Linear Algebra and Differential Equations Using MATLAB® - Hardcover

Dellnitz, Michael; Golubitsky, Martin

Hardcover

ISBN 10: 0534354254 ISBN 13: 9780534354251

Verlag: Brooks/Cole, 1999

Inhaltsangabe

Book by Golubitsky Martin Dellnitz Michael

Über diesen Titel

Críticas

1. PRELIMINARIES Vectors and Matrices / MATLAB(R) / Special Kinds of Matrices / The Geometry of Vector Operations 2. SOLVING LINEAR EQUATIONS Systems of Linear Equations and Matrices / The Geometry of Low-Dimensional Solutions / Gaussian Elimination / Reduction to Echelon Form / Linear Equations with Special Coefficients / Uniqueness of Reduced Echelon Form 3. MATRICES AND LINEARITY Matrix Multiplication of Vectors / Matrix Mappings / Linearity / The Principle of Superposition / Composite and Multiplication of Matrices / Properties of Matrix Multiplication / Solving Linear Systems and Inverses / Determinants of 2 x 2 Matrices 4. SOLVING ORDINARY DIFFERENTIAL EQUATIONS A Single Differential Equation / Graphing Solutions to Differential Equations / Phase Space Pictures and Equilibria / Separation of Variables / Uncoupled Linear Systems of Two Equations / Coupled Linear Systems / The Initial Value Problem and Eigenvectors / Eigenvalues of 2 x 2 Matrices / Initial Value Problems Revisited / Markov Chains 5. VECTOR SPACES Vector Spaces and Subspaces / Construction of Subspaces / Spanning Sets and MATLAB(R) / Linear Dependence and Linear Independence / Dimension and Bases / The Proof of the Main Theorem 6. CLOSED FORM SOLUTIONS FOR PLANAR ODES The Initial Value Problem / Closed Form Solutions by the Direct Method / Solutions Using Matrix Exponentials / Linear Normal Form Planar Systems / Similar Matrices / Formulas for Matrix Exponentials / Second Order Equations 7. QUALITATIVE THEORY OF PLANAR ODES Sinks, Saddles, and Sources / Phase Portraits of Sinks / Phase Portraits of Nonhyperbolic Systems 8. DETERMINANTS AND EIGENVALUES Determinants / Eigenvalues / Appendix: Existence of Determinants 9. LINEAR MAPS AND CHANGES OF COORDINATES Linear Mappings and Bases / Row Rank Equals Column Rank / Vectors and Matrices in Coordinates / Matrices of Linear Maps on a Vector Space 10. ORTHOGONALITY Orthonormal Bases / Least Squares Approximations / Least Squares Fitting of Data / Symmetric Matrices / Orthogonal Matrices of QR Decompositions 11. AUTONOMOUS PLANAR NONLINEAR SYSTEMS Introduction / Equilibria and Linearization / Periodic Solutions / Stylized Phase Portraits 12. BIFURCATION THEORY Two Species Population Models / Examples of Bifurcations / The Continuous Flow Stirred Tank Reactor / The Remaining Global Bifurcations / Saddle-Node Bifurcations Revisited / Hopf Bifurcations Revisited 13. MATRIX NORMAL FORMS Real Diagonalizable Matrices / Simple Complex Eigenvalues / Multiplicity and Generalized Eigenvectors / The Jordan Normal Form Theorem / Appendix: Markov Matrix Theory / Appendix: Proof of Jordan Normal Form 14. HIGHER DIMENSIONAL SYSTEMS Linear Systems in Jordan Normal Form / Qualitative Theory Near Equilibria / MATLAB(R) ODE45 in One Dimension / Higher Dimensional Systems Using ODE45 / Quasiperiodic Motions and Tori / Chaos and the Lorenz Equation 15. LINEAR DIFFERENTIAL EQUATIONS Solving Systems in Original Coordinates / Higher Order Equations / Linear Differential Operators / Undetermined Coefficients / Periodic Forcing and Resonance 16. LAPLACE TRANSFORMS The Method of Laplace Transforms / Laplace Transforms and Their Computation / Partial Fractions / Discontinuous Forcing / RLC Circuits 17. ADDITIONAL TECHNIQUES FOR SOLVING ODES Nonconstant Coefficient Linear Equations / Variation of Parameters for Systems / The Wronskian / Higher Order Equations / Simplification by Substitution / Exact Differential Equations / Hamiltonian Systems 18. NUMERICAL SOLUTIONS OF ODES A Description of Numerical Methods / Error Bounds for Euler's Method / Local and Global Error Bounds / APPENDIX: VARIABLE STEP METHODS / MATLAB(R) COMMANDS / ANSWERS TO SELECTED ODD-NUMBERED PROBLEMS / INDEX

Reseña del editor

These world-renowned authors integrate linear algebra and ordinary differential equations in this unique book, interweaving instructions on how to use MATLAB (R) with examples and theory. They use computers in two ways: in linear algebra, computers reduce the drudgery of calculations to help students focus on concepts and methods; in differential equations, computers display phase portraits graphically for students to focus on the qualitative information embodied in solutions, rather than just to learn to develop formulas for solutions.

Details zur Ausgabe

• Verlag: Brooks/Cole
• Erscheinungsdatum: 1999
• ISBN 10: 0534354254
• ISBN 13: 9780534354251
• Einband: Tapa dura
• Sprache: Englisch
• Anzahl der Seiten: 600