Numerical analysis using MATLAB and Excel
This text is written primarily for students/readers who have a good background of high-school algebra, geometry, trigonometry, and the fundamentals of differential and integral calculus.
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| Format: | eBook Book |
| Language: | English |
| Published: |
Fremont, Calif.
Orchard Publications
2007
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| Edition: | 3 |
| Subjects: | |
| ISBN: | 9781934404034, 1934404039 |
| Online Access: | Get full text |
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Table of Contents:
- where is the norm of the matrix defined in relation (C.1) above. Matrices with condition number close to unity are said to be well-conditioned matrices, and those with very large condition number are said to be ill-conditioned matrices. -- The condition number of a matrix is computed with the MATLAB function cond(A). -- Example C.2 -- Using the MATLAB function cond(A), compute the condition number of the matrix defined as -- Solution: -- At the MATLAB command prompt, we enter -- A=[-2 5 -4 9 -- -3 -6 8 1 -- 7 -5 3 2 -- 4 -9 -8 -1] -- cond(A) -- and MATLAB outputs -- ans = -- 2.3724 -- This condition number is relatively close to unity and thus we classify matrix A as a well-condi tioned matrix. -- We recall from Chapter 4 that if the determinant of a square matrix A is singular, that is, if , the inverse of A is undefined. Please refer to Chapter 4, Page 4-22. -- Now, let us consider that the coefficient matrix is very small, i.e., almost singular. Accordingly, we classify such a matrix as ill-conditioned. -- C.3 Hilbert Matrices -- Let be a positive integer. A unit fraction is the reciprocal of this integer, that is, . Thus, are unit fractions. A Hilbert matrix is a matrix with unit fraction elements -- (C.3) -- (C.4) -- MATLAB's function hilb(n) displays the Hilbert matrix. -- Example C.3 -- Compute the determinant and the condition number of the Hilbert matrix using MATLAB. -- Solution: -- At the MATLAB command prompt, we enter -- det(hilb(6)) -- and MATLAB outputs -- ans = -- 5.3673e-018 -- This is indeed a very small number and for all practical purposes this matrix is singular. -- We can find the condition number of a matrix A with the cond(A) MATLAB function. Thus, for the Hilbert matrix, -- cond(hilb(6)) -- ans = -- 1.4951e+007
- Numerical Analysis Front Cover THIRD Edition.pdf -- Numerical Analysis THIRD Edition Front Matter.pdf -- Numerical Analysis THIRD Edition Preface.pdf -- Numerical Analysis THIRD Edition TOC All Chapters.pdf -- Numerical Analysis THIRD Edition Chapter 01.pdf -- Chapter 1 -- 1.1 Command Window -- 1.2 Roots of Polynomials -- 1.3 Polynomial Construction from Known Roots -- 1.4 Evaluation of a Polynomial at Specified Values -- 1.5 Rational Polynomials -- 1.6 Using MATLAB to Make Plots -- 1.7 Subplots -- 1.8 Multiplication, Division and Exponentiation -- 1.9 Script and Function Files -- 1.10 Display Formats -- 1.11 Summary -- 1.12 Exercises -- 1.13 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 02.pdf -- Chapter 2 -- 2.1 Newton's Method for Root Approximation -- 2.2 Approximations with Spreadsheets -- 2.3 The Bisection Method for Root Approximation -- 2.4 Summary -- 2.5 Exercises -- 2.6 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 03.pdf -- Chapter 3 -- 3.1 Alternating Voltages and Currents -- 3.2 Characteristics of Sinusoids -- 3.3 Inverse Trigonometric Functions -- 3.4 Phasors -- 3.5 Addition and Subtraction of Phasors -- 3.6 Multiplication of Phasors -- 3.7 Division of Phasors -- 3.8 Exponential and Polar Forms of Phasors -- 3.9 Summary -- 3.10 Exercises -- 3.11 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 04.pdf -- Chapter 4 -- 4.1 Matrix Definition -- 4.2 Matrix Operations -- 4.3 Special Forms of Matrices -- 4.4 Determinants -- 4.5 Minors and Cofactors -- 4.6 Cramer's Rule -- 4.7 Gaussian Elimination Method -- 4.8 The Adjoint of a Matrix -- 4.9 Singular and Non-Singular Matrices -- 4.10 The Inverse of a Matrix -- 4.11 Solution of Simultaneous Equations with Matrices -- 4.12 Summary -- 4.13 Exercises
- This is a large number and if the coefficient matrix is multiplied by this number, seven decimal places might be lost.
