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Scientific Computing : For Scientists and Engineers.

By: Contributor(s): Series: De Gruyter Textbook SerPublisher: Berlin/Boston : De Gruyter, Inc., 2015Copyright date: ©2015Description: 1 online resource (150 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9783110359428
Subject(s): Genre/Form: Additional physical formats: Print version:: Scientific Computing : For Scientists and EngineersDDC classification:
  • 502.85
LOC classification:
  • QA76.9 .C65
Online resources:
Contents:
Intro -- Preface -- Contents -- 1 Introduction -- 1.1 Why study numerical methods? -- 1.2 Terminology -- 1.3 Convergence terminology -- 1.4 Exercises -- 2 Computer representation of numbers and roundoff error -- 2.1 Examples of the effects of roundoff error -- 2.2 Binary numbers -- 2.3 64 bit floating point numbers -- 2.3.1 Avoid adding large and small numbers -- 2.3.2 Subtracting two nearly equal numbers is bad -- 2.4 Exercises -- 3 Solving linear systems of equations -- 3.1 Linear systems of equations and solvability -- 3.2 Solving triangular systems -- 3.3 Gaussian elimination -- 3.4 The backslash operator -- 3.5 LU decomposition -- 3.6 Exercises -- 4 Finite difference methods -- 4.1 Approximating the first derivative -- 4.1.1 Forward and backward differences -- 4.1.2 Centered difference -- 4.1.3 Three point difference formulas -- 4.1.4 Further notes -- 4.2 Approximating the second derivative -- 4.3 Application: Initial value ODE’s using the forward Euler method -- 4.4 Application: Boundary value ODE's -- 4.5 Exercises -- 5 Solving nonlinear equations -- 5.1 The bisection method -- 5.2 Newton's method -- 5.3 Secant method -- 5.4 Comparing bisection, Newton, secant method -- 5.5 Combining secant and bisection and the fzero command -- 5.6 Equation solving in higher dimensions -- 5.7 Exercises -- 6 Accuracy in solving linear systems -- 6.1 Gauss-Jordan elimination and finding matrix inverses -- 6.2 Matrix and vector norms and condition number -- 6.3 Sensitivity in linear system solving -- 6.4 Exercises -- 7 Eigenvalues and eigenvectors -- 7.1 Mathematical definition -- 7.2 Power method -- 7.3 Application: Population dynamics -- 7.4 Exercises -- 8 Fitting curves to data -- 8.1 Interpolation -- 8.1.1 Interpolation by a single polynomial -- 8.1.2 Piecewise polynomial interpolation -- 8.2 Curve fitting -- 8.2.1 Line of best fit.
8.2.2 Curve of best fit -- 8.3 Exercises -- 9 Numerical integration -- 9.1 Newton-Cotes methods -- 9.2 Composite rules -- 9.3 MATLAB's integral function -- 9.4 Gauss quadrature -- 9.5 Exercises -- 10 Initial value ODEs -- 10.1 Reduction of higher order ODEs to first order -- 10.2 Common methods and derivation from integration rules -- 10.2.1 Backward Euler -- 10.2.2 Crank-Nicolson -- 10.2.3 Runge-Kutta 4 -- 10.3 Comparison of speed of implicit versus explicit solvers -- 10.4 Stability of ODE solvers -- 10.4.1 Stability of forward Euler -- 10.4.2 Stability of backward Euler -- 10.4.3 Stability of Crank-Nicolson -- 10.4.4 Stability of Runge-Kutta 4 -- 10.5 Accuracy of ODE solvers -- 10.5.1 Forward Euler -- 10.5.2 Backward Euler -- 10.5.3 Crank-Nicolson -- 10.5.4 Runge-Kutta 4 -- 10.6 Summary, general strategy, and MATLAB ODE solvers -- 10.7 Exercises -- A Getting started with Octave and MATLAB -- A.1 Basic operations -- A.2 Arrays -- A.3 Operating on arrays -- A.4 Script files -- A.5 Function files -- A.5.1 Inline functions -- A.5.2 Passing functions to other functions -- A.6 Outputting information -- A.7 Programming in MATLAB -- A.8 Plotting -- A.9 Exercises.
Summary: Nowadays most mathematics done in practice is done on a computer. In engineering it is necessary to solve more than 1 million equations simultaneously, and computers can be used to reduce the calculation time from years to minutes or even seconds. This book explains: How can we approximate these important mathematical processes? How accurate are our approximations? How efficient are our approximations?.
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Intro -- Preface -- Contents -- 1 Introduction -- 1.1 Why study numerical methods? -- 1.2 Terminology -- 1.3 Convergence terminology -- 1.4 Exercises -- 2 Computer representation of numbers and roundoff error -- 2.1 Examples of the effects of roundoff error -- 2.2 Binary numbers -- 2.3 64 bit floating point numbers -- 2.3.1 Avoid adding large and small numbers -- 2.3.2 Subtracting two nearly equal numbers is bad -- 2.4 Exercises -- 3 Solving linear systems of equations -- 3.1 Linear systems of equations and solvability -- 3.2 Solving triangular systems -- 3.3 Gaussian elimination -- 3.4 The backslash operator -- 3.5 LU decomposition -- 3.6 Exercises -- 4 Finite difference methods -- 4.1 Approximating the first derivative -- 4.1.1 Forward and backward differences -- 4.1.2 Centered difference -- 4.1.3 Three point difference formulas -- 4.1.4 Further notes -- 4.2 Approximating the second derivative -- 4.3 Application: Initial value ODE’s using the forward Euler method -- 4.4 Application: Boundary value ODE's -- 4.5 Exercises -- 5 Solving nonlinear equations -- 5.1 The bisection method -- 5.2 Newton's method -- 5.3 Secant method -- 5.4 Comparing bisection, Newton, secant method -- 5.5 Combining secant and bisection and the fzero command -- 5.6 Equation solving in higher dimensions -- 5.7 Exercises -- 6 Accuracy in solving linear systems -- 6.1 Gauss-Jordan elimination and finding matrix inverses -- 6.2 Matrix and vector norms and condition number -- 6.3 Sensitivity in linear system solving -- 6.4 Exercises -- 7 Eigenvalues and eigenvectors -- 7.1 Mathematical definition -- 7.2 Power method -- 7.3 Application: Population dynamics -- 7.4 Exercises -- 8 Fitting curves to data -- 8.1 Interpolation -- 8.1.1 Interpolation by a single polynomial -- 8.1.2 Piecewise polynomial interpolation -- 8.2 Curve fitting -- 8.2.1 Line of best fit.

