation by Newton's Forward Interpolation Formula
  • 8.1.3 Numerical Differentiation by Newton's Backward Interpolation Formula
  • 8.1.4 Numerical Differentiation by Stirling's Formula
  • 8.2 Errors in Numerical Differentiation
  • 8.2.1 Truncation Error
  • 8.2.2 Rounding Error.
  • 8.3 Maximum and Minimum Values of a Tabulated Function.

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