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فهرست محتوای دیجیتالی
فهرست مطالب
فهرست نقدها
1157 مرتبه مشاهده شده
Numerical analysis using MATLAB and Excel
Karris, Steven T.
- ISBN:9781934404041
- ISBN:1934404047
- ISBN:1934404039
- ISBN:9781934404034
- DOI:9781934404034
- Call Number : QA 297 .K37 2007
- Main Entry: Karris, Steven T.
- Title:Numerical analysis using MATLAB and Excel [electronic resource] / Steven T. Karris.
- Edition:3rd ed.
- Publisher:[Fremont, Calif.?] : Orchard Publications, 2007.
- Physical Description:1v. (various pagings): ill
- Notes:Includes bibliographical references and index
- Subject:MATLAB
- Subject:Microsoft Excel (Computer file)
- Subject:Numerical analysis -- Data processing.
- Subject:Mathematical analysis.
- Reproduction no./Source:OverDrive, Inc http://www.overdrive.com
- Additional formats:Print version Karris, Steven T Numerical analysis using MATLAB and Excel 3rd ed [Fremont, Calif.?] : Orchard Publications, 2007 9781934404034
فهرست محتوای دیجیتالی
Numerical analysis using MATLAB and Excelمحتواي کتاب
فهرست مطالب
- Cover Page
- Back Page
- Title Page
- ISBN 1934404047
- Preface
- Table of Contents
- 1 Introduction to MATLAB
- 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
- 2 Root Approximations
- 3 Sinusoids and Phasors
- 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
- 4 Matrices and Determinants
- 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
- 4.14 Solutions to End-of-Chapter Exercises
- 5 Differential Equations, State Variables, and State Equations
- 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
- 6 Fourier, Taylor, and Maclaurin Series
- 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
- 7 Finite Differences and Interpolation
- 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
- 8 Linear and Parabolic Regression
- 9 Solution of Differential Equations by Numerical Methods
- 10 Integration by Numerical Methods
- 11 Difference Equations
- 12 Partial Fraction Expansion
- 13 The Gamma and Beta Functions and Distributions
- 14 Orthogonal Functions and Matrix Factorizations
- 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
- 15 Bessel, Legendre, and Chebyshev Functions
- 16 Optimization Methods
- Appendix A Difference Equations in Discrete−Time Systems
- Appendix B Introduction to Simulink®
- Appendix C Ill-Conditioned Matrices
- 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
- Solution:
- C.2 Condition Number of a Matrix
- The condition number of a matrix is defined as
- (C.2)
- 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
- 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
- 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
- This is a large number and if the coefficient matrix is multiplied by this number, seven decimal places might be lost.
- Let us consider another example.
- Example C.4
- Solution:
- Here, we are asked to find the values of and of the linear system
- Using MATLAB, we define and , and we use the left division operation, i.e.,
- A=[0.585 0.378; 0.728 0.464]; b=[0.187 0.256]'; x=b\A
- x =
- 2.9428 1.8852
- Check:
- A=[0.585 0.378; 0.728 0.464]; x=[2.9428 1.8852]'; b=A*x
- b =
- 2.4341
- 3.0171
- but these are not the given values of the vector , so let us check the determinant and the condi tion number of the matrix .
- determinant = det(A)
- determinant =
- -0.0037
- condition=cond(A)
- condition =
- 328.6265
- Therefore, we conclude that this system of equations is ill-conditioned and the solution is invalid.
- Example C.4 above should serve as a reminder that when we solve systems of equations using matrices, we should check the determinants and the condition number to predict possible floating point and roundoff errors.
- Ill-Conditioned Matrices
- References and Suggestions for Further Study
- Index
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