- 8.4 Regression with Power Series Approximations -- 8.5 Summary -- 8.6 Exercises -- 8.7 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 09.pdf -- Chapter 9 -- 9.1 Taylor Series Method -- 9.2 Runge-Kutta Method -- 9.3 Adams' Method -- 9.4 Milne's Method -- 9.5 Summary -- 9.6 Exercises -- 9.7 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 10.pdf -- Chapter 10 -- 10.1 The Trapezoidal Rule -- 10.2 Simpson's Rule -- 10.3 Summary -- 10.4 Exercises -- 10.5 Solution to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 11.pdf -- Chapter 11 -- 11.1 Introduction -- 11.2 Definition, Solutions, and Applications -- 11.3 Fibonacci Numbers -- 11.4 Summary -- 11.5 Exercises -- 11.6 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 12.pdf -- Chapter 12 -- 12.1 Partial Fraction Expansion -- 12.2 Alternate Method of Partial Fraction Expansion -- 12.3 Summary -- 12.4 Exercises -- 12.5 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 13.pdf -- Chapter 13 -- 13.1 The Gamma Function -- 13.2 The Gamma Distribution -- 13.3 The Beta Function -- 13.4 The Beta Distribution -- 13.5 Summary -- 13.6 Exercises -- 13.7 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 14.pdf -- Chapter 14 -- 14.1 Orthogonal Functions -- 14.2 Orthogonal Trajectories -- 14.3 Orthogonal Vectors -- 14.4 The Gram-Schmidt Orthogonalization Procedure -- 14.5 The LU Factorization -- 14.6 The Cholesky Factorization -- 14.7 The QR Factorization -- 14.8 Singular Value Decomposition -- 14.9 Summary -- 14.10 Exercises -- 14.11 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 15.pdf -- Chapter 15 -- 15.1 The Bessel Function -- 15.2 Legendre Functions -- 15.3 Laguerre Polynomials
- 15.4 Chebyshev Polynomials -- 15.5 Summary -- 15.6 Exercises -- 15.7 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 16.pdf -- Chapter 16 -- 16.1 Linear Programming -- 16.2 Dynamic Programming -- 16.3 Network Analysis -- 16.4 Summary -- 16.5 Exercises -- 16.6 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Appendix A Difference Equations.pdf -- Appendix A -- A.1 Recursive Method for Solving Difference Equations -- A.2 Method of Undetermined Coefficients -- Numerical Analysis THIRD Edition Appendix B Introduction to Simulink.pdf -- Appendix B -- B.1 Simulink and its Relation to MATLAB -- B.2 Simulink Demos -- Numerical Analysis THIRD Edition Appendix C Ill-Conditioned Matrices.pdf -- Appendix C -- Ill-Conditioned Matrices -- his appendix supplements Chapters 4 and 14 with concerns when the determinant of the coefficient matrix is small. We will introduce a reference against which the determinant can be measured to classify a matrix as a well- or ill-conditioned. -- C.1 The Norm of a Matrix -- A norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. An example is the two-dimensional Euclidean space denoted as . The elements of the Euclidean vector space (e.g., (2,5))... -- The Euclidean norm of a matrix , denoted as , is defined as -- (C.1) -- and it is computed with the MATLAB function norm(A). -- Example C.1 -- Using the MATLAB function norm(A), compute the Euclidean norm of the matrix , defined as -- Solution: -- At the MATLAB command prompt, we enter -- A=[-2 5 -4 9 -- -3 -6 8 1 -- 7 -5 3 2 -- 4 -9 -8 -1] -- norm(A) -- and MATLAB outputs -- ans = -- 14.5539 -- C.2 Condition Number of a Matrix -- The condition number of a matrix is defined as -- (C.2)
- 4.14 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 05.pdf -- Chapter 5 -- 5.1 Simple Differential Equations -- 5.2 Classification -- 5.3 Solutions of Ordinary Differential Equations (ODE) -- 5.4 Solution of the Homogeneous ODE -- 5.5 Using the Method of Undetermined Coefficients for the Forced Response -- 5.6 Using the Method of Variation of Parameters for the Forced Response -- 5.7 Expressing Differential Equations in State Equation Form -- 5.8 Solution of Single State Equations -- 5.9 The State Transition Matrix -- 5.10 Computation of the State Transition Matrix -- 5.11 Eigenvectors -- 5.12 Summary -- 5.13 Exercises -- 5.14 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 06.pdf -- Chapter 6 -- 6.1 Wave Analysis -- 6.2 Evaluation of the Coefficients -- 6.3 Symmetry -- 6.4 Waveforms in Trigonometric Form of Fourier Series -- 6.5 Alternate Forms of the Trigonometric Fourier Series -- 6.6 The Exponential Form of the Fourier Series -- 6.7 Line Spectra -- 6.8 Numerical Evaluation of Fourier Coefficients -- 6.9 Power Series Expansion of Functions -- 6.10 Taylor and Maclaurin Series -- 6.11 Summary -- 6.12 Exercises -- 6.13 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 07.pdf -- Chapter 7 -- 7.1 Divided Differences -- 7.2 Factorial Polynomials -- 7.3 Antidifferences -- 7.4 Newton's Divided Difference Interpolation Method -- 7.5 Lagrange's Interpolation Method -- 7.6 Gregory-Newton Forward Interpolation Method -- 7.7 Gregory-Newton Backward Interpolation Method -- 7.8 Interpolation with MATLAB -- 7.9 Summary -- 7.10 Exercises -- 7.11 Solutions to End-of-Chapter Exercises -- Numerical Analysis THIRD Edition Chapter 08.pdf -- Chapter 8 -- 8.1 Curve Fitting -- 8.2 Linear Regression -- 8.3 Parabolic Regression
- Intro -- Preface -- Table of Contents -- Chapter 1 Introduction to MATLAB -- Chapter 2 Root Approximations -- Chapter 3 Sinusoids and Phasors -- Chapter 4 Matrices and Determinants -- Chapter 5 Differential Equations, State Variables, and State Equations -- Chapter 6 Fourier, Taylor, and Maclaurin Series -- Chapter 7 Finite Differences and Interpolation -- Chapter 8 Linear and Parabolic Regression -- Chapter 9 Solution of Differential Equations by Numerical Methods -- Chapter 10 Integration by Numerical Methods -- Chapter 11 Difference Equations -- Chapter 12 Partial Fraction Expansion -- Chapter 13 The Gamma and Beta Functions and Distributions -- Chapter 14 Orthogonal Functions and Matrix Factorizations -- Chapter 15 Bessel, Legendre, and Chebyshev Functions -- Chapter 16 Optimization Methods -- Appendix A Difference Equations in Discrete-Time Systems -- Appendix B Introduction to Simulink -- Appendix C Ill-Conditioned Matrices -- References and Suggestions for Further Study -- Index