8.2.2 Curve of best fit -- 8.3 Exercises -- 9 Numerical integration -- 9.1 Newton-Cotes methods -- 9.2 Composite rules -- 9.3 MATLAB's integral function -- 9.4 Gauss quadrature -- 9.5 Exercises -- 10 Initial value ODEs -- 10.1 Reduction of higher order ODEs to first order -- 10.2 Common methods and derivation from integration rules -- 10.2.1 Backward Euler -- 10.2.2 Crank-Nicolson -- 10.2.3 Runge-Kutta 4 -- 10.3 Comparison of speed of implicit versus explicit solvers -- 10.4 Stability of ODE solvers -- 10.4.1 Stability of forward Euler -- 10.4.2 Stability of backward Euler -- 10.4.3 Stability of Crank-Nicolson -- 10.4.4 Stability of Runge-Kutta 4 -- 10.5 Accuracy of ODE solvers -- 10.5.1 Forward Euler -- 10.5.2 Backward Euler -- 10.5.3 Crank-Nicolson -- 10.5.4 Runge-Kutta 4 -- 10.6 Summary, general strategy, and MATLAB ODE solvers -- 10.7 Exercises -- A Getting started with Octave and MATLAB -- A.1 Basic operations -- A.2 Arrays -- A.3 Operating on arrays -- A.4 Script files -- A.5 Function files -- A.5.1 Inline functions -- A.5.2 Passing functions to other functions -- A.6 Outputting information -- A.7 Programming in MATLAB -- A.8 Plotting -- A.9 Exercises.

Nowadays most mathematics done in practice is done on a computer. In engineering it is necessary to solve more than 1 million equations simultaneously, and computers can be used to reduce the calculation time from years to minutes or even seconds. This book explains: How can we approximate these important mathematical processes? How accurate are our approximations? How efficient are our approximations?.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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